Last updated: 2017-02-20

Code version: 92d7daa

Overview

The purpose of this vignette is to introduce the Dirichlet distribution. You should be familiar with the Beta distribution since the Dirichlet can be thought of as a generalization of the Beta distribution.

If you want more details you could look at Wikipedia.

The Dirichlet Distribution

You can think of the $$J$$-dimensional Dirichlet distribution as a distribution on probability vectors, $$q=(q_1,\dots,q_J)$$, whose elements are non-negative and sum to 1. It is perhaps the most commonly-used distribution for probability vectors, and plays a central role in Bayesian inference from multinomial data.

The Dirichlet distribution has $$J$$ parameters, $$\alpha_1,\dots,\alpha_J$$ that control the mean and variance of the distribution. If $$q \sim \text{Dirichlet}(\alpha_1,\dots,\alpha_J)$$ then:

• The expectation of $$q_j$$ is $$\alpha_j/(\alpha_1 + \dots + \alpha_J)$$.

• The variance of $$q_j$$ becomes smaller as the sum $$\sum_j \alpha_j$$ increases.

As a generalization of the Beta distribution

The 2-dimensional Dirichlet distribution is essentially the Beta distribution. Specifically, let $$q=(q_1,q_2)$$. Then $$q \sim Dirichlet(\alpha_1,\alpha_2)$$ implies that $q_1 \sim \text{Beta}(\alpha_1,\alpha_2)$ and $$q_2 = 1-q_1$$.

Other connections to the Beta distribution

More generally, the marginals of the Dirichlet distribution are also beta distributions.

That is, if $$q \sim \text{Dirichlet}(\alpha_1, \dots,\alpha_J)$$ then $$q_j \sim \text{Beta}(\alpha_j,\sum_{j' \neq j} \alpha_{j'})$$.

Density

The density of the Dirichlet distribution is most conveniently written as $p(q | \alpha) = \frac{\Gamma(\alpha_1+\dots+\alpha_J)}{\Gamma(\alpha_1)\dots \Gamma(\alpha_J)}\prod_{j=1}^J q_j^{\alpha_j-1} \qquad (q_j \geq 0; \quad \sum_j q_j =1).$ where $$Gamma$$ here denotes the gamma function.

Actually when writing the density this way, a little care needs to be taken to make things formally correct. Specifically, if you perform standard (Lebesgue) integration of this “density” over the $$J$$ dimensional space $$q_1,\dots, q_J$$ it integrates to 0, and not 1 as a density should. This problem is caused by the constraint that the $$q$$s must sum to 1, which means that the Dirichlet distribution is effectively a $$J-1$$-dimensional distribution and not a $$J$$ dimensional distribution.

The simplest resolution to this is to think of the $$J$$ dimensional Dirichlet distribution as a distribution on the $$J-1$$ numbers $$(q_1, \dots, q_{J-1})$$, satisfying $$\sum_{j=1}^{J-1} q_j \leq 1$$, and then define $$q_J := (1-q_1-q_2-\dots - q_{J-1})$$. Then, if we integrate the density $p(q_1,\dots,q_{J-1} | \alpha) = \frac{\Gamma(\alpha_1+\dots+\alpha_J)}{\Gamma(\alpha_1)\dots \Gamma(\alpha_J)} \prod_{j=1}^{J-1} q_j^{\alpha_j-1} (1-q_1-\dots - q_{J-1})^{\alpha_J} \qquad (q_j \geq 0; \quad \sum_{j=1}^{J-1} q_j \leq 1).$ over $$(q_1,\dots,q_{J-1})$$, it integrates to 1 as a density should.

Examples

Session Information

sessionInfo()
R version 3.3.2 (2016-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X El Capitan 10.11.6

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base

loaded via a namespace (and not attached):
[1] backports_1.0.5 magrittr_1.5    rprojroot_1.2   htmltools_0.3.5
[5] tools_3.3.2     rstudioapi_0.6  yaml_2.1.14     Rcpp_0.12.9
[9] stringi_1.1.2   rmarkdown_1.3   knitr_1.15.1    git2r_0.18.0
[13] stringr_1.1.0   digest_0.6.12   workflowr_0.3.0 gtools_3.5.0
[17] evaluate_0.10  

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