# poll 10 people, get 6 yes answers
# with a uniform prior, the posterior is Beta(7,5)
x=seq(from=0,to=1,by=.001)
plot(x,dbeta(x,7,5),type="l")      # posterior density plot
7/12                               # posterior mean
# with a noninformative "Beta(0,0)" prior, the posterior is Beta(6,4)
lines(x,dbeta(x,6,4),lty=2)
6/10
# Jeffreys prior is Beta(0.5,0.5), and gives a Beta(6.5,4.5) posterior
lines(x,dbeta(x,6.5,4.5),lty=3)
6.5/11
# what if we started with a Beta(1,3) prior?  posterior is Beta(7,7)
lines(x,dbeta(x,7,7),lty=4)
7/14
# what about a Beta(3,7) prior?  posterior is Beta(9,11)
lines(x,dbeta(x,9,11),lty=5)
9/20

# now suppose we had polled 100 people and got 60 yes answers
#  what are the resulting posterior densities and posterior means?
plot(x,dbeta(x,61,41),type="l")   # Beta(1,1) prior
61/102
lines(x,dbeta(x,60,40),lty=2)     # Beta(0,0)
60/100
lines(x,dbeta(x,60.5,40.5),lty=3) # Beta(0.5,0.5)
60.5/101
lines(x,dbeta(x,61,43),lty=4)       # Beta(1,3)
61/104
lines(x,dbeta(x,63,47),lty=5)      # Beta(3,7)
63/110

# for the cookies example, we counted 855 chips in 39 cookies for an
#  average of 21.92 chips per cookie
# class prior was Gamma(10,1)
x=seq(from=.5,to=30,by=.1)
x2=seq(from=18,to=26,by=.01)
plot(x,dgamma(x,10,1),type="l",ylim=c(0,.55))
likelihood=numeric(length(x2))
for(i in 1:length(x2)) likelihood[i]=prod(dpois(cookies,x2[i]))
lines(x2,likelihood*3.5e48)
lines(x2,dgamma(x2,10+855,1+39),lty=2)

# compare posteriors from different priors
plot(x2,dgamma(x2,10+855,1+39),type="l",ylim=c(0,1)) # class Gamma(10,1) prior
lines(x2,dgamma(x2,0+855,0+39),lty=2)       # noninformative "Gamma(0,0)" prior
lines(x2,dgamma(x2,100+855,10+39),lty=3) # more informative class Gamma(100,10)
lines(x2,dgamma(x2,220+855,10+39),lty=4) # more informative Gamma(220,10)
lines(x2,dgamma(x2,2200+855,100+39),lty=5) #way more informative Gamma(2200,100)