Calculating the Normalized Maximum Likelihood Distribution for Bayesian Forests

When learning Bayesian network structures from sample data, an important issue is how to evaluate the goodness of alternative network structures. Perhaps the most commonly used model (class) selection criterion is the marginal likelihood, which is obtained by integrating over a prior distribution for the model parameters. However, the problem of determining a reasonable prior for the parameters is a highly controversial issue, and no completely satisfying Bayesian solution has yet been presented in the non-informative setting. The normalized maximum likelihood (NML), based on Rissanen’s information-theoretic MDL methodology, offers an alternative, theoretically solid criterion that is objective and non-informative, while no parameter prior is required. It has been previously shown that for discrete data, this criterion can be computed in linear time for Bayesian networks with no arcs, and in quadratic time for the so called Naive Bayes network structure. Here we extend the previous results by showing how to compute the NML criterion in polynomial time for tree-structured Bayesian networks. The order of the polynomial depends on the number of values of the variables, but neither on the number of variables itself, nor on the sample size.