  Repulsive mixture models have recently gained popularity for Bayesian cluster
detection. Compared to more traditional mixture models, repulsive mixture
models produce a smaller number of well separated clusters. The most commonly
used methods for posterior inference either require to fix a priori the number
of components or are based on reversible jump MCMC computation. We present a
general framework for mixture models, when the prior of the `cluster centres'
is a finite repulsive point process depending on a hyperparameter, specified by
a density which may depend on an intractable normalizing constant. By
investigating the posterior characterization of this class of mixture models,
we derive a MCMC algorithm which avoids the well-known difficulties associated
to reversible jump MCMC computation. In particular, we use an ancillary
variable method, which eliminates the problem of having intractable normalizing
constants in the Hastings ratio. The ancillary variable method relies on a
perfect simulation algorithm, and we demonstrate this is fast because the
number of components is typically small. In several simulation studies and an
application on sociological data, we illustrate the advantage of our new
methodology over existing methods, and we compare the use of a determinantal or
a repulsive Gibbs point process prior model.
