Some Properties of joint Probability Distributions

Several Artificial Intelligence schemes for reasoning under uncertainty explore either explicitly or implicitly asymmetries among probabilities of various states of their uncer­ tain domain models. Even though the correct working of these schemes is practically con­ tingent upon the existence of a small number of probable states, no formal justification has been proposed of why this should be the case. This paper attempts to fill this apparent gap by studying asymmetries among probabili­ ties of various states of uncertain models. By rewriting the joint probability distribu­ tion over a model's variables into a product of individual variables' prior and conditional probability distributions and applying cen­ tral limit theorem to this product, we can demonstrate that the probabilities of indi­ vidual states of the model can be expected to be drawn from highly skewed lognormal distributions. With sufficient asymmetry in individual prior and conditional probability distributions, a small fraction of states can be expected to cover a large portion of the total probability space with the remaining states having practically negligible probabil­ ity. Theoretical discussion is supplemented by simulation results and an illustrative real­ world example.