Probability Distributions > Dirichlet Distribution

**Contents:**

- What is a Dirichlet Distribution?
- What are Random PMFs?
- The Dirichlet Process.
- PDF/Mean/Variance.
- Similarity to Other Distributions.

## 1. What is a Dirichlet Distribution?

A Dirichlet distribution (pronounced *Deer-eesh-lay*) is a way to model random probability mass functions (PMFs) for finite sets. It is also sometimes used as a prior in Bayesian statistics. The distribution creates *n* positive numbers (a set of random *vectors *X_{1}…X_{n}) that add up to 1; Therefore, it is closely related to the multinomial distribution, which also requires n numbers that sum to 1.

The distribution is named after the 19th century Belgian mathematician Johann Dirichlet.

## 2. What are Random PMFs?

When probability is introduced in basic statistics, one of the common topics to come up is rolling a fair die. The “fair die” is almost certainly a myth; Manufacturing processes are pretty good, but they aren’t perfect. If you roll 1000 dice, the theoretical odds of any particular number showing up (i.e. a 1, 2, 3, 4, 5, or 6) are 1/6. However, you won’t get that *exact *distribution in a real experiment due to manufacturing defects. No die is perfectly weighted—there will always be a tiny bit of sway to one side of a die or another. If you have ten dice, each die will have its own probability mass function (PMF).

Another example of a random PMF is the distribution of words in books and other documents; A book of length *k *words can be modeled by a Dirichlet distribution with a PMF of length *k*.

## 3. The Dirichlet Process

The Dirichlet process is a way to model randomness of a probability mass function(PMF) with unlimited options (e.g. an unlimited amount of dice in a bag). The process is similar Polya’s Urn, **only instead of having a set number of ball colors you have an unlimited amount**.

- Start out with an empty urn.
- Randomly pick a colored ball and place it in the urn.
- Then choose one option:
- Randomly pick a colored ball and place it in the urn.
- Randomly remove a colored ball from the urn, then put it back with another ball of the same color.

As the number of balls in the urn increase, the probability of picking a new color *decreases*. The proportion of balls in the urn after an infinite amount of draws is a Dirichlet process. For an example of a Dirichlet process, see: Chinese Restaurant Process.

## 4. PDF/Mean/Variance

The explanation above gives an outline of a Dirichlet distribution. The actual math behind the distribution is a little more complex. In order to fully understand the distribution, you should have an idea about:

- Beta Distributions.
- Change of Variables.
- Gamma Functions.
- Multivariate Distributions.

The probability density function (PDF) is:

Where:

and a_{1}, …, am are parameters with a_{i} > 0 for i=1,…,m.

## Mean

The mean of θ_{j} is:

E(θ_{j}) = a_{j} / A.

## Variance

The variance of θ_{j} is:

var(θ_{j}) = a_{j} / A(A + 1) – a_{j} / A(A + 1).

## 5. Similarity to Other Distributions

- The Dirichlet is the multivariate generalization of the
**beta distribution**. It is an extension of the beta distribution for modeling probabilities for two or more disjoint events; when m=2 (see PDF below), the Dirichlet distribution is equal to the PDF of the beta distribution. - The Dirichlet equals the
**uniform distribution**when all parameters (α_{1}…α_{k}) are equal. - The Dirichlet distribution is a
**conjugate prior**to the categorical distribution and multinomial distributions. A compound variant is the Dirichlet-multinomial.

**References**:

Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Polya urn schemes. Ann. Statist. 2, 1, 353-355.

Cifarelli, D. M. and Regazzini, E. (1990). Distribution functions of means of a Dir. process. Ann. Statist. 18 429-442

Emilion, R. (2005). Process of random distributions. Afrika Stat, vol 1, 1, pp. 27-46, http://www.ufrsat.org/jas (contenus).

Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics, Cambridge University Press.

Ferguson, T.S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2 615-629.

Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.

Kingman, J. F. C. (1975). Random discrete distributions. J. Roy. Statist. Soc. B, 37, 1-22.