  Mixture models represent the key modelling approach for Bayesian cluster
analysis. Different likelihood and prior specifications are required to capture
the prototypical shape of the clusters. In addition, the mixture modelling
approaches also crucially differ in the specification of the prior on the
number of components and the prior on the component weight distribution. We
investigate how these specifications impact on the implicitly induced prior on
the number of 'filled' components, i.e., data clusters, and the prior on the
partitions. We derive computationally feasible calculations to obtain these
implicit priors for reasonable data analysis settings and make a reference
implementation available in the R package 'fipp'. In many applications the
implicit priors are of more practical relevance than the explicit priors
imposed and thus suitable prior specifications depend on the implicit priors
induced. We highlight the insights which may be gained from inspecting these
implicit priors by analysing them for three different modelling approaches
previously proposed for Bayesian cluster analysis. These modelling approaches
consist of the Dirichlet process mixture and the static and dynamic mixture of
finite mixtures model. The default priors suggested in the literature for these
modelling approaches are used and the induced priors compared. Based on the
implicit priors, we discuss the suitability of these modelling approaches and
prior specifications when aiming at sparse cluster solutions and flexibility in
the prior on the partitions.
