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 depth3 circuits.
We also consider circuits that are dlocal, in the sense that each output bit depends on at most d input bits. We give lineartime algorithms for deciding whether a 2local sampler's joint distribution is fully independent, and whether it is exchangeable.
We also show that for general circuits, certain approximation versions of the problems of deciding full independence and exchangeability are SZKcomplete.
We also introduce a boundederror version of C_{=P}, which we call BC_{=P}, and we investigate its structural properties.
BibTeX  Entry
@InProceedings{watson:LIPIcs:2014:4496,
author = {Thomas Watson},
title = {{The Complexity of Deciding Statistical Properties of Samplable Distributions}},
booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
pages = {663674},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897651},
ISSN = {18688969},
year = {2014},
volume = {25},
editor = {Ernst W. Mayr and Natacha Portier},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2014/4496},
URN = {urn:nbn:de:0030drops44960},
doi = {10.4230/LIPIcs.STACS.2014.663},
annote = {Keywords: Complexity, statistical properties, samplable distributions}
}
Keywords: 

Complexity, statistical properties, samplable distributions 
Collection: 

31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014) 
Issue Date: 

2014 
Date of publication: 

05.03.2014 