Context tree selection: A unifying view
Context tree models have been introduced by Rissanen in  as a parsimonious generalization of Markov models. Since then, they have been widely used in applied probability and statistics. The present paper investigates non-asymptotic properties of two popular procedures of context tree estimation: Rissanenâs algorithm Context and penalized maximum likelihood. First showing how they are related, we prove finite horizon bounds for the probability of over- and under-estimation. Concerning over-estimation, no boundedness or loss-of-memory conditions are required: the proof relies on new deviation inequalities for empirical probabilities of independent interest. The under-estimation properties rely on classical hypotheses for processes of infinite memory. These results improve on and generalize the bounds obtained in Duarte etÂ al. (2006) , Galves etÂ al. (2008) , Galves and Leonardi (2008) , Leonardi (2010) , refining asymptotic results of Böhlmann and Wyner (1999)  and Csiszár and Talata (2006) .
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Volume (Year): 121 (2011)
Issue (Month): 11 (November)
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