Suppose that we have an observable X that can take only two values, X1 and X2, for the situation in Exercise 11.5.7. The distribution of X depends on the states of nature and is as follows:
|
|
θ1
|
θ2
|
θ3
|
|
X1
|
0.1
|
0.5
|
0.6
|
|
X2
|
0.9
|
0.5
|
0.4
|
That is, P (X = x1|θ1) = 0.1 or P (X = x2|θ3) = 0.4, and so forth.
Suppose you observe X1; what is the updated prior? What is the Bayes decision?