Technical Report CIS9506

TR#:CIS9506
Class:CIS
Title: A CHARACTERIZATION OF THE DIRICHLET DISTRIBUTION THROUGH GLOBAL AND LOCAL INDEPENDENCE.
Authors: D. Geiger and D. Heckerman
PDFCIS9506.pdf
Abstract: We provide a new characterization of the Dirichlet distribution. Let $\theta_{i,j}, 1 \leq i \leq k,1 \leq j \leq n$, be positive random variables that sum to unity. Define $\theta_i = \sum^n_{j=1} \theta_{ij}, \theta_I = \{\theta_i\}^{k-1}_{i=1}, \theta_{j|i} = \theta_{ij}/\sum_j \theta_{ij}$, and $\theta_{J|i} = \{\theta_{j|i}\}^{n-1}_{j=1}$. We prove that if $\{\theta_I , \theta_{J|1},...,\theta_{J|k}\}$ are mutually independent and $\{\theta_J, \theta_{I|1},..., \theta_{I|n}\}$ are mutually independent (where $\theta_J$ and $\theta_{I|j}$ are defined analogously), and assuming strictly positive pdfs, then the pdf of $\theta_{ij}$ is Dirichlet.
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