Tenability and Computability of Generalized Polya Urns

by Scott R Konzem

Institution: The George Washington University
Year: 2017
Keywords: Statistics; Computer science
Posted: 02/01/2018
Record ID: 2154934
Full text PDF: http://pqdtopen.proquest.com/#viewpdf?dispub=10263413


Urn models have a storied part in the history of probability and have been studied extensively over the past century for their wide range of applications. We analyze a generalized class of urn models introduced in the past decade, the so-called "multiset" class, in which more than one ball is sampled at a time. We investigate sufficient conditions for a multiset urn process to be tenable, meaning the process can continue indefinitely without getting stuck. We fully characterize the "strongly tenable" class of Plya urn schemes, which is tenable under any starting conditions that allow the process to begin. We find several "weakly tenable" classes of Plya urn schemes that are tenable only under restricted starting conditions. We enumerate the size of some of these tenable classes using combinatorics, probabilistically analyze them, and provide an algorithm to assess the tenability of an arbitrary urn scheme using breadth-first search. We further analyze the computational complexity of the tenability problem itself. By showing how to encode the Boolean satisfiability problem within a Plya urn scheme, we find that the problem of determining whether a multiset urn scheme is untenable is in the complexity class NP-hard, and this places constraints on the kinds of tenability theorems we can hope to find. Finally, we analyze a generalized fault tolerant urn model that can take action to avoid getting stuck, and by showing that this model is Turing-equivalent, we show that the tenability problem for this model is undecidable.