Information-Theoretical Methods in Statistics

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

The investigator studies statistical model selection problems using information criteria approach. The research concentrates on two related problems: statistical estimation of context trees and statistical estimation of basic neighborhoods of Markov random fields. Statistical estimation of context trees means estimating the memory structure of a discrete-time and finitely-valued stochastic process from a realization of finite length. As the memory length may depend on the actual past, the memory structure can be represented by a tree graph. In the literature, the memory length is usually assumed to be bounded. Then the process is a Markov chain, but the context tree model provides a more efficient description of the memory structure. The research focuses on dropping the finite memory assumption, therefore arbitrary stationary ergodic processes are considered. Model selection methods using information criteria are studied to estimate context trees, using information theoretical, statistical and probability tools. The notion of strong consistency is generalized and new statistical aspects are considered. Markov random fields can be regarded as generalizations of Markov chains to higher dimensional index sets. Statistical estimation of basic neighborhoods of Markov random fields means estimating the interaction structure of finitely valued random variables on an integer lattice, from a realization in a finite region. The basic neighborhood is the smallest region around a site that affects the distribution at the site. The research focuses on modified information criteria to develop consistent estimators of the basic neighborhood. In both of the investigated areas, computational complexity aspects and applications are also considered. Context tree models are used in statistics, information theory, bioinformatics and various other disciplines. Specific areas of applications include lossless data compression, universal prediction of individual sequences and genetics to model DNA and protein sequences. Markov random fields are special Gibbs fields, therefore they provide essential models in statistical physics for modeling interactive particle systems. They are also used in several other fields, including image processing and pattern recognition. The project aims at two directions. On one hand, it aims to achieve progress in the theory of model selection methods using information criteria, in particular, in context tree estimation and in estimation of basic neighborhoods of Markov random fields. On the other hand, it aims to bring the theoretical achievements to the fields of applications. The research includes collaborative work with researchers in the area, and it also involves the potential of broadening the applications of the results in collaboration with researchers from the areas of applications, namely, from bioinformatics and engineering. A goal is to involve graduate students to the research.