渡辺澄夫ベイズ理論 with R
鈴木 讓
2024-08-12
第1章
問題 5
f <- function(x, mu, sig2) (2*pi*sig2)^(-1/2) * exp(-(x-mu)^2/2)
## 正規分布の定義
phi.seq <- c(-3, -1, 1, 3)
n.seq <- c(1, 5, 10)
m <- length(phi.seq)
l <- length(n.seq)
n <- 30 ## サンプルサイズ n
x <- rnorm(n) ## 正規乱数をn 個発生
par(mfrow = c(2, 2)) ## グラフを4 個発生
for(phi in phi.seq){
plot(0, 0, xlim=c(-5, 7), ylim=c(0, 0.5), type="n")
for(k in 1:l){
nn <- n.seq[k] ## サンプルの最初のn 個
mu <- (phi+sum(x)) / (nn+1)
sig2 <- 1
curve(f(x, mu, sig2), col=k+1, add=TRUE) ## 曲線を描く
title(paste("phi=", phi))
}
## 真の分布の曲線を描く
curve(dnorm(x), lwd=2, lty=2, col=1, add=TRUE)
legend("topright",c("真", n.seq), lty=c(2, rep(1,4)),
lwd=c(2,rep(1,4)), col=1:(l+1))
}
par(mfrow = c(1,1))問題 6
plot(0, 0, xlim=c(-5, 5), ylim=c(0, 0.4), type="n", xlab="t", ylab="Density")
curve(dnorm(x), -5, 5, lwd=2, add=TRUE)
for(i in 1:5) curve(dt(x, i), -5, 5, col=i+1, add=TRUE)
legend("topright", paste("degree", 1:5), lwd=1, col=2:6)
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。
L <- 2
M <- 100
eps <- 0.1
#L=3; M=30; eps=0.3
euler <- function(p, q){
r <- p + eps*q
s <- q - eps*p
return(list(p=r,q=s))
}
leapfrog <- function(p, q){
p <- p + eps/2*q
q <- q - eps*p
p <- p + eps/2*q
return(list(p=p, q=q))
}
draw <- function(proc, pch, col, P=0, Q=1){
p <- rep(0,M)
q <- rep(0,M)
p[1] <- P
q[1] <- Q
for(i in 1:(M-1)){
res <- proc(p[i],q[i])
p[i+1] <- res$p
q[i+1] <- res$q
}
points(p,q,pch=pch,col=col)
lines(p,q)
}
plot(0,xlim=c(-L, L),ylim=c(-L, L))
draw(euler, 1, 2)
draw(leapfrog, 4, 4)
legend("bottomright", legend=c("Euler", "Leapfrog"), lwd=1, pch=c(1, 4), col=c(2, 4))
問題 19
例4では、\(\sigma^2=1\)を仮定しているため、\(\verb@model4.stan@ではなく、下記の\)19.stan@$を用いる。
data{
int N; // データ数
array[N] real; // データ
}
parameters{
real mu; // 平均値
}
model{
mu ~ normal(0,100);
for(n in 1:N)
y[n] ~ normal(mu, 1);
}#n <- 10
n <- 100
y <- rnorm(n)
library(rstan)## 要求されたパッケージ StanHeaders をロード中です##
## rstan version 2.26.21 (Stan version 2.26.1)## For execution on a local, multicore CPU with excess RAM we recommend calling
## options(mc.cores = parallel::detectCores()).
## To avoid recompilation of unchanged Stan programs, we recommend calling
## rstan_options(auto_write = TRUE)
## For within-chain threading using `reduce_sum()` or `map_rect()` Stan functions,
## change `threads_per_chain` option:
## rstan_options(threads_per_chain = 1)## Do not specify '-march=native' in 'LOCAL_CPPFLAGS' or a Makevars filemodel <- stan_model("model19.stan")
data_list <- list(N=n, y=y)
fit <- sampling(model, data=data_list)fit## Inference for Stan model: anon_model.
## 4 chains, each with iter=2000; warmup=1000; thin=1;
## post-warmup draws per chain=1000, total post-warmup draws=4000.
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
## mu -0.08 0.00 0.10 -0.28 -0.14 -0.08 -0.01 0.11 1375 1
## lp__ -62.78 0.02 0.69 -64.73 -62.95 -62.50 -62.34 -62.29 1949 1
##
## Samples were drawn using NUTS(diag_e) at Wed Aug 14 07:11:23 2024.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).stan_dens(fit)
問題 22
説明変数を\(\verb@rm@\)以外に\(\verb@lstat@\)を加えて、\(\verb@model10@\)で\(\verb@beta@\)と\(\verb@sigma@\)の事後分布を求める。
library(rstan)
library(MASS)
n <- nrow(Boston)
X <- cbind(rep(1,n), Boston$rm, Boston$lstat)
data_list <- list(N=n, M=3, y=Boston$medv, x=X) # M=3 に注意
model <- stan_model("model10.stan")
fit <- sampling(model, data=data_list)print(fit, probs=c(0.025, 0.5, 0.975))## Inference for Stan model: anon_model.
## 4 chains, each with iter=2000; warmup=1000; thin=1;
## post-warmup draws per chain=1000, total post-warmup draws=4000.
##
## mean se_mean sd 2.5% 50% 97.5% n_eff Rhat
## beta[1] -1.35 0.09 3.13 -7.37 -1.40 4.93 1121 1
## beta[2] 5.10 0.01 0.44 4.23 5.10 5.94 1190 1
## beta[3] -0.64 0.00 0.04 -0.73 -0.64 -0.56 1288 1
## sigma 5.55 0.00 0.18 5.21 5.55 5.92 1925 1
## lp__ -1118.89 0.04 1.42 -1122.42 -1118.54 -1117.13 1245 1
##
## Samples were drawn using NUTS(diag_e) at Wed Aug 14 07:12:51 2024.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).stan_dens(fit)
問題 23
library(rstan)
library(MASS)
n <- nrow(Boston)
X <- cbind(rep(1,n), Boston$rm)
data_list <- list(N =n, M=2, y = Boston$medv, x = X)
fit <- stan("model10.stan", data = data_list)print(fit, probs = c(0.025, 0.5, 0.975))## Inference for Stan model: anon_model.
## 4 chains, each with iter=2000; warmup=1000; thin=1;
## post-warmup draws per chain=1000, total post-warmup draws=4000.
##
## mean se_mean sd 2.5% 50% 97.5% n_eff Rhat
## beta[1] -34.75 0.08 2.65 -40.08 -34.72 -29.67 1049 1
## beta[2] 9.12 0.01 0.42 8.28 9.11 9.96 1041 1
## sigma 6.62 0.01 0.21 6.22 6.62 7.06 1493 1
## lp__ -1208.81 0.04 1.23 -1212.00 -1208.48 -1207.38 1086 1
##
## Samples were drawn using NUTS(diag_e) at Wed Aug 14 07:13:00 2024.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).問題 24
\(\verb@model24.stan@\)は下記のようになる。
data {
int<lower = 0> N;
vector[N] y;
}
parameters {
ordered[2] mu;
real<lower=0> sigma[2];
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]));
}library(rstan)
N <- 100
y <- rnorm(100)
data_list <- list(N = N, y = y)
fit <- stan(file = "model24.stan", data = data_list)## Warning: There were 1 divergent transitions after warmup. See
## https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
## to find out why this is a problem and how to eliminate them.## Warning: Examine the pairs() plot to diagnose sampling problemsstan_dens(fit)
問題 25
\(\verb@generated quantities@\)を書き入れて、\(\verb@model_X@\)を完成させた。
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;
}y <- rnorm(100, 5, 2)
x <- rnorm(100, 7, 6)
fit_X <- stan("model_X.stan", data=list(N=N, y=y, x=x))## Warning in validityMethod(object): The following variables have undefined
## values: prob. Many subsequent functions will not work correctly.diff <- rstan::extract(fit_X)$diffplot(density(diff))
問題 26
テキストのStan Code(下記)を\(\verb@model26.stan@\)とおいた。
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);
}N <- 100
M <- 3
beta <- rnorm(M)
x <- matrix(rnorm(N*M), N, M)
y <- c()
for(i in 1:N){
p <- as.numeric(1/(1+exp(x[i,]%*%beta)))
if(runif(1,0,1) < p)
y[i] <- 1
else
y[i] <- 0
}
data_list <- list(N = N, M = M, x = x, y = y)
fit <- stan("model26.stan", data = data_list)stan_dens(fit)
第3章
問題 35
n <- 100
a <- NULL
sum <- 0
for(i in 1:n){
x <- rbinom(1, 1, 0.5)
sum <- sum + x
a[i] <- 1/i * sum
}
plot(1:n, a, type="l")
問題 37
CLT <- function (m, n, df) {
S <- NULL
mu <- df
sigma <- sqrt(2 * df)
for (j in 1:m) {
x <- rchisq(n, df)
S <- c(S, (sum(x)-n*mu) / sqrt(n) / sigma)
}
plot(density(S), xlab="Y_n", ylab="確率密度", main="自由度2のカイ2乗分布から得られた正規分布")
}
## 正規分布
m <- 300
n <- 100
df <- 2
CLT(m, n, df)
第4章
問題 48
mu <- 0
sigma <- 1
G <- function(y) 0.5*log(2*pi*(n+2)/(n+1))+(n+1)/(n+2)/2*(sigma^2+(mu-sum(y)/(n+1))**2)
T <- function(y) 0.5*log(2*pi*(n+2)/(n+1))+(n+1)/(n+2)/2*mean((y-sum(y)/(n+1))**2)
n <- 100
y <- rnorm(n)
G(y)## [1] 1.419894T(y)## [1] 1.390267第5章
問題 61
\(\verb@model13.stan@\)に\(\verb@generated quantities@\)を加えた、下記の\(\verb@model61.stan@\)を用いる。
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));
}library(rstan)
library(bayesplot)## This is bayesplot version 1.10.0## - Online documentation and vignettes at mc-stan.org/bayesplot## - bayesplot theme set to bayesplot::theme_default()## * Does _not_ affect other ggplot2 plots## * See ?bayesplot_theme_set for details on theme settingV_n <- function(log_likelihood) mean(colMeans(log_likelihood^2) - colMeans(log_likelihood)^2)
T_n <- function(log_likelihood) -mean(log(colMeans(exp(log_likelihood))))
WAIC <- function(log_likelihood) T_n(log_likelihood) + V_n(log_likelihood)
CV <- function(log_lik) mean(log(colMeans(1/exp(log_lik))))
generator <- function(n) {
data1 <- rnorm(n)-4 # N(-4,1) に従う乱数
data2 <- rnorm(n)+2 # N(2, 1) に従う乱数
data3 <- (runif(n) <= 0.6) # 確率 0.6 で 1 となる論理ベクトル
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
}
N <- 1000
y <- generator(N)
fit <- stan(file="model61.stan", data = list(N = N, y = y))
m2 <- extract(fit)
WAIC <- WAIC(m2$log_lik)
AIC <- 1/2*log(2*pi*exp(1)) + 1/2*log((N-1)/N*var(y)) + 3/2/NWAIC## [1] 2.071598AIC## [1] 2.551454問題 62
library(rstan)
library(MASS)
library(loo)## This is loo version 2.5.1## - Online documentation and vignettes at mc-stan.org/loo## - As of v2.0.0 loo defaults to 1 core but we recommend using as many as possible. Use the 'cores' argument or set options(mc.cores = NUM_CORES) for an entire session.## - Windows 10 users: loo may be very slow if 'mc.cores' is set in your .Rprofile file (see https://github.com/stan-dev/loo/issues/94).##
## 次のパッケージを付け加えます: 'loo'## 以下のオブジェクトは 'package:rstan' からマスクされています:
##
## loolibrary(bayesplot)
V_n <- function(log_likelihood) mean(colMeans(log_likelihood^2) - colMeans(log_likelihood)^2)
T_n <- function(log_likelihood) -mean(log(colMeans(exp(log_likelihood))))
WAIC <- function(log_likelihood) T_n(log_likelihood) + V_n(log_likelihood)
index <- c(1, 3, 5, 6, 8, 10, 11, 12, 13, 14)
lm <- formula(medv~.-medv,data=Boston)
df <- Boston[,index]
X <- model.matrix(lm,df)
N <- nrow(df)
K <- length(index)
Y <- df$medv
data_list <- list(N=N, M=K, y=Y, x=X)
fit <-stan(file="model11.stan", data=data_list, seed=1)
m1 <- extract(fit)
m2 <- extract_log_lik(fit)2*N*WAIC(m1$log_lik) ## [1] 3070.385waic(m2)## Warning:
## 11 (2.2%) p_waic estimates greater than 0.4. We recommend trying loo instead.##
## Computed from 4000 by 506 log-likelihood matrix
##
## Estimate SE
## elpd_waic -1535.2 33.4
## p_waic 18.0 3.6
## waic 3070.4 66.9
##
## 11 (2.2%) p_waic estimates greater than 0.4. We recommend trying loo instead.問題 64
Fn <- function(x) {
k <- sum(x)
log(gamma(n+2))-log(gamma(n-k+1))-log(gamma(k+1))
}
m <- 500
n <- 100
T <- NULL
for(j in 1:m){
x <- rbinom(n, 1, 0.25)
T <- c(T,Fn(x))
}
plot(density(T), main="自由エネルギー", col="red",
xlab="$F_n$", ylab="確率密度関数")
問題 66
library(rstan)
wbic <- function(log_likelihood) -mean(rowSums(log_likelihood))
bic <- function(x,y){
beta2 <- as.vector(solve(t(x)%*%x)%*%t(x)%*%y)
sigma2 <- sum((y-x%*%beta2)^2)/n
return(0.5*n*log(2*pi*exp(1)*sigma2)+0.5*(p+2)*log(n))
}
model15 <- stan_model("model15.stan")
library(MASS)
index <- c(1, 3, 5, 6, 8, 10, 11, 12, 13, 14)
lm <- formula(medv~.-medv, data=Boston)
df <- Boston[, index]
x <- model.matrix(lm, df)
x <- as.matrix(x)
p <- ncol(x)-1
n <- nrow(x)
y <- Boston[,14]
data_list <- list(N=n, M=p+1, y=y, x=x, beta=1/log(n))
fit <- sampling(model15, data = data_list, iter=3000)
mm <- rstan::extract(fit)wbic(mm$log_lik)## [1] 1557.152bic(x,y)## [1] 1554.374第6章
問題 68
a <- 0
b <- 0
x.min <- -1
x.max <- 5
f <- function(x) sqrt(max(x^3+a*x+b,0))
x.seq <- seq(x.min, x.max, 0.001)
y.seq <- NULL
for(x in x.seq) y.seq <- c(y.seq, f(x))
y.max <- max(y.seq)
plot(0,xlab = "x", ylab="y", xlim=c(x.min,x.max), ylim=c(-y.max,y.max), type="n",
main=paste("a=",a,", b=",b))
lines(x.seq, y.seq)
lines(x.seq, -y.seq)
abline(h=0)
abline(v=0)
a <- -3
b <- 2
x.min <- -3
x.max <- 3
x.seq <- seq(x.min, x.max,0.001)
y.seq <- NULL
for(x in x.seq) y.seq <- c(y.seq,f(x))
y.max <- max(y.seq)
plot(0, xlab = "x", ylab="y", xlim=c(x.min,x.max), ylim=c(-y.max,y.max), type="n",
main=paste("a=",a,", b=",b))
lines(x.seq,y.seq)
lines(x.seq,-y.seq)
abline(h=0)
abline(v=0)
第7章
問題 77
delta <- function(a,j)lines(c(-a, a, a, -a, -a),c(0, 0, 1/a, 1/a, 0), col=j)
a.seq <- c(10^(-3))
plot(0, xlim=c(-0.05, 0.05), xlab="x", ylab="fa(x)", ylim=c(0,1005), type="n",
main="Uniform Distribution")
for(a in a.seq) delta(a,a*100+1)
問題 86
まず、本文にある例76のプログラム(下記)を実行する(1-2時間程度要する)。
library(rstan)
library(MASS)
V_n <- function(log_likelihood) mean(colMeans(log_likelihood^2) - colMeans(log_likelihood)^2)
T_n <- function(log_likelihood) -mean(log(colMeans(exp(log_likelihood))))
WAIC <- function(log_likelihood, beta) T_n(log_likelihood) + beta * V_n(log_likelihood)
CV <- function(log_likelihood, beta)
- mean(log(colMeans(exp((1-beta) * log_likelihood)))
- log(colMeans(exp(-beta * log_likelihood))))
data(Boston)
index <- c(1, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14)
lm <- formula(medv ~ . -medv, data=Boston)
df <- Boston[, index]
X <- model.matrix(lm, df)
N <- nrow(df)
K <- length(index)
Y <- df$medv
waic_values <- NULL
cv_values <- NULL
beta.seq <- seq(0.1, 1.6, 0.1)
for(beta in beta.seq){
data_list <- list(N=N, M=K, y=Y, x=X, beta=beta)
fit <- stan(file="model15.stan", data=data_list, seed=1)
m2 <- extract(fit)
waic_values <- c(waic_values, N*WAIC(m2$log_lik, beta))
cv_values <- c(cv_values, N*CV(m2$log_lik, beta))
}##
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## Chain 4:その上で、下記のプログラムを実行する。
plot(beta.seq, waic_values,xlab="beta", ylab="WAIC/CV", ylim=c(1520,1560),col="red", type="l")
lines(beta.seq, cv_values, col="blue", type="l")
legend("topright", legend=c("WAIC","CV"), col=c("red","blue"), lwd=1)
title("WAICとCVの値の変化")
第8章
問題 98
下記のコードを\(\verb@model98.stan@\)とする。
data {
int<lower=1> K; // number of mixture components
int<lower=1> N; // number of data points
vector[2] y[N]; // observations
real beta;
}
parameters {
simplex[K] theta;
vector[2] mu[K];
real<lower=0> sigma; // Add a new parameter for the common standard deviation
}
transformed parameters{
vector[K] log_theta = log(theta); // cache log calculation
}
model {
mu ~ multi_normal(rep_vector(0.0, 2), diag_matrix(rep_vector(sigma, 2))); // Use sigma for the variance instead of 1.0
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(rep_vector(sigma, 2))); // Use sigma for the variance instead of 1.0
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(rep_vector(sigma,2)));
log_lik[n] = log_sum_exp(lps);
}
}下記の実行は、数時間かかる。
wbic <- function(log_likelihood) - mean(rowSums(log_likelihood))
library(rstan)
b.seq <- c(1, 10, 100, 250)
K.seq <- c(1, 2, 3, 4)
x <- list()
n <- 100
for(i in 1:n) x[[i]] <- c(rnorm(1, -2, 1), rnorm(1, -2, 1))
for(i in (n+1):(2*n)) x[[i]] <- c(rnorm(1, 2, 1), rnorm(1, 2, 1))
for(i in (2*n+1):(3*n)) x[[i]] <- c(rnorm(1, 0, 1), rnorm(1, 0, 1))
WBIC <- NULL
for(b in b.seq)for(k in K.seq){
data_list <- list(K = k, N = length(x), y=x, beta=b/log(n))
fit <- stan(file = "model98.stan", data=data_list, warmup=2500, seed=1, iter=5000)
mm <- rstan::extract(fit)
wbic.1 <- wbic(mm$log_lik)
WBIC <- c(WBIC, wbic.1)
}##
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## Chain 4:## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess##
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## Chain 4:## Warning: The largest R-hat is 1.53, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#r-hat
## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#tail-ess##
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#r-hat## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#tail-ess##
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
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## Chain 4:## Warning: The largest R-hat is 1.41, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#r-hat## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#tail-ess##
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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## Chain 4:## Warning: The largest R-hat is 1.53, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#r-hat## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#tail-ess##
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## Chain 4:## Warning: The largest R-hat is 1.53, indicating chains have not mixed.
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## https://mc-stan.org/misc/warnings.html#r-hat## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
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## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#tail-ess##
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## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#r-hat## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#tail-ess##
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## Chain 4:## Warning: The largest R-hat is 1.53, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#r-hat## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#tail-ess##
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## Chain 4:## Warning: The largest R-hat is 2.31, indicating chains have not mixed.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#r-hat## Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#bulk-ess## Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
## Running the chains for more iterations may help. See
## https://mc-stan.org/misc/warnings.html#tail-essWBIC## [1] 1239.456 1133.445 1128.979 1128.260 1233.269 1120.001 1103.050 1102.836
## [9] 1232.646 1118.727 1101.084 1098.820 1232.604 1118.644 1100.958 1098.646