Bounds on estimated Markov orders of individual sequences

Abstract

We study the maximal values estimated by commonly used Markov model order estimators on individual sequences. We start with penalized maximum likelihood (PML) estimators with cost functions of the form - log P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> (x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) + f (n)α <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> , where P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> (x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) is the ML probability of the input sequence x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> under a Markov model of order k, a is the size of the input alphabet, and f(n) is an increasing (penalization) function of n (the popular BIC estimator corresponds to f(n) = α - 1/2 log n). Comparison with a memoryless model yields a known upper bound k(n) on the maximum order that x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> can estimate. We show that, under mild conditions on f that are satisfied by commonly used penalization functions, this simple bound is not far from tight, in the following sense: for sufficiently large n, and any k<;k̅(n), there are sequences x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> that estimate order k; moreover, for all but a vanishing fraction of the values of n such that k = k̅(n), there are sequences x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> that estimate order k. We also study KT-based MDL Markov order estimators, and show that in this case, there are sequences x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> that estimate order n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2-ϵ</sup> , which is much larger than the maximum log n/log α(l + o(1)) attainable by BIC, or the order o(log n) required for consistency of the KT estimator. In fact, for these sequences, limiting the allowed estimated order might incur in a significant asymptotic penalty in description length. All the results are constructive, and in each case we exhibit explicit sequences that attain the claimed estimated orders.