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# Bayesian model selection and statistical modeling 
# Chapman & Hall/CRC Taylor and Francis Group 
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# Chapter3: Section3.3.3
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# Bernoulli distribution with a uniform prior
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# Author : Tomohiro Ando
#
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# Data generation

lambda <- 5

n <- 5

x <- rpois(n,lambda)


# Setting hyperparameters

alpha <- beta <- 0.1


# Posterior inference

alphahat <- alpha+sum(x)

betahat  <- beta+n

mode <- (sum(x)+alpha-1)/(beta+n)



# Exact posterior density

l.range <- seq(2,10,len=100)

Exactposterior <- dgamma(l.range, shape=alphahat, scale=1/betahat)


# Approximated posterior density

s <- (beta+n)^2/(sum(x)+alpha-1)

Approximatedposterior <- dnorm(l.range,mean=mode,sd=sqrt(1/s))


# Plot

plot(l.range, Exactposterior, type="l",xlab="",ylab="") 
lines(l.range, Approximatedposterior,lwd=3,lty=2)