Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Calculation of Exact Estimators by Integration Over the Surface of an n-Dimensional Sphere (1305.0764v1)

Published 3 May 2013 in math.ST and stat.TH

Abstract: 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.

Summary

We haven't generated a summary for this paper yet.