## The Annals of Statistics

- Ann. Statist.
- Volume 18, Number 3 (1990), 1317-1327.

### On the Uniform Consistency of Bayes Estimates for Multinomial Probabilities

#### Abstract

A $k$-sided die is thrown $n$ times, to estimate the probabilities $\theta_1, \ldots, \theta_k$ of landing on the various sides. The MLE of $\theta$ is the vector of empirical proportions $p = (p_1, \ldots, p_k)$. Consider a set of Bayesians that put uniformly positive prior mass on all reasonable subsets of the parameter space. Their posterior distributions will be uniformly concentrated near $p$. Sharp bounds are given, using entropy. These bounds apply to all sample sequences: There are no exceptional null sets.

#### Article information

**Source**

Ann. Statist., Volume 18, Number 3 (1990), 1317-1327.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347751

**Digital Object Identifier**

doi:10.1214/aos/1176347751

**Mathematical Reviews number (MathSciNet)**

MR1062710

**Zentralblatt MATH identifier**

0788.62005

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62A15

Secondary: 62E20: Asymptotic distribution theory

**Keywords**

Bayes estimates consistency Laplace's method multinomial Bernstein-von Mises theorem

#### Citation

Diaconis, P.; Freedman, D. On the Uniform Consistency of Bayes Estimates for Multinomial Probabilities. Ann. Statist. 18 (1990), no. 3, 1317--1327. doi:10.1214/aos/1176347751. https://projecteuclid.org/euclid.aos/1176347751