3.6 A Bayesian Test of the Fairness of Coin (pp. 52-56).txt

#A Bayesian Test of the Fairness of Coin
#(Albert pp. 52-56)

library(LearnBayes)

#probability of 5 or fewer heads in n=20 and prob of heads=.5 (fair)
#non-bayesian, p-value approach
#we are testing if coin is FAIR

pbinom(5, 20, 0.5)

#p-value is 2*min{P(Y<=y), P(Y>=y)}
#or p-value =2*pbinom()

#the bayesian approach follows (see p. 53 for the math)

#assume beta(10, 10) prioir for p when the coin is not fair
#we observe y=5 heads in n=20 tosses
#posterior probability is in lambda

n=20
y=5
a=10
p=.5
m1=dbinom(y, n, p)* dbeta(p, a, a)/dbeta(p, a+y, a+n-y)
lambda=dbinom(y, n, p)/(dbinom(y, n, p) + m1)
lambda 

#here posterior probability of the hypothesis of fairness is .28
#NOT .042, leading to reversal of statistical decision!

#Sensitivity of this posterior calculation to the (arbitrary) 
#choice of parameter a=10

prob.fair=function(log.a)
{
a=exp(log.a)
m2=dbinom(y, n, p)* dbeta(p, a, a)/dbeta(p, a+y, a+n-y)
dbinom(y, n, p)/(dbinom(y, n, p) + m2)
}

#Using the curve function, we graph the posterior probability for values of log a

n=20; y=5; p=.5
curve(prob.fair(x), from=-4, to=5, xlab="log a",
ylab="Prob(coin is fair)", lwd=2)

#We see from this graph that the probability of fairness appears to be greater
#than .2 for all choices of a.

#Note that Bayesian calculation was 
#based solely on the likelihood based on the event "exactly 5 heads" 
#NOT 5 heads or fewer as the p-value approach did

#Bayesian computation for 5 heads or fewer (under binomial model p=.5)
#see pp.54-55 for the math - cummulative probability

n=20
y=5
a=10
p=.5
m2=0

for (k in 0:y)

#code of P(sub1)(Y,=5)  -> the predictive probability of (this event) under
#the alternative model with beta(10, 10)

m2=m2+dbinom(k, n, p)*dbeta(p, a, a)/dbeta(p, a+k, a+n-k)

#see m1 in p.54 (predictive density)

#pbinom computes cummulative probability (here of 5 heads under the binomial mdel p=.5)

lambda=pbinom(y, n, p)/(pbinom(y, n, p)+m2)
lambda