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We announce two results on the problem of estimating the order of a Markov chain from observation of a sample path. First is that the Bayesian Information Criterion (BIC) leads to an almost surely consistent estimator. Second is that the Bayesian minimum description length estimator, of which the BIC estimator is an approximation, fails to be consistent for the uniformly distributed i.i.d. process. A key tool is a strong ratio-typicality result for empirical $k$-block distributions. Complete proofs are given in the authors' article to appear in The Annals of Statistics.

Copyright 1999 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 05 (1999), pp. 123-127
• Publisher Identifier: S 1079-6762(99)00070-0
• 1991 Mathematics Subject Classification. Primary 62F12, 62M05; Secondary 62F13, 60J10
• Key words and phrases. Bayesian information criterion, order estimation, ratio-typicality, Markov chains
• Received by the editors February 25, 1999
• Posted on October 19, 1999
• Communicated by Yitzhak Katznelson
• Comments (When Available)

Imre Csiszár

A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, 1364 Budapest, Hungary

E-mail address: csiszar@math-inst.hu

Paul C. Shields

Mathematics Department, The University of Toledo, Toledo, OH 43606

E-mail address: paul.shields@utoledo.edu

First author supported in part by a joint NSF-Hungarian Academy grant 92
Second author supported in part by a joint NSF-Hungarian Academy grant INT-9515485