## The Annals of Statistics

- Ann. Statist.
- Volume 25, Number 3 (1997), 917-1369

### A characterization of the Dirichlet distribution through global and local parameter independence

Dan Geiger and David Heckerman

#### Abstract

We provide a new characterization of the Dirichlet distribution. Let $\theta_{ij}, 1 \leq i \leq k, 1 \leq j \leq n$, be positive random variables that sum to unity. Define $\theta_{i \cdot} = \Sigma_{j=1}^n \theta_{ij}, \theta_{I \cdot} = {\theta_{i \cdot}_{i=1}^{k-1}, \theta_{j|i} = \theta_{ij}/ \Sigma_j \theta_{ij}$ and \theta_{J|i} = {\theta_{j|i}}_{j=1}^{n-1}$. We prove that if ${\theta_{I \cdot}, \theta_{J|1}, \dots, \theta_{J|k}}$ are mutually independent and ${\theta_{\cdot J}, \theta_{I|1}, \dots, \theta_{I|n}}$ are mutually independent (where $\theta_{\cdot J}$ and $\theta_{I|j}$ are defined analogously, and each parameter set has a strictly positive pdf, then the pdf of $\theta_{ij}$ is Dirichlet. This characterization implies that under assumptions made by several previous authors for selecting a Bayesian network structure out of a set of candidate structures, a Dirichlet prior on the parameters is inevitable.

#### Article information

**Source**

Ann. Statist. Volume 25, Number 3 (1997), 1344-1369.

**Dates**

First available: 20 November 2003

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1069362752

**Mathematical Reviews number (MathSciNet)**

MR1447755

**Digital Object Identifier**

doi:10.1214/aos/1069362752

**Zentralblatt MATH identifier**

0885.62009

**Subjects**

Primary: 62E10: Characterization and structure theory 60E05: Distributions: general theory

Secondary: 62A15 62C10: Bayesian problems; characterization of Bayes procedures 39B99: None of the above, but in this section

**Keywords**

Bayesian network characterization Dirichlet distribution functional equation graphical model hyper-Markov law

#### Citation

Geiger, Dan; Heckerman, David. A characterization of the Dirichlet distribution through global and local parameter independence. The Annals of Statistics 25 (1997), no. 3, 1344--1369. doi:10.1214/aos/1069362752. http://projecteuclid.org/euclid.aos/1069362752.