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

Observations depending on sums of random variables are common throughout many fields; however, no efficient solution is currently known for performing max-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 "infimal convolution", "min-convolution", or "convolution on the tropical semiring"), for which no O(k log(k)) method is currently known. Here I present a O(k log(k)) numerical method for estimating the max-convolution of two nonnegative vectors (e.g., two probability mass functions), where k is the length of the larger vector. This numerical max-convolution method is then demonstrated by performing fast max-product inference on a convolution tree, a data structure for performing fast inference given information on the sum of n discrete random variables in O(n k log(n k) log(n) ) steps (where each random variable has an arbitrary prior distribution on k contiguous possible states). The numerical max-convolution method can be applied to specialized classes of hidden Markov models to reduce the runtime of computing the Viterbi path from n k^2 to n k log(k), and has potential application to the all-pairs shortest paths problem.

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