Sufficiency, Separability and Temporal Probabilistic Models

Suppose we are given the conditional proba­ bility of one variable given some other vari­ ables. Normally the full joint distribution over the conditioning variables is required to determine the probability of the conditioned variable. Under what circumstances are the marginal distributions over the conditioning variables sufficient to determine the probabil­ ity of the conditioned variable? Sufficiency in this sense is equivalent to additive separa­ bility of the conditional probability distribu­ tion. Such separability structure is natural and can be exploited for efficient inference. Separability has a natural generalization to conditional separability. Separability provides a precise notion of hi­ erarchical decomposition in temporal prob­ abilistic models. Given a system that is decomposed into separable subsystems, ex­ act marginal probabilities over subsystems at future points in time can be computed by propagating marginal subsystem probabil­ ities, rather than complete system joint prob­ abilities. Thus, separability can make exact prediction tractable. However, observations can break separability, so exact monitoring of dynamic systems remains hard.