  This paper reconsiders the problem of calculating the expected set of
probabilities <p_i>, given the observed set of items {m_i}, that are
distributed among n bins with an (unknown) set of probabilities {p_i} for being
placed in the ith bin. The problem is often formulated using Bayes theorem and
the multinomial distribution, along with a constant prior for the values of the
p_i, leading to a Dirichlet distribution for the {p_i}. The moments of the p_i
can then be calculated exactly. Here a new approach is suggested for the
calculation of the moments, that uses a change of variables that reduces the
problem to an integration over a portion of the surface of an n-dimensional
sphere. This greatly simplifies the calculation by allowing a straightforward
integration over (n-1) independent variables, with the constraints on the set
of p_i being automatically satisfied. For the Dirichlet and similar
distributions the problem simplifies even further, with the resulting integrals
subsequently factorising, allowing their easy evaluation in terms of Beta
functions. A proof by induction confirms existing calculations for the moments.
The advantage of the approach presented here is that the methods and results
apply with minimum or no modifications to numerical calculations that involve
more complicated distributions or non-constant prior distributions, for which
cases the numerical calculations will be greatly simplified.
