Lecture 22 : Error exponents in hypothesis testing , GLRT

2. Let Q(X,Y ) be some joint distribution. Suppose (X, Y ) = (X1, Y1), . . . , (Xn, Yn) iid ∼ Q0(X,Y ) = Q(X)Q(Y ), where Q(X) and Q(Y ) are the marginal distributions corresponding to the joint distribution Q(X,Y ). In other words, Xi and Yi are independent. We are interested in the probability that (X, Y ) appear to be depdendent or jointly distributed according to Q(X,Y ). In last lecture, we saw that using Sanov’s theorem we get: Q0 (E) ≈ 2−nD(Q(X,Y )||Q(X)Q(Y )) = 2−nI(X,Y )