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# Testing Product Distributions: A Closer Look

*Proceedings of the 32nd International Conference on Algorithmic Learning Theory*, PMLR 132:367-396, 2021.

#### Abstract

We study the problems of {\em identity} and {\em closeness testing} of $n$-dimensional product distributions. Prior works of Canonne, Diakonikolas, Kane and Stewart (COLT 2017) and Daskalakis and Pan (COLT 2017) have established tight sample complexity bounds for {\em 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 $d_{\mathrm{TV}}(P,Q)>\epsilon$. We build on this prior work to give a more comprehensive map of the complexity of testing of product distributions by investigating {\em 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.