Consistent estimation of the basic neighborhood of Markov random fields

For Markov random fields on $\mathbb{Z}^d$ with finite state space, we address the statistical estimation of the basic neighborhood, the smallest region that determines the conditional distribution at a site on the condition that the values at all other sites are given. A modification of the Bayesian Information Criterion, replacing likelihood by pseudo-likelihood, is proved to provide strongly consistent estimation from observing a realization of the field on increasing finite regions: the estimated basic neighborhood equals the true one eventually almost surely, not assuming any prior bound on the size of the latter. Stationarity of the Markov field is not required, and phase transition does not affect the results.Comment: Published at http://dx.doi.org/10.1214/009053605000000912 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Context Tree Selection: A Unifying View

The present paper investigates non-asymptotic properties of two popular procedures of context tree (or Variable Length Markov Chains) estimation: Rissanen's algorithm Context and the Penalized Maximum Likelihood criterion. First showing how they are related, we prove finite horizon bounds for the probability of over- and under-estimation. Concerning overestimation, no boundedness or loss-of-memory conditions are required: the proof relies on new deviation inequalities for empirical probabilities of independent interest. The underestimation properties rely on loss-of-memory and separation conditions of the process. These results improve and generalize the bounds obtained previously. Context tree models have been introduced by Rissanen as a parsimonious generalization of Markov models. Since then, they have been widely used in applied probability and statistics

Maximum entropy estimation of transition probabilities of reversible Markov chains

In this paper, we develop a general theory for the estimation of the transition probabilities of reversible Markov chains using the maximum entropy principle. A broad range of physical models can be studied within this approach. We use one-dimensional classical spin systems to illustrate the theoretical ideas. The examples studied in this paper are: the Ising model, the Potts model and the Blume-Emery-Griffiths model

Divergence rates of Markov order estimators and their application to statistical estimation of stationary ergodic processes

Stationary ergodic processes with finite alphabets are estimated by finite memory processes from a sample, an n-length realization of the process, where the memory depth of the estimator process is also estimated from the sample using penalized maximum likelihood (PML). Under some assumptions on the continuity rate and the assumption of non-nullness, a rate of convergence in $\bar{d}$-distance is obtained, with explicit constants. The result requires an analysis of the divergence of PML Markov order estimators for not necessarily finite memory processes. This divergence problem is investigated in more generality for three information criteria: the Bayesian information criterion with generalized penalty term yielding the PML, and the normalized maximum likelihood and the Krichevsky-Trofimov code lengths. Lower and upper bounds on the estimated order are obtained. The notion of consistent Markov order estimation is generalized for infinite memory processes using the concept of oracle order estimates, and generalized consistency of the PML Markov order estimator is presented.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ468 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On estimating the memory for finitarily markovian processes

Finitarily Markovian processes are those processes $\{X_n\}_{n=-\infty}^{\infty}$ for which there is a finite $K$ ($K = K(\{X_n\}_{n=-\infty}^0$) such that the conditional distribution of $X_1$ given the entire past is equal to the conditional distribution of $X_1$ given only $\{X_n\}_{n=1-K}^0$. The least such value of $K$ is called the memory length. We give a rather complete analysis of the problems of universally estimating the least such value of $K$, both in the backward sense that we have just described and in the forward sense, where one observes successive values of $\{X_n\}$ for $n \geq 0$ and asks for the least value $K$ such that the conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=n-K+1}^n$ is the same as the conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=-\infty}^n$. We allow for finite or countably infinite alphabet size

On the minimal penalty for Markov order estimation

We show that large-scale typicality of Markov sample paths implies that the likelihood ratio statistic satisfies a law of iterated logarithm uniformly to the same scale. As a consequence, the penalized likelihood Markov order estimator is strongly consistent for penalties growing as slowly as log log n when an upper bound is imposed on the order which may grow as rapidly as log n. Our method of proof, using techniques from empirical process theory, does not rely on the explicit expression for the maximum likelihood estimator in the Markov case and could therefore be applicable in other settings.Comment: 29 page