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We derive bounds on the probability of a goal node given a set of acquired input nodes. The bounds apply to decomposable networks; a class of Bayesian networks encompassing causal trees and causal polytrees. The difficulty of computing the bounds depends on the characteristics of the decomposable network. For directly connected networks with binary goal nodes, tight bounds can be computed in polynomial time. For other kinds of decomposable networks, the derivation of the bounds requires solving an integer program with a nonlinear objective function, a computationally intractable problem in the worst case. We provide a relaxation technique that computes looser bounds in polynomial time for more complex decomposable networks. We briefly describe an application of the probability bounds to a record linkage problem.