  In this paper, we show how a complete and exact Bayesian analysis of a
parametric mixture model is possible in some cases when components of the
mixture are taken from exponential families and when conjugate priors are used.
This restricted set-up allows us to show the relevance of the Bayesian approach
as well as to exhibit the limitations of a complete analysis, namely that it is
impossible to conduct this analysis when the sample size is too large, when the
data are not from an exponential family, or when priors that are more complex
than conjugate priors are used.
