  We use statistical mechanics to study model-based Bayesian data clustering.
In this approach, each partition of the data into clusters is regarded as a
microscopic system state, the negative data log-likelihood gives the energy of
each state, and the data set realisation acts as disorder. Optimal clustering
corresponds to the ground state of the system, and is hence obtained from the
free energy via a low `temperature' limit. We assume that for large sample
sizes the free energy density is self-averaging, and we use the replica method
to compute the asymptotic free energy density. The main order parameter in the
resulting (replica symmetric) theory, the distribution of the data over the
clusters, satisfies a self-consistent equation which can be solved by a
population dynamics algorithm. From this order parameter one computes the
average free energy, and all relevant macroscopic characteristics of the
problem. The theory describes numerical experiments perfectly, and gives a
significant improvement over the mean-field theory that was used to study this
model in past.
