Many problems require repeated inference on probabilistic graphical models, with different values for evidence variables or other changes. Examples of such problems include utility maximization, MAP inference, online and interactive inference, parameter and structure learning, and dynamic inference. Since small changes to the evidence typically only affect a small region of the network, repeatedly performing inference from scratch can be massively redundant. In this paper, we propose expanding frontier belief propagation (EFBP), an efficient approximate algorithm for probabilistic inference with incremental changes to the evidence (or model). EFBP is an extension of loopy belief propagation (BP) where each run of inference reuses results from the previous ones, instead of starting from scratch with the new evidence; messages are only propagated in regions of the network affected by the changes. We provide theoretical guarantees bounding the difference in beliefs generated by EFBP and standard BP, and apply EFBP to the problem of expected utility maximization in influence diagrams. Experiments on viral marketing and combinatorial auction problems show that EFBP can converge much faster than BP without significantly affecting the quality of the solutions.