Dirichlet distribution

Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4).

In probability and statistics, the Dirichlet distribution (after Johann Peter Gustav Lejeune Dirichlet), often denoted Dir(α), is a family of continuous multivariate probability distributions parametrized by the vector α of positive reals. It is the multivariate generalization of the beta distribution, and conjugate prior of the categorical distribution and multinomial distribution in Bayesian statistics. That is, its probability density function returns the belief that the probabilities of K rival events are $x_{i}$ given that each event has been observed $\alpha _{i}-1$ times.

Probability density function

Illustrating how the log of the density function changes when K=3 as we change the vector α from α=(0.3, 0.3, 0.3) to (2.0, 2.0, 2.0), keeping all the individual $\alpha _{i}$'s equal to each other.

The Dirichlet distribution of order K ≥ 2 with parameters α₁, ..., α_K > 0 has a probability density function with respect to Lebesgue measure on the Euclidean space R^K–1 given by

$f(x_{1},\dots ,x_{K};\alpha _{1},\dots ,\alpha _{K})={\frac {1}{\mathrm {B} (\alpha )}}\prod _{i=1}^{K}x_{i}^{\alpha _{i}-1}$

for all x₁, ..., x_K–1 > 0 satisfying x₁ + ... + x_K–1 < 1, where x_K is an abbreviation for 1 – x₁ – ... – x_K–1. The density is zero outside this open (K − 1)-dimensional simplex.

The normalizing constant is the multinomial beta function, which can be expressed in terms of the gamma function:

$\mathrm {B} (\alpha )={\frac {\prod _{i=1}^{K}\Gamma (\alpha _{i})}{\Gamma {\bigl (}\sum _{i=1}^{K}\alpha _{i}{\bigr )}}},\qquad \alpha =(\alpha _{1},\dots ,\alpha _{K}).$

Properties

Let $X=(X_{1},\ldots ,X_{K})\sim \operatorname {Dir} (\alpha )$, meaning that the first K – 1 components have the above density and

$X_{K}=1-X_{1}-\cdots -X_{K-1}.$

Define $\textstyle \alpha _{0}=\sum _{i=1}^{K}\alpha _{i}$. Then

$\mathrm {E} [X_{i}]={\frac {\alpha _{i}}{\alpha _{0}}},$

$\mathrm {Var} [X_{i}]={\frac {\alpha _{i}(\alpha _{0}-\alpha _{i})}{\alpha _{0}^{2}(\alpha _{0}+1)}}={\frac {\mathrm {E} [X_{i}](1-\mathrm {E} [X_{i}])}{(\alpha _{0}+1)}}.$

in fact, the marginals are Beta distributions:

$X_{i}\sim \operatorname {Beta} (\alpha _{i},\alpha _{0}-\alpha _{i}).$

Furthermore, if $i\neq j$

$\mathrm {Cov} [X_{i},X_{j}]={\frac {-\alpha _{i}\alpha _{j}}{\alpha _{0}^{2}(\alpha _{0}+1)}}.$

The mode of the distribution is the vector (x₁, ..., x_K) with

$x_{i}={\frac {\alpha _{i}-1}{\alpha _{0}-K}},\quad \alpha _{i}>1.$

The Dirichlet distribution is conjugate to the multinomial distribution in the following sense: if

$\beta \mid X=(\beta _{1},\ldots ,\beta _{K})\mid X\sim \operatorname {Mult} (X),$

where β_i is the number of occurrences of i in a sample of n points from the discrete distribution on {1, ..., K} defined by X, then

$X\mid \beta \sim \operatorname {Dir} (\alpha +\beta ).$

This relationship is used in Bayesian statistics to estimate the hidden parameters, X, of a categorical distribution (discrete probability distribution) given a collection of n samples. Intuitively, if the prior is represented as Dir(α), then Dir(α + β) is the posterior following a sequence of observations with histogram β.

Entropy

If X is a Dir(α) random variable, then we can use the exponential family differential identities to get an analytic expression for the expectation of $\log X_{i}$:

$\mathrm {E} [\log X_{i}]=\psi (\alpha _{i})-\psi (\alpha _{0})$

where $\psi$ is the digamma function. This yields the following formula for the information entropy of X:

$H(X)=\log \mathrm {B} (\alpha )+(\alpha _{0}-K)\psi (\alpha _{0})-\sum _{j=1}^{K}(\alpha _{j}-1)\psi (\alpha _{j})$

Aggregation

If $X=(X_{1},\ldots ,X_{K})\sim \operatorname {Dir} (\alpha _{1},\ldots ,\alpha _{K})$, then $X'=(X_{1},\ldots ,X_{i}+X_{j},\ldots ,X_{K})\sim \operatorname {Dir} (\alpha _{1},\ldots ,\alpha _{i}+\alpha _{j},\ldots ,\alpha _{K})$. This aggregation property may be used to derive the marginal distribution of $X_{i}$ mentioned above.

Neutrality

Main article: Neutral vector

If $X=(X_{1},\ldots ,X_{K})\sim \operatorname {Dir} (\alpha )$, then the vector~$X$ is said to be neutral in the sense that $X_{K}$ is independent of $X_{1}/(1-X_{K}),X_{2}/(1-X_{K}),\ldots ,X_{K-1}/(1-X_{K})$ and similarly for $X_{2},\ldots ,X_{K-1}$.

Observe that any permutation of $X$ is also neutral (a property not possessed by samples drawn from a generalized Dirichlet distribution).

The derivation of the neutrality property:

Let $X_{1},X_{2},\cdots ,X_{K}\sim Dir(\alpha _{1},\alpha _{2},\cdots ,\alpha _{K})$. And let

$X_{K}^{\prime }=X_{K}$ ,   $X_{1}^{\prime }={\frac {X_{1}}{1-X_{K}^{\prime }}}$ ,   $X_{2}^{\prime }={\frac {X_{2}}{1-X_{K}^{\prime }}}$ ,   $\cdots$,   $X_{K-2}^{\prime }={\frac {X_{K-2}}{1-X_{K}^{\prime }}}$

For the purpose of convenience, we set $X_{K-1}^{\prime }=1-\sum _{i=1}^{K-2}{X_{i}^{\prime }}$. Here we aim to derive that $X_{1}^{\prime },X_{2}^{\prime },\cdots ,X_{K-1}^{\prime }$ also follow a Dirichlet distribution as $X_{1}^{\prime },X_{2}^{\prime },\cdots ,X_{K-1}^{\prime }\sim Dir(\alpha _{1},\alpha _{2},\cdots ,\alpha _{K-1})$.

We start the derivation with change of variables from $X_{1},X_{2},\cdots ,X_{K-1}$ to $X_{1}^{\prime },X_{2}^{\prime },X_{3}^{\prime },\cdots ,X_{K-2}^{\prime },X_{K}^{\prime }$. The Jacobian can be calculated easily:

$Jacobian\left({\frac {X_{1},X_{2},\cdots ,X_{K-2},X_{K}}{X_{1}^{\prime },X_{2}^{\prime },X_{3}^{\prime },\cdots ,X_{K-2}^{\prime },X_{K}^{\prime }}}\right)={\begin{vmatrix}1-X_{K}^{\prime }&0&0&\cdots &0&-X_{1}^{\prime }\\0&1-X_{k}^{\prime }&0&\cdots &0&-X_{2}^{\prime }\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&0&\cdots &1-X_{K}^{\prime }&-X_{K-2}^{\prime }\\0&0&0&\cdots &0&1\\\end{vmatrix}}=(1-X_{K}^{\prime })^{K-2}$

Thus, the probability density function of $X_{1}^{\prime },X_{2}^{\prime },X_{3}^{\prime },\cdots ,X_{K}^{\prime }$ is the following:

${\begin{aligned}&f(X_{1}^{\prime },X_{2}^{\prime },\dots ,X_{K}^{\prime };\alpha _{1},\dots ,\alpha _{K})\\&={\frac {1}{\mathrm {B} (\alpha )}}X_{K}^{\prime \alpha _{K}-1}(1-X_{K}^{\prime })^{\sum _{i=1}^{K-1}{\alpha _{i}}-1}\prod _{i=1}^{K-1}{X_{i}^{\prime \alpha _{i}-1}}\\&=\left({\frac {\Gamma (\sum _{i=1}^{K}\alpha _{i})}{\Gamma (\alpha _{K})\Gamma \left(\sum _{i=1}^{K-1}\alpha _{i}\right)}}X_{K}^{\prime \alpha _{K}-1}(1-X_{K}^{\prime })^{\sum _{i=1}^{K-1}{\alpha _{i}}-1}\right)\left({\frac {\Gamma \left(\sum _{i=1}^{K-1}\alpha _{i}\right)}{\prod _{i=1}^{K-1}\Gamma (\alpha _{i})}}\prod _{i=1}^{K-1}{X_{i}^{\prime \alpha _{i}-1}}\right)\\&=Beta\left(\alpha _{K},\sum _{i=1}^{K-1}\alpha _{i}\right)\times Dir(\alpha _{1},\alpha _{2},\cdots ,\alpha _{K-1})\end{aligned}}$

From the above equation, it is obvious that the derived probability density function is actually a joint distribution of two independent parts, a Beta distributed $X_{K}^{\prime }$ part and a Dirichlet distributed $X_{1}^{\prime },X_{2}^{\prime },\cdots ,X_{K-1}^{\prime }$ part. By trivially integrating out $X_{K}$, the result is obvious.

Related distributions

• If, for $\scriptstyle i\in \{1,2,\ldots ,K\},$

$Y_{i}\sim \operatorname {Gamma} ({\textrm {shape}}=\alpha _{i},{\textrm {scale}}=\theta ){\text{ independently}},$

then

$V=\sum _{i=1}^{K}Y_{i}\sim \operatorname {Gamma} ({\textrm {shape}}=\sum _{i=1}^{K}\alpha _{i},{\textrm {scale}}=\theta ),$

and

$X=(X_{1},\ldots ,X_{K})=(Y_{1}/V,\ldots ,Y_{K}/V)\sim \operatorname {Dir} (\alpha _{1},\ldots ,\alpha _{K}).$

Though the X_is are not independent from one another, they can be seen to be generated from a set of $K$ independent gamma random variables. Unfortunately, since the sum $V$ is lost in forming X, it is not possible to recover the original gamma random variables from these values alone. Nevertheless, because independent random variables are simpler to work with, this reparametrization can still be useful for proofs about properties of the Dirichlet distribution.

The following is a derivation of Dirichlet distribution from Gamma distribution.

Let Y_i, i=1,2,...K be a list of i.i.d variables, following Gamma distributions with the same scale parameter θ

$Y_{i}\sim \operatorname {Gamma} ({\textrm {shape}}=\alpha _{i},{\textrm {scale}}=\theta ){\text{ independently}},$

then the joint distribution of Y_i, i=1,2,...K is

$f(Y_{1},Y_{2},\cdots ,Y_{K};\alpha _{1},\alpha _{2},\cdots ,\alpha _{K},\theta )=\prod _{i=1}^{K}Y_{i}^{\alpha _{i}-1}{\frac {e^{-Y_{i}/\theta }}{\theta ^{\alpha _{i}}\Gamma (\alpha _{i})}}.$

Through the change of variables, set

$\gamma =\sum _{i=1}^{K}Y_{i},\quad X_{1}={\frac {Y_{1}}{\sum _{i=1}^{K}Y_{i}}},\quad X_{2}={\frac {Y_{2}}{\sum _{i=1}^{K}Y_{i}}},\quad \cdots ,\quad X_{K-1}={\frac {Y_{K-1}}{\sum _{i=1}^{K}Y_{i}}}.$

Then, it's easy to derive that

$Y_{1}=\gamma X_{1},\quad Y_{2}=\gamma X_{2},\quad \cdots ,\quad Y_{K-1}=\gamma X_{K-1},\quad Y_{K}=\gamma (1-X_{1}-X_{2}\cdots -X_{K-1}).$

Then, the Jacobian is

$Jacobian\left({\frac {Y_{1},Y_{2},\cdots ,Y_{K}}{\gamma ,X_{1},X_{2},\cdots ,X_{K-1}}}\right)={\begin{vmatrix}X_{1}&\gamma &0&\cdots &0\\X_{2}&0&\gamma &\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\X_{K-1}&0&0&\cdots &\gamma \\1-\sum _{i=1}^{K-1}X_{i}&-\gamma &-\gamma &\cdots &-\gamma \\\end{vmatrix}}=\gamma ^{K-1}$

It means

$dY_{1}dY_{2}\cdots dY_{K}=\gamma ^{K-1}d\gamma dX_{1}\cdots dX_{K-1}$

So,

$f(X_{1},X_{2},\cdots ,X_{K-1},\gamma ;\alpha _{1},\alpha _{2},\cdots ,\alpha _{K},\theta )$

$={\frac {e^{-\gamma /\theta }}{\theta ^{\sum _{i=1}^{K}\alpha _{i}}\prod _{i=1}^{K}\Gamma (\alpha _{i})}}\gamma ^{\sum _{i=1}^{K}\alpha _{i}-1}X_{1}^{\alpha _{1}-1}X_{2}^{\alpha _{2}-1}\cdots X_{K-1}^{\alpha _{K-1}-1}(1-\sum _{i=1}^{K-1}X_{i})^{\alpha _{K}-1}.$

By integrating out γ, we can get the Dirichlet distribution as the following.

$\int _{0}^{\infty }f(X_{1},X_{2},\cdots ,X_{K-1},\gamma ;\alpha _{1},\alpha _{2},\cdots ,\alpha _{K},\theta )d\gamma$

$={\frac {1}{\prod _{i=1}^{K}\Gamma (\alpha _{i})}}X_{1}^{\alpha _{1}-1}X_{2}^{\alpha _{2}-1}\cdots X_{K-1}^{\alpha _{K-1}-1}(1-\sum _{i=1}^{K-1}X_{i})^{\alpha _{K}-1}\int _{0}^{\infty }{\frac {e^{-\gamma /\theta }}{\theta ^{\sum _{i=1}^{K}\alpha _{i}}}}\gamma ^{\sum _{i=1}^{K}\alpha _{i}-1}d\gamma$

According to the Gamma distribution,

$\int _{0}^{\infty }{\frac {e^{-\gamma /\theta }}{\theta ^{\sum _{i=1}^{K}\alpha _{i}}}}\gamma ^{\sum _{i=1}^{K}\alpha _{i}-1}d\gamma =\Gamma (\sum _{i=1}^{K}\alpha _{i})$

Finally, we get the following Dirichlet distribution

$(X_{1},\ldots ,X_{K})\sim \operatorname {Dir} (\alpha _{1},\ldots ,\alpha _{K}).$

where X_K is (1-X₁ - X₂... -X_K-1)

• Multinomial opinions in subjective logic are equivalent to Dirichlet distributions.

Random number generation

Gamma distribution

A fast method to sample a random vector $x=(x_{1},\ldots ,x_{K})$ from the K-dimensional Dirichlet distribution with parameters $(\alpha _{1},\ldots ,\alpha _{K})$ follows immediately from this connection. First, draw K independent random samples $y_{1},\ldots ,y_{K}$ from gamma distributions each with density

${\textrm {Gamma}}(\alpha _{i},1)={\frac {y_{i}^{\alpha _{i}-1}\;e^{-y_{i}}}{\Gamma (\alpha _{i})}},\!$

and then set

$x_{i}=y_{i}/\sum _{j=1}^{K}y_{j}.\!$

Marginal beta distributions

A less efficient algorithm relies on the univariate marginal and conditional distributions being beta and proceeds as follows. Simulate $x_{1}$ from a ${\textrm {Beta}}(\alpha _{1},\sum _{i=2}^{K}\alpha _{i})$ distribution. Then simulate $x_{2},\ldots ,x_{K-1}$ in order, as follows. For $j=2,\ldots ,K-1$, simulate $\phi _{j}$ from a ${\textrm {Beta}}(\alpha _{j},\sum _{i=j+1}^{K}\alpha _{i})$ distribution, and let $x_{j}=(1-\sum _{i=1}^{j-1}x_{i})\phi _{j}$. Finally, set $x_{K}=1-\sum _{i=1}^{K-1}x_{i}$.

Intuitive interpretations of the parameters

String cutting

One example use of the Dirichlet distribution is if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had a designated average length, but allowing some variation in the relative sizes of the pieces. The α/α₀ values specify the mean lengths of the cut pieces of string resulting from the distribution. The variance around this mean varies inversely with α₀.

Example of Dirichlet(1/2,1/3,1/6) distribution

Pólya's urn

Consider an urn containing balls of K different colors. Initially, the urn contains α₁ balls of color 1, α₂ balls of color 2, and so on. Now perform N draws from the urn, where after each draw, the ball is placed back into the urn with an additional ball of the same color. In the limit as N approaches infinity, the proportions of different colored balls in the urn will be distributed as Dir(α₁,...,α_K).

For a formal proof, note that the proportions of the different colored balls form a bounded [0,1]^K-valued martingale, hence by the martingale convergence theorem, these proportions converge almost surely and in mean to a limiting random vector. To see that this limiting vector has the above Dirichlet distribution, check that all mixed moments agree.

Note that each draw from the urn modifies the probability of drawing a ball of any one color from the urn in the future. This modification diminishes with the number of draws, since the relative effect of adding a new ball to the urn diminishes as the urn accumulates increasing numbers of balls. This "diminishing returns" effect can also help explain how large α values yield Dirichlet distributions with most of the probability mass concentrated around a single point on the simplex.

See also

• beta distribution
• binomial distribution
• categorical distribution
• generalized Dirichlet distribution
• latent Dirichlet allocation
• multinomial distribution
• multivariate Polya distribution

External links

• Dirichlet Distribution
• Estimating the parameters of the Dirichlet distribution
• Non-Uniform Random Variate Generation, Luc Devroye