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Observations depending on sums of random variables are common throughout manyfields; however, no efficient solution is currently known for performingmax-product inference on these sums of general discrete distributions(max-product inference can be used to obtain maximum a posteriori estimates).The limiting step to max-product inference is the max-convolution problem(sometimes presented in log-transformed form and denoted as "infimalconvolution", "min-convolution", or "convolution on the tropical semiring"),for which no O(k log(k)) method is currently known. Here I present a O(klog(k)) numerical method for estimating the max-convolution of two nonnegativevectors (e.g., two probability mass functions), where k is the length of thelarger vector. This numerical max-convolution method is then demonstrated byperforming fast max-product inference on a convolution tree, a data structurefor performing fast inference given information on the sum of n discrete randomvariables in O(n k log(n k) log(n) ) steps (where each random variable has anarbitrary prior distribution on k contiguous possible states). The numericalmax-convolution method can be applied to specialized classes of hidden Markovmodels to reduce the runtime of computing the Viterbi path from n k^2 to n klog(k), and has potential application to the all-pairs shortest paths problem.

A fast numerical method for max-convolution and the application to efficient max-product inference in Bayesian networks