  Many datasets give partial information about an ordering or ranking by
indicating which team won a game, which item a user prefers, or who infected
whom. We define a continuous spin system whose Gibbs distribution is the
posterior distribution on permutations, given a probabilistic model of these
interactions. Using the cavity method we derive a belief propagation algorithm
that computes the marginal distribution of each node's position. In addition,
the Bethe free energy lets us approximate the number of linear extensions of a
partial order and perform model selection between competing probabilistic
models, such as the Bradley-Terry-Luce model of noisy comparisons and its
cousins.
