Bayesian Network Structure Learning using Factorized NML Universal Models.pdf

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Bayesian Network Structure Learning using Factorized NML Universal Models.pdf

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Bayesian Network Structure Learning using Factorized NML Universal Models

Bayesian Network Structure Learning using Factorized NML Universal Models Teemu Roos, Tomi Silander, Petri Kontkanen, and Petri Myllym?ki Complex Systems Computation Group, Helsinki Institute for Information Technology HIIT University of Helsinki Helsinki University of Technology P.O.Box 68 (Department of Computer Science) FIN-00014 University of Helsinki, Finland Email: firstname.lastname@cs.helsinki.fi Abstract— Universal codes/models can be used for data com- pression and model selection by the minimum description length (MDL) principle. For many interesting model classes, such as Bayesian networks, the minimax regret optimal normalized max- imum likelihood (NML) universal model is computationally very demanding. We suggest a computationally feasible alternative to NML for Bayesian networks, the factorized NML universal model, where the normalization is done locally for each variable. This can be seen as an approximate sum-product algorithm. We show that this new universal model performs extremely well in model selection, compared to the existing state-of-the-art, even for small sample sizes. I. INTRODUCTION The stochastic complexity of a sequence under a given model class is a central concept in the minimum description length (MDL) principle [1], [2], [3], [4]. Its interpretation as the length of the shortest achievable encoding makes it a yardstick for the comparison of different model classes. In recent formulations of MDL, stochastic complexity is defined using the so called normalized maximum likelihood (NML) universal model, originally introduced by Shtarkov [5] for data compression; for the role of NML in MDL model selection, see [6], [7], [3], [4], [8]. Since the introduction of the NML universal model in the context of MDL, there has been significant interest in the evaluation of NML stochastic complexity for different practi- cally relevant model classes, both exactly and asymptotically. For discrete models, exact evaluation is often computationally in