7.2.1 For the model discussed in Example 7.1.1, Let x_1, x_n to be independent Bernuolli variables with unknown parameter theta. We assume a uniform(0,1) prior over theta. If the data set turns out to be that all x_i's are 1, then what is the posterior distribution of theta? What is the posterior mean of theta*theta?

EXAMPLE 7.1.1 Bernoulli Modell Suppose that we observe a sample (x1, . . . , x") from the Bernoulli(9) distribution with 6 e [0, 1] unknown. For the prior, we take it to be equal to a Beta(a, 3) density (see Problem 2.4.16). Then the posterior of 9 is proportional to the likelihood n \"0;.- (1 _9)1x,- = ai (1 _ ariaf) i=1 times the prior 3-1 (a. m 6\"\" (I cop1 . This product is proportional to afl (1 _ 9)(lf)+l . We recognize this as the unnormalized density of a Beta(n.f + a, n (l i) + l?) dis- tribution. So in this example, we did not need to compute m(x1 , . . . , x") to obtain the posterior. As a specic case, suppose that we observe m? = 10 in a sample of n = 40 and a = = 1, i.e., we have a uniform prior one. Then the posterior oft9 is given by the Beta(11, 31) distribution. We plot the posterior density in Figure 7.1.3 as well as the PI'IOI'. 0.0 0.1 0.2 0.3 0.4 05 0.6 0.7 0.8 0.9 1.0 theta Figure 7.1.3: Prior (dashed line) and posterior densities (solid line) in Example 7.1.1. The spread of the posterior distribution gives us some idea of the precision of any probability statements we make about 9. Note how much information the data have added, as reected in the graphs of the prior and posterior densities