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December2000 The consistency of the BIC Markov order estimator
Imre Csiszár, Paul C. Shields
Ann. Statist. 28(6): 1601-1619 (December2000). DOI: 10.1214/aos/1015957472


The Bayesian Information Criterion (BIC) estimates the order of a Markov chain (with finite alphabet $A$) from observation of a sample path $x_1, x_2,\dots, x_n$, as that value $k = \hat{k}$ that minimizes the sum of the negative logarithm of the $k$th order maximum likelihood and the penalty term $\frac{|A|^k(|A|-1)}{2}\log n$. We show that $\hat{k}$ equals the correct order of the chain, eventually almost surely as $n \rightarrow \infty$, thereby strengthening earlier consistency results that assumed an apriori bound on the order. A key tool is a strong ratio-typicality result for Markov sample paths.We also show that the Bayesian estimator or minimum description length estimator, of which the BIC estimator is regarded as an approximation, fails to be consistent for the uniformly distributed i.i.d. process.


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Imre Csiszár. Paul C. Shields. "The consistency of the BIC Markov order estimator." Ann. Statist. 28 (6) 1601 - 1619, December2000.


Published: December2000
First available in Project Euclid: 12 March 2002

zbMATH: 1105.62311
MathSciNet: MR1835033
Digital Object Identifier: 10.1214/aos/1015957472

Primary: 62F12 , 62M05
Secondary: 60J10 , 62F13

Keywords: Bayesian Information Criterion , Markov chains , order estimation , ratio-typicality

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.28 • No. 6 • December2000
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