Computing Optimal Bayesian Decisions for Rank Aggregation via MCMC Sampling

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 , dmax, and the input size, where φ is the dispersion of the model, kmax measures agents’ largest total bias in bipartitions of alternatives, and dmax is the maximum ratio between prior probabilities. We also show that in some cases the mixing time is at least Θ(φ−kmax/2). For Condorcet’s model, our Markov chain is rapid mixing for moderate prior distributions. Efficiency of our algorithms are illustrated by experiments on real-world datasets.