  The mutual information of two random variables i and j with joint
probabilities t_ij is commonly used in learning Bayesian nets as well as in
many other fields. The chances t_ij are usually estimated by the empirical
sampling frequency n_ij/n leading to a point estimate I(n_ij/n) for the mutual
information. To answer questions like "is I(n_ij/n) consistent with zero?" or
"what is the probability that the true mutual information is much larger than
the point estimate?" one has to go beyond the point estimate. In the Bayesian
framework one can answer these questions by utilizing a (second order) prior
distribution p(t) comprising prior information about t. From the prior p(t) one
can compute the posterior p(t|n), from which the distribution p(I|n) of the
mutual information can be calculated. We derive reliable and quickly computable
approximations for p(I|n). We concentrate on the mean, variance, skewness, and
kurtosis, and non-informative priors. For the mean we also give an exact
expression. Numerical issues and the range of validity are discussed.
