  The Bradley-Terry model is a popular approach to describe probabilities of
the possible outcomes when elements of a set are repeatedly compared with one
another in pairs. It has found many applications including animal behaviour,
chess ranking and multiclass classification. Numerous extensions of the basic
model have also been proposed in the literature including models with ties,
multiple comparisons, group comparisons and random graphs. From a computational
point of view, Hunter (2004) has proposed efficient iterative MM
(minorization-maximization) algorithms to perform maximum likelihood estimation
for these generalized Bradley-Terry models whereas Bayesian inference is
typically performed using MCMC (Markov chain Monte Carlo) algorithms based on
tailored Metropolis-Hastings (M-H) proposals. We show here that these MM\
algorithms can be reinterpreted as special instances of
Expectation-Maximization (EM) algorithms associated to suitable sets of latent
variables and propose some original extensions. These latent variables allow us
to derive simple Gibbs samplers for Bayesian inference. We demonstrate
experimentally the efficiency of these algorithms on a variety of applications.
