## The Annals of Statistics

### 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

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