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January, 1975 The Bayes Factor Against Equiprobability of a Multinomial Population Assuming a Symmetric Dirichlet Prior
I. J. Good
Ann. Statist. 3(1): 246-250 (January, 1975). DOI: 10.1214/aos/1176343015

Abstract

A sample $(n_1, n_2,\cdots, n_t)$ is taken from a $t$-category multinomial population. The hypothesis of equiprobability, that the $t$ physical probabilities associated with the cells are all equal to $1/t$, is called the null hypothesis. Conditional on the non-null hypothesis, a symmetric Dirichlet prior of parameter $k$ is assumed $(k \geqq 0)$ and the Bayes factor against the null hypothesis, with this assumption, is denoted by $F(k)$. A conjecture made in 1965 is almost proved, namely that $F(k)$ has a unique local maximum and that this occurs for a finite value of $k$ if and only if Pearson's $X^2$ exceeds its number of degrees of freedom. The result is required for the calculation of $\max F(k)$, which provides a non-Bayesian significance criterion whose simple asymptotic distribution is good even in the extreme tail, and even for sample sizes less than $t$. This criterion arose from an attitude involving a Bayes/non-Bayes compromise.

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I. J. Good. "The Bayes Factor Against Equiprobability of a Multinomial Population Assuming a Symmetric Dirichlet Prior." Ann. Statist. 3 (1) 246 - 250, January, 1975. https://doi.org/10.1214/aos/1176343015

Information

Published: January, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0303.62030
MathSciNet: MR375594
Digital Object Identifier: 10.1214/aos/1176343015

Subjects:
Primary: 62E15
Secondary: 62G10

Keywords: Bayes factor , Bayes/non-Bayes compromise , hierarchical probability judgments , Multinomial significance test , symmetric Dirichlet distribution , Type II likelihood ratio , weight of evidence

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 1 • January, 1975
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