. 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 ; : : : ; x n , as that value k = k that minimizes the sum of the negative logarithm of the k-th order maximum likelihood and the penalty term jAj k (jAj\Gamma1) 2 log n: We show that k equals the correct order of the chain, eventually almost surely as n ! 1, 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 an approximation, fails to be consistent for the uniformly distributed i.i.d. process. AMS 1991 subject classification: Primary 62F12, 62M05; Secondary 62F13, 60J10 Key words and phrases: Bayesian Information Criterion, order estimation, ratiotypicality, Markov chains. 1 Supported in part by a joint N...

We consider the sequence comparison problem, also known as "hidden" pattern problem, where one searches for a given subsequence in a text (rather than a string understood as a sequence of consecutive symbols). A characteristic parameter is...

. 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. 1. Introduction Let M k denote the class of Markov chains of order at most k, with values drawn from a finite set A, and let M = S 1 k=0 M k . An important problem is to estimate the order of a Markov chain from observation of a finite sample path. A popular method is the so-called Bayesian Information Criterion (BIC), first introduced by Schwarz, [12], which gives the estimator defined by k BIC = k BIC (x n 1 ) ...