Bayesian logic

We combine probabilistic logic and Bayesian networks to obtain the advantages of each in what we call Bayesian logic. Like probabilistic logic, it is a theoretically grounded way of representing and reasoning with uncertainty that uses only as much probabilistic information as one has, since it permits one to specify probabilities as intervals rather than precise values. Like Bayesian networks, it can capture conditional independence relations, which are probably our richest source of probabilistic knowledge. The inference problem in Bayesian logic can be solved as a nonlinear program (which becomes a linear program in ordinary probabilistic logic). We show that Benders decomposition, applied to the nonlinear program, allows one to use the same column generation methods in Bayesian logic that are now being used to solve inference problems in probabilistic logic. We also show that if the independence conditions are properly represented, the number of nonlinear constraints grows only linearly with the number of nodes in a large class of networks (rather than exponentially, as in the general case).