$G$ method in action: Fast exact sampling from set of permutations of order $n$ according to Mallows model through Cayley metric

Abstract

Using $G$ method, we give a fast exact (not approximate) Markovian method for sampling from $\mathbb{S}_{n}$, the set of permutations of order $n$, according to the Mallows model through Cayley metric (a model for ranked data). This method has something in common with the cyclic Gibbs sampler and something in common with the swapping method. The number of steps of our method is equal to the number of steps of swapping method, that is, $n-1$; moreover, both methods use the best probability distributions on sampling, the swapping method uses uniform probability distributions while our method uses almost uniform probability distributions (all the components of an almost uniform probability distribution are, here, identical, excepting at most one of them). But, besides sampling, we can do other things for the Mallows model through Cayley metric—we compute the normalizing constant and, by Uniqueness theorem, certain important probabilities.