2013/05/03 by Webster, Anthony J
#FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1305.0764
This paper reconsiders the problem of calculating the expected set of probabilities , given the observed set of items mi, that are distributed among n bins with an (unknown) set of probabilities pi 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 pi, leading to a Dirichlet distribution for the pi. The moments of the pi 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 pi 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.