Information-Theoretic Demensionality Reduction for Poisson Models: Supplementary Material

Proof of Theorem 1. We first establish the following Lemma. Lemma 1. Consider random variables X ∈ R and Y ∈ R. Let f Y |X be the Radon-Nikodym derivatives of probability measure P θ Y |X with respect to arbitrary measures QY provided that P θ Y |X QY . θ ∈ R is a parameter. f Y is the Radon-Nikodym derivatives of probability measure P θ Y with respect to QY provided that P θ Y QY . Note that in the continuous or discrete case, f Y |X and f θ Y are simply probability density or mass functions with QY chosen to be the Lebesgue measure or the counting measure, respectively. We have