EE 226 a - Summary of Lecture 10 Hypothesis Testing

So far we have considered the Bayesian detection problem of minimizing E(c(X, g(Y))) when both pX and fY|X are known. Recall that, when X ∈ {0, 1} the optimal decision has the form g(Y) = h(Λ(Y)) where Λ(y) = fY|X [y|1]/fY|X [y|0] is the likelihood ratio. In particular, if Λ(y) = f(k(y)), then k(Y) is a sufficient statistic for detecting X given Y. In this section we explore the detection problem when pX , the prior, is unknown. We start with the case when X ∈ {0, 1}.