Let Θ1 and Θ2 be some unobserved Bernoulli random variables and let X be an observation. Conditional on X=x, the posterior joint PMF of Θ1 and Θ2 is given by

Question

pΘ1,Θ2∣X(θ1,θ2∣x)=
0.26, if θ1=0,θ2=0,
0.26, if θ1=0,θ2=1,
0.21, if θ1=1,θ2=0,
0.27, if θ1=1,θ2=1,
0, otherwise.

We can view this as a hypothesis testing problem where we choose between four alternative hypotheses: the four possible values of (Θ1,Θ2).

a) What is the estimate of (Θ1,Θ2) provided by the MAP rule?

b) Once you calculate the estimate (θ^1,θ^2) of (Θ1,Θ2), you may report the first component, θ^1, as your estimate of Θ1. With this procedure, your estimate of Θ1 will be

c) What is the probability that Θ1 is estimated incorrectly (the probability of error) when you use the procedure in part (b)?

d) What is the MAP estimate of Θ1 based on X, that is, the one that maximizes pΘ1|X(θ1∣x)?

e) The moral of this example is that an estimate of Θ1 obtained by identifying the maximum of the joint PMF of all unknown random variables is

the MAP estimate of Θ1.

Answer 1

a) (1,1)
b) 1
c) 0.52
d) 1
e) is always the same as

Answer 2

a) (1,1)
b) 1
c) 0.52
d) 0
e) can be different from