{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7543","name":"The Complexity of Deciding Statistical Properties of Samplable Distributions","abstract":"We consider the problems of deciding whether the joint distribution sampled by a given circuit satisfies certain statistical properties such as being i.i.d., being exchangeable, being pairwise independent, having two coordinates with identical marginals, having two uncorrelated coordinates, and many other variants. We give a proof that simultaneously shows all these problems are C_{=P}-complete, by showing that the following promise problem (which is a restriction of all the above problems) is C_{=P}-complete: Given a circuit, distinguish the case where the output distribution is uniform and the case where every pair of coordinates is neither uncorrelated nor identically distributed. This completeness result holds even for samplers that are depth-3 circuits.\r\n\r\nWe also consider circuits that are d-local, in the sense that each output bit depends on at most d input bits. We give linear-time algorithms for deciding whether a 2-local sampler's joint distribution is fully independent, and whether it is exchangeable.\r\n\r\nWe also show that for general circuits, certain approximation versions of the problems of deciding full independence and exchangeability are SZK-complete.\r\n\r\nWe also introduce a bounded-error version of C_{=P}, which we call BC_{=P}, and we investigate its structural properties.","keywords":["Complexity","statistical properties","samplable distributions"],"author":{"@type":"Person","name":"Watson, Thomas","givenName":"Thomas","familyName":"Watson"},"position":57,"pageStart":663,"pageEnd":674,"dateCreated":"2014-03-05","datePublished":"2014-03-05","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Watson, Thomas","givenName":"Thomas","familyName":"Watson"},"copyrightYear":"2014","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2014.663","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6228","volumeNumber":25,"name":"31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)","dateCreated":"2014-03-05","datePublished":"2014-03-05","editor":[{"@type":"Person","name":"Mayr, Ernst W.","givenName":"Ernst W.","familyName":"Mayr"},{"@type":"Person","name":"Portier, Natacha","givenName":"Natacha","familyName":"Portier"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7543","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6228"}}}