各種インストール

!pip install pystan

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!pip install arviz

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!pip install nest_asyncio

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第1章

問題5

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm

# 正規分布の定義
def f(x, mu, sig2):
    return (2 * np.pi * sig2) ** (-1/2) * np.exp(-(x - mu) ** 2 / (2 * sig2))

phi_seq = [-3, -1, 1, 3]
n_seq = [1, 5, 10]
m = len(phi_seq)
l = len(n_seq)
n = 30  # サンプルサイズ n
x = np.random.normal(size=n)  # 正規乱数をn個発生

# グラフを4個発生
fig, axes = plt.subplots(2, 2, figsize=(10, 8))
axes = axes.flatten()

for i, phi in enumerate(phi_seq):
    ax = axes[i]
    ax.set_xlim(-5, 7)
    ax.set_ylim(0, 0.5)
    ax.set_title(f"phi={phi}")

    for k, nn in enumerate(n_seq):
        mu = (phi + np.sum(x[:nn])) / (nn + 1)
        sig2 = 1
        x_vals = np.linspace(-5, 7, 400)
        y_vals = f(x_vals, mu, sig2)
        ax.plot(x_vals, y_vals, label=f"n={nn}", color=f"C{k+1}")

    # 真の分布の曲線を描く
    ax.plot(x_vals, norm.pdf(x_vals), 'k--', linewidth=2, label="真")
    ax.legend(loc="upper right")

plt.tight_layout()
plt.show()

<ipython-input-13-e497f5b1e700>:37: UserWarning: Glyph 30495 (\N{CJK UNIFIED IDEOGRAPH-771F}) missing from current font.
  plt.tight_layout()

問題6

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm, t

# グラフの設定
plt.figure(figsize=(8, 6))
plt.xlim(-5, 5)
plt.ylim(0, 0.4)
plt.xlabel("t")
plt.ylabel("Density")

# 正規分布の曲線を描く
x_vals = np.linspace(-5, 5, 400)
plt.plot(x_vals, norm.pdf(x_vals), 'k-', linewidth=2, label="Normal")

# t分布の曲線を描く
for i in range(1, 6):
    plt.plot(x_vals, t.pdf(x_vals, df=i), label=f"degree {i}", color=f"C{i}")

# 凡例を追加
plt.legend(loc="upper right")

# グラフを表示
plt.show()

第2章

問題18

テキストでは、$\theta(t)=\cos t$を$\verb@p@$, $p(t)=-\sin t$を$\verb@q@$とおいている。 それを、$\theta(t)=\sin t$を$\verb@p@$, $p(t)=\cos t$を$\verb@q@$に変えると、 $\theta'(t)=\cos t$, $p'(t)=-\sin t$をを$\verb@q@$となるため、例22と比較して、$\verb@euler@$, $\verb@leapfrog@$とも、 $\epsilon>0$の前の符号が逆になる。

import numpy as np
import matplotlib.pyplot as plt

L = 2
M = 100
eps = 0.1
# L = 3; M = 30; eps = 0.3

def euler(p, q):
    r = p + eps * q
    s = q - eps * p
    return r, s

def leapfrog(p, q):
    p = p + eps/2 * q
    q = q - eps * p
    p = p + eps/2 * q
    return p, q

def draw(proc, marker, color, P=0, Q=1):
    p = np.zeros(M)
    q = np.zeros(M)
    p[0] = P
    q[0] = Q
    for i in range(M - 1):
        p[i+1], q[i+1] = proc(p[i], q[i])
    plt.plot(p, q, marker=marker, color=color, label=proc.__name__)
    plt.scatter(p, q, marker=marker, color=color)

# グラフの設定
plt.figure(figsize=(8, 8))
plt.xlim(-L, L)
plt.ylim(-L, L)

# Euler法の描画
draw(euler, 'o', 'blue')

# Leapfrog法の描画
draw(leapfrog, 'x', 'red')

# 凡例を追加
plt.legend(loc="lower right")
plt.show()

問題19

// model19.stan

data{
  int N;
  array[N] real y;
}
parameters{
  real mu;
}
model{
  mu ~ normal(0,100);
  for(n in 1:N)
    y[n] ~ normal(mu, 1);
}

import numpy as np
import matplotlib.pyplot as plt
import nest_asyncio
nest_asyncio.apply()
import stan
import arviz as az

# サンプルサイズ
n = 100

# 正規乱数を生成
y = np.random.normal(size=n)

# データリストの作成
data_list = {'N': n, 'y': y}

# Stanモデルをファイルから読み込む
f = open('model19.stan', 'r', encoding='utf-8')
model19 = f.read()

# モデルをコンパイルして保存
posterior = stan.build(model19, data=data_list)

# サンプリングの実行
fit19 = posterior.sample()

# 結果の表示
az.summary(fit19)

# 密度プロットを表示
az.plot_trace(fit19)
az.plot_density(fit19)

Building...

Building: 38.6s, done.Messages from stanc:
Warning in '/tmp/httpstan_puk2md5x/model_q3qxy56c.stan', line 9, column 16: Argument
    100 suggests there may be parameters that are not unit scale; consider
    rescaling with a multiplier (see manual section 22.12).
Sampling:   0%
Sampling:  25% (2000/8000)
Sampling:  50% (4000/8000)
Sampling:  75% (6000/8000)
Sampling: 100% (8000/8000)
Sampling: 100% (8000/8000), done.
Messages received during sampling:
  Gradient evaluation took 0.000101 seconds
  1000 transitions using 10 leapfrog steps per transition would take 1.01 seconds.
  Adjust your expectations accordingly!
  Gradient evaluation took 6.6e-05 seconds
  1000 transitions using 10 leapfrog steps per transition would take 0.66 seconds.
  Adjust your expectations accordingly!
  Gradient evaluation took 1.8e-05 seconds
  1000 transitions using 10 leapfrog steps per transition would take 0.18 seconds.
  Adjust your expectations accordingly!
  Gradient evaluation took 1.4e-05 seconds
  1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds.
  Adjust your expectations accordingly!

array([[<Axes: title={'center': 'mu'}>]], dtype=object)

問題22

// model10.stan

data{
  int N; // サンプルサイズ
  int M; // 変数の数 ( 切片含む )
  vector[N] y; // 目的変数
  matrix[N,M] x; // 説明変数 , matrix で宣言
}
parameters{
  vector[M] beta; // vector で宣言
  real <lower=0> sigma;
}
model{
  beta ~ normal(0,100);
  sigma ~ cauchy(0,5);
  y ~ normal(x * beta, sigma);
}

import numpy as np
import pandas as pd
import stan
import arviz as az
import nest_asyncio

# 必要なモジュールをインポート
nest_asyncio.apply()

# Bostonデータセットをインターネットから取得
data_url = "http://lib.stat.cmu.edu/datasets/boston"
raw_df = pd.read_csv(data_url, sep="\s+", skiprows=22, header=None)
data = np.hstack([raw_df.values[::2, :], raw_df.values[1::2, :2]])
target = raw_df.values[1::2, 2]
X = data[:, [5,12]]

# データをPandasのDataFrameに変換

# Stanに渡すデータリストの作成
data_list = {
    'N': n,
    'M': X.shape[1],  # M=3
    'y': target,
    'x': X
}

# Stanモデルをファイルから読み込む
f = open('model10.stan', 'r', encoding='utf-8')
model22 = f.read()

# モデルをコンパイル
posterior = stan.build(model22, data=data_list)

# サンプリングの実行
fit22 = posterior.sample()

# 結果の表示
idata = az.from_pystan(posterior=fit22)
az.summary(idata)

# 密度プロットを表示
az.plot_trace(fit)
az.plot_density(fit)
plt.show()

問題23

import numpy as np
import pandas as pd
import stan
import arviz as az
import nest_asyncio

# 必要なモジュールをインポート
nest_asyncio.apply()

# Bostonデータセットをインターネットから取得
data_url = "http://lib.stat.cmu.edu/datasets/boston"
raw_df = pd.read_csv(data_url, sep="\s+", skiprows=22, header=None)
data = np.hstack([raw_df.values[::2, :], raw_df.values[1::2, :2]])
target = raw_df.values[1::2, 2]
X = data[:, [5]]

# データをPandasのDataFrameに変換

# Stanに渡すデータリストの作成
data_list = {
    'N': n,
    'M': X.shape[1],  # M=2
    'y': target,
    'x': X
}

# Stanモデルをファイルから読み込む
f = open('model10.stan', 'r', encoding='utf-8')
model23 = f.read()

# モデルをコンパイル
posterior = stan.build(model23, data=data_list)

# サンプリングの実行
fit23 = posterior.sample()

# 結果の表示
idata = az.from_pystan(posterior=fit23)
az.summary(idata)

問題24

// model24.stan

data {
 int<lower = 0> N;
 vector[N] y;
}
parameters {
  ordered[2] mu;
  array[2] real<lower=0>;
  real<lower=0, upper=1> theta;
}
model {
  mu ~ normal(0, 2);
  theta ~ beta(5, 5);
  for (n in 1:N)
    target += log_mix(theta,
      normal_lpdf(y[n] | mu[1], sigma[1]),
      normal_lpdf(y[n] | mu[2], sigma[2]));
}

import numpy as np
import stan
import arviz as az
import nest_asyncio

# 必要なモジュールをインポート
nest_asyncio.apply()

# サンプルサイズとデータの生成
N = 100
y = np.random.normal(size=N)

# Stanに渡すデータリストの作成
data_list = {'N': N, 'y': y}

# Stanモデルをファイルから読み込む
f = open('model24.stan', 'r', encoding='utf-8')
model24 = f.read()

# モデルをコンパイル
posterior = stan.build(model24, data=data_list)

# サンプリングの実行
fit = posterior.sample()

# 結果の密度プロットを表示
az.plot_density(fit)
plt.show()

問題25

p26の2番目のセルの11行目の$\verb@fit_X['diff']@$は、$\verb@fit_X['diff'].flatten()@$にしないと動きません。

// model_X.stan

data{
  int N;
  vector[N] y;
  vector[N] x;
}

parameters{
  real mu_y;
  real<lower=0> sigma_y;
  real mu_x;
  real<lower=0> sigma_x;
}

model{
  mu_y ~ normal(0,100);
  sigma_y ~ cauchy(0,5);
  mu_x ~ normal(0,100);
  sigma_x ~ cauchy(0,5);
  y ~ normal(mu_y,sigma_y);
  x ~ normal(mu_x,sigma_x);
}

generated quantities{
  real diff;
  real prob;
  diff = mu_x - mu_y;
}

import numpy as np
import stan
import arviz as az
import nest_asyncio
import matplotlib.pyplot as plt
from scipy.stats import gaussian_kde

y = np.random.normal(5,2,100)
x = np.random.normal(7,6,100)
data_list = {"N": 100, "y": y, "x": x}
f = open('model_X.stan', 'r', encoding='utf-8')
model_X=f.read()
posterior = stan.build(model_X, data=data_list)
fit_X = posterior.sample()
S = fit_X['diff'].flatten()
kde = gaussian_kde(S)
x = np.linspace(min(S), max(S), 100)
y = kde(x)
plt.plot(x, y)

問題26

// model26.stan

data{
  int N;
  int M;
  int y[N];
  matrix[N,M] x;
}

parameters{
  vector[M] beta;
}

model{
  beta ~ normal(0,100);
  y ~ bernoulli_logit(x*beta);
}

import numpy as np
import stan
import arviz as az
import nest_asyncio
import matplotlib.pyplot as plt

# 必要なモジュールをインポート
nest_asyncio.apply()

# パラメータの設定
N = 100
M = 3

# ベータ係数の生成
beta = np.random.normal(size=M)

# 説明変数xの生成
x = np.random.normal(size=(N, M))

# 目的変数yの生成
y = np.zeros(N, dtype=int)
for i in range(N):
    p = 1 / (1 + np.exp(-np.dot(x[i, :], beta)))
    y[i] = 1 if np.random.uniform(0, 1) < p else 0

# Stanに渡すデータリストの作成
data_list = {"N": N, "M": M, "x": x, "y": y}

# Stanモデルをファイルから読み込む
f = open('model26.stan', 'r', encoding='utf-8')
model26 = f.read()

# モデルのビルドとサンプリング
posterior = stan.build(model26, data=data_list)
fit = posterior.sample()

# 密度プロットを表示
az.plot_density(fit)
plt.show()

第3章

問題35

import numpy as np
import matplotlib.pyplot as plt

# パラメータの設定
n = 100
a = np.zeros(n)
sum_x = 0

# シミュレーションと累積平均の計算
for i in range(1, n+1):
    x = np.random.binomial(1, 0.5)
    sum_x += x
    a[i-1] = sum_x / i

# プロットの表示
plt.plot(range(1, n+1), a, linestyle='-')
plt.xlabel('Iteration')
plt.ylabel('Cumulative Mean')
plt.title('Cumulative Mean of Binomial Samples')
plt.show()

問題37

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import chi2, norm

def CLT(m, n, df):
    S = []
    mu = df
    sigma = np.sqrt(2 * df)

    for _ in range(m):
        x = np.random.chisquare(df, n)
        S.append((np.sum(x) - n * mu) / (np.sqrt(n) * sigma))

    # 密度プロットの描画
    density = plt.hist(S, bins=30, density=True, alpha=0.6, color='g')
    xmin, xmax = plt.xlim()
    x = np.linspace(xmin, xmax, 100)
    p = norm.pdf(x, 0, 1)
    plt.plot(x, p, 'k', linewidth=2)
    title = "自由度2のカイ2乗分布から得られた正規分布"
    plt.title(title)
    plt.xlabel("Y_n")
    plt.ylabel("確率密度")
    plt.show()

# パラメータの設定
m = 300
n = 100
df = 2

# CLT関数の実行
CLT(m, n, df)

第4章

問題48

import numpy as np

# パラメータの設定
mu = 0
sigma = 1
n = 100

# G関数とT関数の定義
def G(y):
    return 0.5 * np.log(2 * np.pi * (n + 2) / (n + 1)) + (n + 1) / (n + 2) / 2 * (sigma**2 + (mu - np.sum(y) / (n + 1))**2)

def T(y):
    return 0.5 * np.log(2 * np.pi * (n + 2) / (n + 1)) + (n + 1) / (n + 2) / 2 * np.mean((y - np.sum(y) / (n + 1))**2)

# サンプルデータの生成
y = np.random.normal(mu, sigma, n)

# 関数の実行
G_value = G(y)
T_value = T(y)

# 結果の表示
print("G(y):", G_value)
print("T(y):", T_value)

第5章

問題61

// model61.stan

data {
 int<lower = 0> N;
 vector[N] y;
}
parameters {
  ordered[2] mu;
  real<lower=0, upper=1> theta;
  real<lower=0> sigma;
}
model {
  mu ~ normal(0, 2);
  theta ~ beta(5, 5);
  for (n in 1:N)
    target += log_mix(theta,
      normal_lpdf(y[n] | mu[1], sigma),
      normal_lpdf(y[n] | mu[2], sigma));
}
generated quantities{
  vector[N] log_lik;
  for (n in 1:N)
    log_lik[n]= log_mix(theta,
      normal_lpdf(y[n] | mu[1], sigma),
      normal_lpdf(y[n] | mu[2], sigma));
}

import numpy as np
import stan
import arviz as az
import nest_asyncio

# 必要なモジュールをインポート
nest_asyncio.apply()

# WAICとその他の関数の定義
def V_n(log_likelihood):
    return np.mean(np.mean(log_likelihood**2, axis=1) - np.mean(log_likelihood, axis=1)**2)

def T_n(log_likelihood):
    return -np.mean(np.log(np.mean(np.exp(log_likelihood), axis=1)))

def WAIC(log_likelihood):
    return T_n(log_likelihood) + V_n(log_likelihood)

# データ生成関数
def generator(n):
    data1 = np.random.normal(-4, 1, n)  # N(-4, 1) に従う乱数
    data2 = np.random.normal(2, 1, n)   # N(2, 1) に従う乱数
    data3 = (np.random.uniform(0, 1, n) <= 0.6)  # 確率 0.6 で 1 となる論理ベクトル
    return data1 * data3 + data2 * (1 - data3)  # 確率 0.6 で N(-4,1), 確率0.4で N(2,1) 乱数を採択

# データの生成
N = 1000
y = generator(N)

# Stanモデルをファイルから読み込む
with open('model61.stan', 'r', encoding='utf-8') as f:
    model_code = f.read()

# モデルのビルドとサンプリング
posterior = stan.build(model_code, data={'N': N, 'y': y})
fit = posterior.sample()

# Stanから生成されたパラメータの抽出
log_lik = fit['log_lik']

# WAICの計算
waic_value = WAIC(log_lik)
print("WAIC:", waic_value)

# AICの計算
aic_value = 1/2 * np.log(2 * np.pi * np.exp(1)) + 1/2 * np.log((N - 1) / N * np.var(y)) + 3/2/N
print("AIC:", aic_value)

問題62

// model11.stan

data{
  int N; // サンプルサイズ
  int M; // 変数の数 ( 切片含む )
  vector[N] y; // 目的変数
  matrix[N,M] x; // 説明変数 , matrix で宣言
}
parameters{
  vector[M] beta; // vector で宣言
  real <lower=0> sigma;
}
model{
  beta ~ normal(0,100);
  sigma ~ cauchy(0,5);
  y ~ normal(x * beta, sigma);
}
generated quantities{
  array[N] real log_lik;
  for(n in 1:N)
    log_lik[n]= normal_lpdf(y[n]|x[n]*beta, sigma);
}

import numpy as np
import pandas as pd
import stan
import arviz as az
import nest_asyncio
from scipy.special import logsumexp

# 必要なモジュールをインポート
nest_asyncio.apply()

# WAICとその他の関数の定義
def V_n(log_likelihood):
    return np.mean(np.mean(log_likelihood**2, axis=1) - np.mean(log_likelihood, axis=1)**2)

def T_n(log_likelihood):
    return -np.mean(np.log(np.mean(np.exp(log_likelihood), axis=1)))

def WAIC(log_likelihood):
    return T_n(log_likelihood) + V_n(log_likelihood)

# Bostonデータセットをインターネットから取得
data_url = "http://lib.stat.cmu.edu/datasets/boston"
raw_df = pd.read_csv(data_url, sep="\s+", skiprows=22, header=None)

# データの整形
data = np.hstack([raw_df.values[::2, :], raw_df.values[1::2, :2]])
target = raw_df.values[1::2, 2]

# 選択するインデックス（Pythonでは0ベース）
index = [0, 2, 4, 5, 7, 9, 10, 11, 12]  # 1, 3, 5, 6, 8, 10, 11, 12, 13 のRに対応

# 正しい列名を使用（13列のみ）
df = pd.DataFrame(data, columns=['CRIM', 'ZN', 'INDUS', 'CHAS', 'NOX', 'RM', 'AGE', 'DIS', 'RAD', 'TAX', 'PTRATIO', 'B', 'LSTAT'])

# 必要な列を選択
df = df.iloc[:, index]
X = np.column_stack((np.ones(df.shape[0]), df))
N = df.shape[0]
K = X.shape[1]
Y = target

# Stanに渡すデータリストの作成
data_list = {'N': N, 'M': K, 'y': Y, 'x': X}

# Stanモデルをファイルから読み込む
with open('model11.stan', 'r', encoding='utf-8') as f:
    model_code = f.read()

# モデルのビルドとサンプリング
posterior = stan.build(model_code, data=data_list, random_seed=1)
fit = posterior.sample()

# Stanから生成されたパラメータの抽出
log_lik = fit['log_lik']

# WAICの計算
waic_value = 2 * N * WAIC(log_lik)
print("2*N*WAIC:", waic_value)

# ArviZを使用してWAICの計算
idata = az.from_pystan(posterior=fit)
waic_result = az.waic(idata)
print("WAIC (via ArviZ):", waic_result)

問題64

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import gamma
from scipy.stats import gaussian_kde

# Fn関数の定義
def Fn(x, n):
    k = np.sum(x)
    return np.log(gamma(n+2)) - np.log(gamma(n-k+1)) - np.log(gamma(k+1))

# パラメータの設定
m = 500
n = 100
T = []

# シミュレーションの実行
for j in range(m):
    x = np.random.binomial(1, 0.25, n)
    T.append(Fn(x, n))

# カーネル密度推定の実行
T = np.array(T)
kde = gaussian_kde(T)
x_vals = np.linspace(min(T), max(T), 1000)
y_vals = kde(x_vals)

# 密度プロットの描画
plt.figure(figsize=(8, 6))
plt.plot(x_vals, y_vals, color='red')
plt.title("自由エネルギー")
plt.xlabel("$F_n$")
plt.ylabel("確率密度関数")
plt.show()

問題66

// model15.stan

data {
  int N;                // サンプルサイズ
  int M;                // 説明変数の数＋１(切片)
  vector[N] y;          // 目的変数
  matrix[N, M] x;       // デザイン行列
  real beta;            // 逆温度
}
parameters {
  vector[M] b;          // M-1個の傾きと切片
  real<lower=0> sigma;  // 標準偏差
}
model {
  for(n in 1:N)
    target += beta * normal_lpdf(y[n] | x[n] * b, sigma);
}
generated quantities{
  vector[N] log_lik;
  for (n in 1:N)
    log_lik[n]= normal_lpdf(y[n] | x[n]*b, sigma);
}

import numpy as np
import pandas as pd
import stan
import arviz as az
import nest_asyncio

# 必要なモジュールをインポート
nest_asyncio.apply()

# WBICとBICの計算関数
def wbic(log_likelihood):
    return -np.mean(np.sum(log_likelihood, axis=0))

def bic(x, y):
    n, p = x.shape
    beta2 = np.linalg.inv(x.T @ x) @ (x.T @ y)
    sigma2 = np.sum((y - x @ beta2) ** 2) / n
    return 0.5 * n * np.log(2 * np.pi * np.exp(1) * sigma2) + 0.5 * (p + 2) * np.log(n)

# Bostonデータセットをインターネットから取得
data_url = "http://lib.stat.cmu.edu/datasets/boston"
raw_df = pd.read_csv(data_url, sep="\s+", skiprows=22, header=None)

# データの整形
data = np.hstack([raw_df.values[::2, :], raw_df.values[1::2, :2]])
target = raw_df.values[1::2, 2]

# 正しい列名を使用（13列のみ）
df = pd.DataFrame(data, columns=['CRIM', 'ZN', 'INDUS', 'CHAS', 'NOX', 'RM', 'AGE', 'DIS', 'RAD', 'TAX', 'PTRATIO', 'B', 'LSTAT'])

# 選択するインデックス（Pythonでは0ベース）
index = [0, 2, 4, 5, 7, 9, 10, 11, 12]  # 1, 3, 5, 6, 8, 10, 11, 12, 13 のRに対応

# 必要な列を選択
x = df.iloc[:, index].values
x = np.column_stack((np.ones(df.shape[0]), x))  # 切片項を追加
y = target
n, p = x.shape

# Stanに渡すデータリストの作成
data_list = {'N': n, 'M': p, 'y': y, 'x': x, 'beta': 1/np.log(n)}

# Stanモデルをファイルから読み込む
with open('model15.stan', 'r', encoding='utf-8') as f:
    model_code = f.read()

# モデルのビルドとサンプリング
posterior = stan.build(model_code, data=data_list)
fit = posterior.sample(num_chains=4, num_samples=3000)

# Stanから生成されたパラメータの抽出
log_lik = fit['log_lik']

# WBICの計算
wbic_value = wbic(log_lik)
print("WBIC:", wbic_value)

# BICの計算
bic_value = bic(x, y)
print("BIC:", bic_value)

第6章

問題68

import numpy as np
import matplotlib.pyplot as plt

# パラメータの設定
a = 0
b = 0
x_min = -1
x_max = 5

# 関数fの定義
def f(x, a, b):
    return np.sqrt(np.maximum(x**3 + a*x + b, 0))

# xのシーケンスを作成
x_seq = np.arange(x_min, x_max, 0.001)

# yのシーケンスを計算
y_seq = np.array([f(x, a, b) for x in x_seq])

# yの最大値を計算
y_max = np.max(y_seq)

# プロットの設定
plt.figure(figsize=(8, 6))
plt.plot(x_seq, y_seq, label='y=f(x)')
plt.plot(x_seq, -y_seq, label='y=-f(x)')
plt.axhline(0, color='black', linewidth=0.5)
plt.axvline(0, color='black', linewidth=0.5)
plt.xlim(x_min, x_max)
plt.ylim(-y_max, y_max)
plt.xlabel('x')
plt.ylabel('y')
plt.title(f'a={a}, b={b}')
plt.legend()
plt.show()

import numpy as np
import matplotlib.pyplot as plt

# パラメータの設定
a = -3
b = 2
x_min = -3
x_max = 3

# 関数fの定義
def f(x, a, b):
    return np.sqrt(np.maximum(x**3 + a*x + b, 0))

# xのシーケンスを作成
x_seq = np.arange(x_min, x_max, 0.001)

# yのシーケンスを計算
y_seq = np.array([f(x, a, b) for x in x_seq])

# yの最大値を計算
y_max = np.max(y_seq)

# プロットの設定
plt.figure(figsize=(8, 6))
plt.plot(x_seq, y_seq, label='y=f(x)')
plt.plot(x_seq, -y_seq, label='y=-f(x)')
plt.axhline(0, color='black', linewidth=0.5)
plt.axvline(0, color='black', linewidth=0.5)
plt.xlim(x_min, x_max)
plt.ylim(-y_max, y_max)
plt.xlabel('x')
plt.ylabel('y')
plt.title(f'a={a}, b={b}')
plt.legend()
plt.show()

第7章

問題77

import numpy as np
import matplotlib.pyplot as plt

# delta関数の定義
def delta(a, j):
    x_values = [-a, a, a, -a, -a]
    y_values = [0, 0, 1/a, 1/a, 0]
    plt.plot(x_values, y_values, color=j)

# パラメータの設定
a_seq = [10**(-3)]

# プロットの設定
plt.figure(figsize=(8, 6))
plt.xlim(-0.05, 0.05)
plt.ylim(0, 1005)
plt.xlabel("x")
plt.ylabel("fa(x)")
plt.title("Uniform Distribution")

# delta関数を使って線を描画
for a in a_seq:
    delta(a, 'blue')  # 色は固定

plt.show()

問題86

import numpy as np
import pandas as pd
import stan
import nest_asyncio

# 必要なモジュールをインポート
nest_asyncio.apply()

# V_n, T_n, WAIC, CVの関数定義
def V_n(log_likelihood):
    return np.mean(np.mean(log_likelihood**2, axis=1) - np.mean(log_likelihood, axis=1)**2)

def T_n(log_likelihood):
    return -np.mean(np.log(np.mean(np.exp(log_likelihood), axis=1)))

def WAIC(log_likelihood, beta):
    return T_n(log_likelihood) + beta * V_n(log_likelihood)

def CV(log_likelihood, beta):
    return -np.mean(
        np.log(np.mean(np.exp((1-beta) * log_likelihood), axis=1)) -
        np.log(np.mean(np.exp(-beta * log_likelihood), axis=1))
    )

# Bostonデータセットをインターネットから取得
data_url = "http://lib.stat.cmu.edu/datasets/boston"
raw_df = pd.read_csv(data_url, sep="\s+", skiprows=22, header=None)

# データの整形
data = np.hstack([raw_df.values[::2, :], raw_df.values[1::2, :2]])
target = raw_df.values[1::2, 2]

# 使用する変数のインデックス (13列しかないので0から12を使用)
index = [0, 2, 4, 5, 6, 7, 9, 10, 11, 12]
X = data[:, index]
N = data.shape[0]
K = X.shape[1]
Y = target

# パラメータの初期化
waic_values = []
cv_values = []
beta_seq = np.arange(0.1, 1.6, 0.1)

# Stanモデルの実行とWAIC/CVの計算
for beta in beta_seq:
    data_list = {'N': N, 'M': K, 'y': Y, 'x': X, 'beta': beta}

    with open('model15.stan', 'r', encoding='utf-8') as f:
        model_code = f.read()

    # モデルのビルドとサンプリング
    posterior = stan.build(model_code, data=data_list, random_seed=1)
    fit = posterior.sample(num_chains=4, num_samples=1000)

    log_lik = fit['log_lik']

    # WAICとCVの計算
    waic_values.append(N * WAIC(log_lik, beta))
    cv_values.append(N * CV(log_lik, beta))

# 結果の表示
print("WAIC values:", waic_values)
print("CV values:", cv_values)

# プロットの設定
plt.figure(figsize=(8, 6))

# WAICのプロット
plt.plot(beta_seq, waic_values, label="WAIC", color="red", linestyle="-")

# CVのプロット
plt.plot(beta_seq, cv_values, label="CV", color="blue", linestyle="-")

# グラフの詳細設定
plt.xlabel("Beta")
plt.ylabel("WAIC/CV")
plt.ylim(1520, 1560)
plt.legend(loc="upper right")
plt.title("WAICとCVの値の変化")
plt.grid(True)

# グラフの表示
plt.show()

第8章

問題98

// model98.stan

data {
  int<lower=1> K;          // number of mixture components
  int<lower=1> N;          // number of data points
  array[N] vector[2] y;    // observations
  real beta;
}
parameters {
  simplex[K] theta;
  array[K] vector[2] mu;
  vector<lower=0>[2] sigma;
}
transformed parameters{
  vector[K] log_theta = log(theta);  // cache log calculation
}
model {
  theta ~ dirichlet(rep_vector(1.0, K));  // Dirichlet事前分布
  mu ~ multi_normal(rep_vector(0.0, 2), diag_matrix(rep_vector(1.0, 2)));  // 多変量正規分布
  sigma ~ normal(0, 1);  // 事前分布として正規分布
  for (n in 1:N) {
    vector[K] lps = log_theta;
    for (k in 1:K)
      lps[k] += multi_normal_lpdf(y[n] | mu[k], diag_matrix(sigma));
    target += beta*log_sum_exp(lps);
  }
}
generated quantities {
  vector[N] log_lik;
  for (n in 1:N) {
    vector[K] lps = log_theta;
    for (k in 1:K)
      lps[k] += multi_normal_lpdf(y[n] | mu[k], diag_matrix(sigma));
    log_lik[n] = log_sum_exp(lps);
  }
}

import numpy as np
import stan
import nest_asyncio

# 必要なモジュールをインポート
nest_asyncio.apply()

import shutil
shutil.rmtree('/path/to/stan_cache', ignore_errors=True)

# WBICの計算関数
def wbic(log_likelihood):
    return -np.mean(np.sum(log_likelihood, axis=0))

# パラメータの設定
b_seq = [1, 10, 100, 250]
K_seq = [1, 2, 3, 4]
x = []

# データの生成
n = 100
for i in range(n):
    x.append([np.random.normal(-2, 1), np.random.normal(-2, 1)])
for i in range(n, 2 * n):
    x.append([np.random.normal(2, 1), np.random.normal(2, 1)])
for i in range(2 * n, 3 * n):
    x.append([np.random.normal(0, 1), np.random.normal(0, 1)])

# WBICの計算

f = open('model98.stan', 'r', encoding='utf-8')
model_code = f.read()

WBIC = []
for b in b_seq:
    for k in K_seq:
        data_list = {'K': k, 'N': len(x), 'y': x, 'beta': b / np.log(n)}

        # モデルのビルドとサンプリング
        posterior = stan.build(model_code, data=data_list, random_seed=1)
        fit = posterior.sample(num_chains=4, num_samples=5000, num_warmup=2500)
        print(fit)
        log_lik = fit['log_lik']

        # WBICの計算
        wbic_value = wbic(log_lik)
        print(wbic_value)
        WBIC.append(wbic_value)

# WBICの結果を表示
print("WBIC values:", WBIC)

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Messages received during sampling:
  Gradient evaluation took 0.009365 seconds
  1000 transitions using 10 leapfrog steps per transition would take 93.65 seconds.
  Adjust your expectations accordingly!
  Gradient evaluation took 0.000744 seconds
  1000 transitions using 10 leapfrog steps per transition would take 7.44 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.000644 seconds
  1000 transitions using 10 leapfrog steps per transition would take 6.44 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.00065 seconds
  1000 transitions using 10 leapfrog steps per transition would take 6.5 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (1,)
    mu: (1, 2)
    sigma: (2,)
    log_theta: (1,)
    log_lik: (300,)
Draws: 20000
1260.3819645213698
Building...

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Messages received during sampling:
  Gradient evaluation took 0.000742 seconds
  1000 transitions using 10 leapfrog steps per transition would take 7.42 seconds.
  Adjust your expectations accordingly!
  Gradient evaluation took 0.001435 seconds
  1000 transitions using 10 leapfrog steps per transition would take 14.35 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001428 seconds
  1000 transitions using 10 leapfrog steps per transition would take 14.28 seconds.
  Adjust your expectations accordingly!
  Gradient evaluation took 0.001373 seconds
  1000 transitions using 10 leapfrog steps per transition would take 13.73 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (2,)
    mu: (2, 2)
    sigma: (2,)
    log_theta: (2,)
    log_lik: (300,)
Draws: 20000
1141.9540145360638
Building...

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Messages received during sampling:
  Gradient evaluation took 0.001225 seconds
  1000 transitions using 10 leapfrog steps per transition would take 12.25 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002702 seconds
  1000 transitions using 10 leapfrog steps per transition would take 27.02 seconds.
  Adjust your expectations accordingly!
  Gradient evaluation took 0.001905 seconds
  1000 transitions using 10 leapfrog steps per transition would take 19.05 seconds.
  Adjust your expectations accordingly!
  Gradient evaluation took 0.001868 seconds
  1000 transitions using 10 leapfrog steps per transition would take 18.68 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (3,)
    mu: (3, 2)
    sigma: (2,)
    log_theta: (3,)
    log_lik: (300,)
Draws: 20000
1128.447539949412
Building...

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Messages received during sampling:
  Gradient evaluation took 0.001623 seconds
  1000 transitions using 10 leapfrog steps per transition would take 16.23 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002632 seconds
  1000 transitions using 10 leapfrog steps per transition would take 26.32 seconds.
  Adjust your expectations accordingly!
  Gradient evaluation took 0.00284 seconds
  1000 transitions using 10 leapfrog steps per transition would take 28.4 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002404 seconds
  1000 transitions using 10 leapfrog steps per transition would take 24.04 seconds.
  Adjust your expectations accordingly!

<stan.Fit>
Parameters:
    theta: (4,)
    mu: (4, 2)
    sigma: (2,)
    log_theta: (4,)
    log_lik: (300,)
Draws: 20000
1127.2309779110658
Building...

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Sampling: 100% (30000/30000), done.
Messages received during sampling:
  Gradient evaluation took 0.000533 seconds
  1000 transitions using 10 leapfrog steps per transition would take 5.33 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.000671 seconds
  1000 transitions using 10 leapfrog steps per transition would take 6.71 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.000459 seconds
  1000 transitions using 10 leapfrog steps per transition would take 4.59 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.000774 seconds
  1000 transitions using 10 leapfrog steps per transition would take 7.74 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (1,)
    mu: (1, 2)
    sigma: (2,)
    log_theta: (1,)
    log_lik: (300,)
Draws: 20000
1246.8354052352738
Building...

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Messages received during sampling:
  Gradient evaluation took 0.000722 seconds
  1000 transitions using 10 leapfrog steps per transition would take 7.22 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001379 seconds
  1000 transitions using 10 leapfrog steps per transition would take 13.79 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001605 seconds
  1000 transitions using 10 leapfrog steps per transition would take 16.05 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001371 seconds
  1000 transitions using 10 leapfrog steps per transition would take 13.71 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (2,)
    mu: (2, 2)
    sigma: (2,)
    log_theta: (2,)
    log_lik: (300,)
Draws: 20000
1128.0460159450765
Building...

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Sampling: 100% (30000/30000), done.
Messages received during sampling:
  Gradient evaluation took 0.001057 seconds
  1000 transitions using 10 leapfrog steps per transition would take 10.57 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001952 seconds
  1000 transitions using 10 leapfrog steps per transition would take 19.52 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.00436 seconds
  1000 transitions using 10 leapfrog steps per transition would take 43.6 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001948 seconds
  1000 transitions using 10 leapfrog steps per transition would take 19.48 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (3,)
    mu: (3, 2)
    sigma: (2,)
    log_theta: (3,)
    log_lik: (300,)
Draws: 20000
1102.0869903589153
Building...

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Messages received during sampling:
  Gradient evaluation took 0.001343 seconds
  1000 transitions using 10 leapfrog steps per transition would take 13.43 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002517 seconds
  1000 transitions using 10 leapfrog steps per transition would take 25.17 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002542 seconds
  1000 transitions using 10 leapfrog steps per transition would take 25.42 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002585 seconds
  1000 transitions using 10 leapfrog steps per transition would take 25.85 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (4,)
    mu: (4, 2)
    sigma: (2,)
    log_theta: (4,)
    log_lik: (300,)
Draws: 20000
1102.1097299016808
Building...

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Sampling: 100% (30000/30000), done.
Messages received during sampling:
  Gradient evaluation took 0.000451 seconds
  1000 transitions using 10 leapfrog steps per transition would take 4.51 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.000839 seconds
  1000 transitions using 10 leapfrog steps per transition would take 8.39 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.000418 seconds
  1000 transitions using 10 leapfrog steps per transition would take 4.18 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.000712 seconds
  1000 transitions using 10 leapfrog steps per transition would take 7.12 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (1,)
    mu: (1, 2)
    sigma: (2,)
    log_theta: (1,)
    log_lik: (300,)
Draws: 20000
1245.8732933339513
Building...

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Sampling: 100% (30000/30000), done.
Messages received during sampling:
  Gradient evaluation took 0.004445 seconds
  1000 transitions using 10 leapfrog steps per transition would take 44.45 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001277 seconds
  1000 transitions using 10 leapfrog steps per transition would take 12.77 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001284 seconds
  1000 transitions using 10 leapfrog steps per transition would take 12.84 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002465 seconds
  1000 transitions using 10 leapfrog steps per transition would take 24.65 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (2,)
    mu: (2, 2)
    sigma: (2,)
    log_theta: (2,)
    log_lik: (300,)
Draws: 20000
1126.6031659530333
Building...

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Sampling: 100% (30000/30000), done.
Messages received during sampling:
  Gradient evaluation took 0.001191 seconds
  1000 transitions using 10 leapfrog steps per transition would take 11.91 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001947 seconds
  1000 transitions using 10 leapfrog steps per transition would take 19.47 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001913 seconds
  1000 transitions using 10 leapfrog steps per transition would take 19.13 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002043 seconds
  1000 transitions using 10 leapfrog steps per transition would take 20.43 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (3,)
    mu: (3, 2)
    sigma: (2,)
    log_theta: (3,)
    log_lik: (300,)
Draws: 20000
1099.9886816454798
Building...

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Sampling: 100% (30000/30000), done.
Messages received during sampling:
  Gradient evaluation took 0.002478 seconds
  1000 transitions using 10 leapfrog steps per transition would take 24.78 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.008536 seconds
  1000 transitions using 10 leapfrog steps per transition would take 85.36 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002529 seconds
  1000 transitions using 10 leapfrog steps per transition would take 25.29 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002519 seconds
  1000 transitions using 10 leapfrog steps per transition would take 25.19 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (4,)
    mu: (4, 2)
    sigma: (2,)
    log_theta: (4,)
    log_lik: (300,)
Draws: 20000
1098.8097110796136
Building...

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Sampling: 100% (30000/30000), done.
Messages received during sampling:
  Gradient evaluation took 0.000498 seconds
  1000 transitions using 10 leapfrog steps per transition would take 4.98 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001066 seconds
  1000 transitions using 10 leapfrog steps per transition would take 10.66 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001268 seconds
  1000 transitions using 10 leapfrog steps per transition would take 12.68 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.000803 seconds
  1000 transitions using 10 leapfrog steps per transition would take 8.03 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (1,)
    mu: (1, 2)
    sigma: (2,)
    log_theta: (1,)
    log_lik: (300,)
Draws: 20000
1245.8160538749798
Building...

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Sampling: 100% (30000/30000)
Sampling: 100% (30000/30000), done.
Messages received during sampling:
  Gradient evaluation took 0.00148 seconds
  1000 transitions using 10 leapfrog steps per transition would take 14.8 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.005485 seconds
  1000 transitions using 10 leapfrog steps per transition would take 54.85 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001315 seconds
  1000 transitions using 10 leapfrog steps per transition would take 13.15 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001395 seconds
  1000 transitions using 10 leapfrog steps per transition would take 13.95 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (2,)
    mu: (2, 2)
    sigma: (2,)
    log_theta: (2,)
    log_lik: (300,)
Draws: 20000
1126.5068149378512
Building...

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Sampling: 100% (30000/30000), done.
Messages received during sampling:
  Gradient evaluation took 0.001033 seconds
  1000 transitions using 10 leapfrog steps per transition would take 10.33 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002151 seconds
  1000 transitions using 10 leapfrog steps per transition would take 21.51 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002103 seconds
  1000 transitions using 10 leapfrog steps per transition would take 21.03 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.001901 seconds
  1000 transitions using 10 leapfrog steps per transition would take 19.01 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (3,)
    mu: (3, 2)
    sigma: (2,)
    log_theta: (3,)
    log_lik: (300,)
Draws: 20000
1099.8498891984248
Building...

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Messages received during sampling:
  Gradient evaluation took 0.002308 seconds
  1000 transitions using 10 leapfrog steps per transition would take 23.08 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is 0. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.002451 seconds
  1000 transitions using 10 leapfrog steps per transition would take 24.51 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.003054 seconds
  1000 transitions using 10 leapfrog steps per transition would take 30.54 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Gradient evaluation took 0.00252 seconds
  1000 transitions using 10 leapfrog steps per transition would take 25.2 seconds.
  Adjust your expectations accordingly!
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.
  Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:
  Exception: multi_normal_lpdf: LDLT_Factor of covariance parameter is not positive definite.  last conditional variance is -nan. (in '/tmp/httpstan_rnfs2vs9/model_7mmttzo2.stan', line 22, column 6 to column 68)
  If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,
  but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

<stan.Fit>
Parameters:
    theta: (4,)
    mu: (4, 2)
    sigma: (2,)
    log_theta: (4,)
    log_lik: (300,)
Draws: 20000
1099.1187638611098
WBIC values: [1260.3819645213698, 1141.9540145360638, 1128.447539949412, 1127.2309779110658, 1246.8354052352738, 1128.0460159450765, 1102.0869903589153, 1102.1097299016808, 1245.8732933339513, 1126.6031659530333, 1099.9886816454798, 1098.8097110796136, 1245.8160538749798, 1126.5068149378512, 1099.8498891984248, 1099.1187638611098]

fit