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# Bayesian model selection and statistical modeling 
# Chapman & Hall/CRC Taylor and Francis Group 
#
# Chapter5: Section5.5.1
#
# Evaluation of the approximation error of BIC
#
# Author : Tomohiro Ando
#
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# Settings

n <- 50   #Sample size
K <- 10^5 # Number of trials

Errors <- rep(0,len=K)

# Setting hyperparameters

alpha <- 2
beta  <- 4


for(k in 1:K){

# Data generation

x <- rbinom(n,size=1,p=0.2)


# Posterior inference

alphahat <- alpha+sum(x)
betahat  <- beta+n-sum(x)


# Marginal likelihood

Marginallikelihood <- choose(n,sum(x))*gamma(alpha+beta)/(gamma(alpha)*gamma(beta))*gamma(alphahat)*gamma(betahat)/(gamma(n+alpha+beta))


#BIC

BIC <- log( dbinom(sum(x),size=n,prob=mean(x)) ) -log(n)/2

Errors[k] <- abs( Marginallikelihood-exp( BIC ) )

}

print(mean(Errors))
print(sd(Errors))