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We propose two efficient and general MCMC algorithms to compute optimal Bayesian decisions for Mallows' model and Condorcet's model w.r.t. any loss function and prior. We show that the mixing time of our Markov chain for Mallows' model is polynomial in ϕ −kmax , d max , and the input size, where ϕ is the dispersion of the model, k max measures agents' largest total bias in bipartitions of alternatives, and d max is the maximum ratio between prior probabilities. We also show that in some casesdblp:conf/uai/HughesHX15 fatcat:5ga5maaedbg2zncekvgjjnk2x4