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PHASE TRANSITIONS FOR THE UNIFORM DISTRIBUTION IN THE PATTERN MAXIMUM LIKELIHOOD PROBLEM AND ITS BETHE APPROXIMATION

Chan, Chun Lam and Fernandes, Winston and Kashyap, Navin and Krishnapur, Manjunath (2017) PHASE TRANSITIONS FOR THE UNIFORM DISTRIBUTION IN THE PATTERN MAXIMUM LIKELIHOOD PROBLEM AND ITS BETHE APPROXIMATION. In: SIAM JOURNAL ON DISCRETE MATHEMATICS, 31 (1). pp. 597-631.

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Official URL: http://dx.doi.org/10.1137/15M1028819

Abstract

The pattern maximum likelihood (PML) estimate, introduced by Orlitsky et al., is an estimate of the multiset of probabilities in an unknown probability distribution p, the estimate being obtained from n independent and identically distributed samples drawn from p. The PML estimate involves solving a difficult optimization problem over the set of all probability mass functions of finite support. In this paper, we describe an interesting phase transition phenomenon in the PML estimate: at a certain sharp threshold, the uniform distribution goes from being a local maximum to being a local minimum for the optimization problem in the estimate. We go on to consider the question of whether a similar phase transition phenomenon also exists in the Bethe approximation of the PML estimate, the latter being an approximation method with origins in statistical physics. We show that the answer to this question is a qualified ``yes.'' Our analysis involves the computation of the mean and variance of the (i,j)th entry, (i,j)th in a random k x k nonnegative integer matrix A with row and column sums all equal to M, drawn according to a distribution that assigns to A a probability proportional to Pi(i,j) (M-a(i,j))!/a(i),(j)!.

Item Type: Journal Article
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Additional Information: Copy right for this article belongs to the SIAM PUBLICATIONS, 3600 UNIV CITY SCIENCE CENTER, PHILADELPHIA, PA 19104-2688 USA
Department/Centre: Division of Electrical Sciences > Electrical Communication Engineering
Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 20 May 2017 05:11
Last Modified: 20 May 2017 05:11
URI: http://eprints.iisc.ernet.in/id/eprint/56885

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