Question

Bob claims to have ESP. To test this claim, you propose the following experiment. You will select one card from 4 large cards with different geometric figures and Bob will try to identify it. Let p denote the probability that Bob is corect in identifying the figure for a single card. You believe that Bob has no ESP ability, so you believe that p.25. However, there is a small chance that p is either larger or smaller than .25. After some thought, you place a beta (3.5,6.5) prior on p. Suppose that the experiment is repeated 10 times and Bob is correct 6 times and incorrect 4 times. a) Find the distribution of the posterior probability of p. b) Graph the prior, likelihood, and posterior distribution of p on the same graph and comment on how the data influences the posterior distribution.

Answer 1

「(10)-- 「(3.5)「(65)p2.5(1-p)5.5: 0 < p < 1

The prior likelihood is

10p*(1 -p)

a) The posterior distribution is

(10) p*.5(1 - p)5s 2.5 9.5 8.5 X) X

Thus the posterior distribution is Beta.

(P|X = 6) ~ Beta(9.5, 10.5)

b) The prior (red), posterior(blue) and likelihood (green) are plotted on the same graph below.

Prior, Posterior Beta Distributions & likelihood 0.4 0.0 0.6 1.0 0.2 0.8

The mode of the posterior distribution shifts to the right indicating .

p > 0.25

R code below.

plot(1:1)
dev.new()
curve(dbeta(x,3.5,6.5), xlim=c(0,1),ylim = c(0,4), main="Prior, Posterior Beta Distributions & likelihood",
lwd=2, col = "red", xlab="p", ylab = "Density")
curve(dbeta(x, 9.5,10.5), lwd=2, col = "blue", add=TRUE)
curve(choose(10,6)*x^6*(1-x)^4, lwd=2, col = "green", add=TRUE)

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