Approximate Factorisation of Probability Trees

Abstract

Abstract. Bayesian networks are efficient tools for probabilistic reasoning over large sets of variables, due to the fact that the joint distribution factorises according to the structure of the network, which captures conditional independence relations among the variables. Beyond conditional independence, the concept of asymmetric (or context specific) independence makes possible the definition of even more efficient reasoning schemes, based on the representation of probability functions through probability trees. In this paper we investigate how it is possible to achieve a finer factorisation by decomposing the original factors for which some conditions hold. We also introduce the concept of approximate factorisation and apply this methodology to the Lazy-Penniless propagation algorithm.

New strategies for finding multiplicative decompositions of probability trees

“…The basic approximate factorisation was introduced in [12], where the formulae for computing the proportionality factors π were given according to several methods generally based on minimising divergence measures. From these formulae an approximate tree is obtained and factorised in the same way as the exact factorisation method introduced in Proposition 3.…”

“…The problem of obtaining multiplicative factorisations has been previously studied in the literature [12], being the most recent contribution the socalled fast-factorisation [4]. Though fast factorisation has the advantage of efficiency, as it can be computed quickly, it is only able to benefit of rather restrictive scenarios, namely those in which a potential can be decomposed as the product of two functions, one of them containing only one variable.…”

“…Though fast factorisation has the advantage of efficiency, as it can be computed quickly, it is only able to benefit of rather restrictive scenarios, namely those in which a potential can be decomposed as the product of two functions, one of them containing only one variable. In this paper, we follow the original ideas in [12] and develop the concept of exact and approximate factorisation of probability trees as a product of smaller trees. Our proposal is more general than fast factorisation and provides more accurate approximations than the method in [12].…”

“…In this paper, we follow the original ideas in [12] and develop the concept of exact and approximate factorisation of probability trees as a product of smaller trees. Our proposal is more general than fast factorisation and provides more accurate approximations than the method in [12].…”

“…Obtain π, the matrix of proportionality factors, from the individual coefficients between the leaves in trees T j respect to the factor subtree, using the formulae in [12].…”