  The integrated completed likelihood (ICL) criterion has proven to be a very
popular approach in model-based clustering through automatically choosing the
number of clusters in a mixture model. This approach effectively maximises the
complete data likelihood, thereby including the allocation of observations to
clusters in the model selection criterion. However for practical implementation
one needs to introduce an approximation in order to estimate the ICL. Our
contribution here is to illustrate that through the use of conjugate priors one
can derive an exact expression for ICL and so avoiding any approximation.
Moreover, we illustrate how one can find both the number of clusters and the
best allocation of observations in one algorithmic framework. The performance
of our algorithm is presented on several simulated and real examples.
