Abstract. The Minimum Description Length (MDL) is an informationtheoretic principle that can be used for model selection and other statistical inference tasks. One way to implement this principle in practice is to compute the Normalized Maximum Likelihood (NML) distribution for a given parametric model class. Unfortunately this is a computationally infeasible task for many model classes of practical importance. In this paper we present a fast algorithm for computing the NML for the Naive Bayes model class, which is frequently used in classification and clustering tasks. The algorithm is based on a relationship between powers of generating functions and discrete convolution. The resulting algorithm has the time complexity of O(n 2), where n is the size of the data. 1

Universal codes/models can be used for data compression and model selection by the minimum description length (MDL) principle. For many interesting model classes, such as Bayesian networks, the minimax regret optimal normalized maximum 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.

Minimum description length (MDL) model selection, in its modern NML formulation, involves a model complexity term which is equivalent to minimax/maximin regret. When the data are discrete-valued, the complexity term is a logarithm of a sum of maximized likelihoods over all possible data-sets. Because the sum has an exponential number of terms, its evaluation is in many cases intractable. In the continuous case, the sum is replaced by an integral for which a closed form is available in only a few cases. We present an approach based on Monte Carlo sampling, which works for all model classes, and gives strongly consistent estimators of the minimax regret. The estimates convergence almost surely to the correct value with increasing number of iterations. For the important class of Markov models, one of the presented estimators is particularly efficient: in empirical experiments, accuracy that is sufficient for model selection is usually achieved already on the first iteration, even for long sequences.

Bayesian networks are parametric models for multidimensional domains exhibiting complex dependencies between the dimensions (domain variables). A central problem in learning such models is how to regularize the number of parameters; in other words, how to determine which dependencies are significant and which are not. The normalized maximum likelihood (NML) distribution or code offers an information-theoretic solution to this problem. Unfortunately, computing it for arbitrary Bayesian network models appears to be computationally infeasible, but we show how it can be computed efficiently for certain restricted type of Bayesian network models.