The Annals of Statistics

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

Dan Geiger and David Heckerman

Full-text: Open access

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.


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