申論題

4. (Total-18%) Consider a binary symmetric channel with input X, output Y, and transition probability a. More specifically, as shown in the figure below, the input and output of the channel may be "0" or "1", P(Y=0|X=0)= P(Y = 1|X = 1) = 1 - a and P(Y = 1|X = 0) = P(Y=0|X=0) = a. The prior probability is P(X = 0) = p.

(c) (5%) For the case of p ＜ a, please give the maximum a posteriori probability (MAP) decision rule to decide which X is sent based on the output Y. (Hint: In this problem, the output Y of the channel can take just two values "0" or "1". Hence, it is sufficient to find the decision result for these two output possibilities.)