Comparing Distributions: Invariance Principles & Mismatched Guesswork

Abstract

We study two different ways of measuring the similarity between distributions over a finite alphabet. The first is an invariance principle which gives a quantitative bound on the expected difference between general functions of two finite sequences of random variables. This result is one way to generalize the foundational basic invariance principle to a particular multivariate setting. The second framework is based on guesswork, which is one way to measure the randomness of a distribution, similar to but notably distinct from the Shannon entropy. Given a bound on the total variation distance between two finite distributions, we give a bound on the difference in guesswork between those distributions and study the geometrical properties of the problem in the non-asymptotic setting.