show the special case for MAP-Based

MAP (Maximum a Posteriori) estimation is a method of estimating an unknown parameter θ based on observed data x, by maximizing the posterior probability P(θ|x) of θ given x. 

In the special case of binary hypothesis testing, where we are testing between two hypotheses H0 and H1, MAP estimation reduces to a simple decision rule. 

Suppose we are given prior probabilities P(H0) and P(H1), and we observe some data x. Let p(x|H0) and p(x|H1) be the likelihood functions for H0 and H1, respectively. Then, the posterior probabilities of H0 and H1 can be computed using Bayes' rule as:

P(H0|x) = (p(x|H0) * P(H0)) / (p(x|H0) * P(H0) + p(x|H1) * P(H1))

P(H1|x) = (p(x|H1) * P(H1)) / (p(x|H0) * P(H0) + p(x|H1) * P(H1))

The MAP estimate of the true hypothesis is then given by:

- Reject H0 and accept H1 if P(H1|x) > P(H0|x) 

- Reject H1 and accept H0 if P(H0|x) > P(H1|x)

In other words, we choose the hypothesis that has the highest posterior probability given the observed data. This decision rule can be expressed in terms of a decision boundary, which separates the two hypotheses and is defined by the equation P(H0|x) = P(H1|x). 

If the prior probabilities P(H0) and P(H1) are equal, then the decision boundary reduces to the likelihood ratio test, where we reject H0 and accept H1 if the likelihood ratio p(x|H1)/p(x|H0) is greater than a threshold value. 

The books [BT] Introduction to Probability and [DS] Probability and Statistics provide more detailed discussions on the topic of MAP estimation and hypothesis testing, including how to choose the appropriate prior probabilities and likelihood functions for a given problem.