  Given the joint chances of a pair of random variables one can compute
quantities of interest, like the mutual information. The Bayesian treatment of
unknown chances involves computing, from a second order prior distribution and
the data likelihood, a posterior distribution of the chances. A common
treatment of incomplete data is to assume ignorability and determine the
chances by the expectation maximization (EM) algorithm. The two different
methods above are well established but typically separated. This paper joins
the two approaches in the case of Dirichlet priors, and derives efficient
approximations for the mean, mode and the (co)variance of the chances and the
mutual information. Furthermore, we prove the unimodality of the posterior
distribution, whence the important property of convergence of EM to the global
maximum in the chosen framework. These results are applied to the problem of
selecting features for incremental learning and naive Bayes classification. A
fast filter based on the distribution of mutual information is shown to
outperform the traditional filter based on empirical mutual information on a
number of incomplete real data sets.
