On Multifractal Property of the Joint Probability Distributions and Its Application to Bayesian Network Inference

This paper demonstrates that the Joint Probability Distribution (JPD) of a Bayesian network is a random multinomial multifractal. With sufficient asymmetry in individual prior and conditional probability distributions, the JPD is not only highly skewed as shown by Druzdzel [3], but also is stochastically self-similar and has clusters of highprobability instantiations at all scales. Based on the discovered multifractal property a two phase hybrid Sampling-And-Search algorithm for finding the Most Probable Explanation (MPE) is developed and tested. The experimental results show that the multifractal property provides a good meta-heuristic for solving the MPE problem. The multifractal properties also strengthen the connections between Bayesian networks and thermodynamics. These connections have recently been exploited in popular Bayesian network inference algorithms based upon models from statistical physics [16, 11], such as free energy minimization.