Testing Product Distributions: A Closer Look
This work provides a more comprehensive understanding of the complexity of testing product distributions for researchers in theoretical computer science and statistics, extending prior non-tolerant testing results.
This paper investigates the identity and closeness testing of n-dimensional product distributions, extending prior work by exploring tolerant testing with various distance measures and arbitrary alphabets. The study provides a detailed understanding of how sample complexity changes with different distance measures for product distributions.
We study the problems of identity and closeness testing of $n$-dimensional product distributions. Prior works by Canonne, Diakonikolas, Kane and Stewart (COLT 2017) and Daskalakis and Pan (COLT 2017) have established tight sample complexity bounds for non-tolerant testing over a binary alphabet: given two product distributions $P$ and $Q$ over a binary alphabet, distinguish between the cases $P = Q$ and $d_{\mathrm{TV}}(P, Q) > ε$. We build on this prior work to give a more comprehensive map of the complexity of testing of product distributions by investigating tolerant testing with respect to several natural distance measures and over an arbitrary alphabet. Our study gives a fine-grained understanding of how the sample complexity of tolerant testing varies with the distance measures for product distributions. In addition, we also extend one of our upper bounds on product distributions to bounded-degree Bayes nets.