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 ${\displaystyle \alpha _{i}}$'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

${\displaystyle f\left(x_{1},\ldots ,x_{K};\alpha _{1},\ldots ,\alpha _{K}\right)={\frac {1}{\mathrm {B} ({\boldsymbol {\alpha }})}}\prod _{i=1}^{K}x_{i}^{\alpha _{i}-1}}$

where ${\displaystyle \{x_{k}\}_{k=1}^{k=K}}$ belong to the standard ${\displaystyle K-1}$ simplex, or in other words: ${\displaystyle \sum _{i=1}^{K}x_{i}=1{\mbox{ and }}x_{i}\in \left[0,1\right]{\mbox{ for all }}i\in \{1,\dots ,K\}}$

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

${\displaystyle \mathrm {B} ({\boldsymbol {\alpha }})={\frac {\prod \limits _{i=1}^{K}\Gamma (\alpha _{i})}{\Gamma \left(\sum \limits _{i=1}^{K}\alpha _{i}\right)}},\qquad {\boldsymbol {\alpha }}=(\alpha _{1},\ldots ,\alpha _{K}).}$

Support

The support of the Dirichlet distribution is the set of K-dimensional vectors ${\displaystyle {\boldsymbol {x}}}$ whose entries are real numbers in the interval [0,1] such that ${\displaystyle \|{\boldsymbol {x}}\|_{1}=1}$, i.e. the sum of the coordinates is equal to 1. These can be viewed as the probabilities of a K-way categorical event. Another way to express this is that the domain of the Dirichlet distribution is itself a set of probability distributions, specifically the set of K-dimensional discrete distributions. The technical term for the set of points in the support of a K-dimensional Dirichlet distribution is the open standard (K − 1)-simplex,^[3] which is a generalization of a triangle, embedded in the next-higher dimension. For example, with K = 3, the support is an equilateral triangle embedded in a downward-angle fashion in three-dimensional space, with vertices at (1,0,0), (0,1,0) and (0,0,1), i.e. touching each of the coordinate axes at a point 1 unit away from the origin.

Special cases

A common special case is the symmetric Dirichlet distribution, where all of the elements making up the parameter vector ${\displaystyle {\boldsymbol {\alpha }}}$ have the same value. The symmetric case might be useful, for example, when a Dirichlet prior over components is called for, but there is no prior knowledge favoring one component over another. Since all elements of the parameter vector have the same value, the symmetric Dirichlet distribution can be parametrized by a single scalar value α, called the concentration parameter. In terms of α, the density function has the form

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

When α=1, the symmetric Dirichlet distribution is equivalent to a uniform distribution over the open standard (K − 1)-simplex, i.e. it is uniform over all points in its support. This particular distribution is known as the flat Dirichlet distribution. Values of the concentration parameter above 1 prefer variates that are dense, evenly distributed distributions, i.e. all the values within a single sample are similar to each other. Values of the concentration parameter below 1 prefer sparse distributions, i.e. most of the values within a single sample will be close to 0, and the vast majority of the mass will be concentrated in a few of the values.

More generally, the parameter vector is sometimes written as the product ${\displaystyle \alpha {\boldsymbol {n}}}$ of a (scalar) concentration parameter α and a (vector) base measure ${\displaystyle {\boldsymbol {n}}=(n_{1},\dots ,n_{K})}$ where ${\displaystyle {\boldsymbol {n}}}$ lies within the (K − 1)-simplex (i.e.: its coordinates ${\displaystyle n_{i}}$ sum to one). The concentration parameter in this case is larger by a factor of K than the concentration parameter for a symmetric Dirichlet distribution described above. This construction ties in with concept of a base measure when discussing Dirichlet processes and is often used in the topic modelling literature.

Moments

Let ${\displaystyle X=(X_{1},\ldots ,X_{K})\sim \operatorname {Dir} ({\boldsymbol {\alpha }})}$.

Let

${\displaystyle \alpha _{0}=\sum _{i=1}^{K}\alpha _{i}.}$

Then^[4][5]

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

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

Furthermore, if ${\displaystyle i\neq j}$

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

The matrix is thus singular.

More generally, moments of Dirichlet-distributed random variables can be expressed as^[6]

${\displaystyle \operatorname {E} \left[\prod _{i=1}^{K}X_{i}^{\beta _{i}}\right]={\frac {B\left({\boldsymbol {\alpha }}+{\boldsymbol {\beta }}\right)}{B\left({\boldsymbol {\alpha }}\right)}}={\frac {\Gamma \left(\sum \limits _{i=1}^{K}\alpha _{i}\right)}{\Gamma \left[\sum \limits _{i=1}^{K}(\alpha _{i}+\beta _{i})\right]}}\times \prod _{i=1}^{K}{\frac {\Gamma (\alpha _{i}+\beta _{i})}{\Gamma (\alpha _{i})}}.}$

Mode

The mode of the distribution is^[7] the vector (x₁, ..., x_K) with

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

Marginal distributions

The marginal distributions are beta distributions:^[8]

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

Conjugate to categorical or multinomial

The Dirichlet distribution is the conjugate prior distribution of the categorical distribution (a generic discrete probability distribution with a given number of possible outcomes) and multinomial distribution (the distribution over observed counts of each possible category in a set of categorically distributed observations). This means that if a data point has either a categorical or multinomial distribution, and the prior distribution of the distribution's parameter (the vector of probabilities that generates the data point) is distributed as a Dirichlet, then the posterior distribution of the parameter is also a Dirichlet. Intuitively, in such a case, starting from what we know about the parameter prior to observing the data point, we then can update our knowledge based on the data point and end up with a new distribution of the same form as the old one. This means that we can successively update our knowledge of a parameter by incorporating new observations one at a time, without running into mathematical difficulties.

Formally, this can be expressed as follows. Given a model

${\displaystyle {\begin{array}{rcccl}{\boldsymbol {\alpha }}&=&\left(\alpha _{1},\ldots ,\alpha _{K}\right)&=&{\text{concentration hyperparameter}}\\\mathbf {p} \mid {\boldsymbol {\alpha }}&=&\left(p_{1},\ldots ,p_{K}\right)&\sim &\operatorname {Dir} (K,{\boldsymbol {\alpha }})\\\mathbb {X} \mid \mathbf {p} &=&\left(\mathbf {x} _{1},\ldots ,\mathbf {x} _{K}\right)&\sim &\operatorname {Cat} (K,\mathbf {p} )\end{array}}}$

then the following holds:

${\displaystyle {\begin{array}{rcccl}\mathbf {c} &=&\left(c_{1},\ldots ,c_{K}\right)&=&{\text{number of occurrences of category }}i\\\mathbf {p} \mid \mathbb {X} ,{\boldsymbol {\alpha }}&\sim &\operatorname {Dir} (K,\mathbf {c} +{\boldsymbol {\alpha }})&=&\operatorname {Dir} \left(K,c_{1}+\alpha _{1},\ldots ,c_{K}+\alpha _{K}\right)\end{array}}}$

This relationship is used in Bayesian statistics to estimate the underlying parameter p of a categorical distribution given a collection of N samples. Intuitively, we can view the hyperprior vector α as pseudocounts, i.e. as representing the number of observations in each category that we have already seen. Then we simply add in the counts for all the new observations (the vector c) in order to derive the posterior distribution.

In Bayesian mixture models and other hierarchical Bayesian models with mixture components, Dirichlet distributions are commonly used as the prior distributions for the categorical variables appearing in the models. See the section on applications below for more information.

Relation to Dirichlet-multinomial distribution

In a model where a Dirichlet prior distribution is placed over a set of categorical-valued observations, the marginal joint distribution of the observations (i.e. the joint distribution of the observations, with the prior parameter marginalized out) is a Dirichlet-multinomial distribution. This distribution plays an important role in hierarchical Bayesian models, because when doing inference over such models using methods such as Gibbs sampling or variational Bayes, Dirichlet prior distributions are often marginalized out. See the article on this distribution for more details.

Entropy

If X is a ${\displaystyle \operatorname {Dir} ({\boldsymbol {\alpha }})}$ random variable, the differential entropy of X (in nat units) is^[9]

${\displaystyle h({\boldsymbol {X}})=\operatorname {E} [-\ln f({\boldsymbol {X}})]=\ln \operatorname {B} ({\boldsymbol {\alpha }})+(\alpha _{0}-K)\psi (\alpha _{0})-\sum _{j=1}^{K}(\alpha _{j}-1)\psi (\alpha _{j})}$

where ${\displaystyle \psi }$ is the digamma function.

The following formula for ${\displaystyle \operatorname {E} [\ln(X_{i})]}$ can be used to derive the differential entropy above. Since the functions ${\displaystyle \ln(X_{i})}$ are the sufficient statistics of the Dirichlet distribution, the exponential family differential identities can be used to get an analytic expression for the expectation of ${\displaystyle \ln(X_{i})}$ (see equation (2.62) in ^[10]) and its associated covariance matrix:

${\displaystyle \operatorname {E} [\ln(X_{i})]=\psi (\alpha _{i})-\psi (\alpha _{0})}$

and

${\displaystyle \operatorname {Cov} [\ln(X_{i}),\ln(X_{j})]=\psi '(\alpha _{i})\delta _{ij}-\psi '(\alpha _{0})}$

where ${\displaystyle \psi }$ is the digamma function, ${\displaystyle \psi '}$ is the trigamma function, and ${\displaystyle \delta _{ij}}$ is the Kronecker delta.

The spectrum of Rényi information for values other than ${\displaystyle \lambda =1}$ is given by^[11]

${\displaystyle F_{R}(\lambda )=(1-\lambda )^{-1}\left(-\lambda \log \mathrm {B} ({\boldsymbol {\alpha }})+\sum _{i=1}^{K}\log \Gamma (\lambda (\alpha _{i}-1)+1)-\log \Gamma (\lambda (\alpha _{0}-K)+K)\right)}$

and the information entropy is the limit as ${\displaystyle \lambda }$ goes to 1.

Another related interesting measure is the entropy of a discrete categorical (one-of-K binary) vector ${\displaystyle {\boldsymbol {Z}}}$ with probability-mass distribution ${\displaystyle {\boldsymbol {X}}}$, i.e., ${\displaystyle P(Z_{i}=1,Z_{j\neq i}=0|{\boldsymbol {X}})=X_{i}}$. The conditional information entropy of ${\displaystyle {\boldsymbol {Z}}}$, given ${\displaystyle {\boldsymbol {X}}}$ is

${\displaystyle S({\boldsymbol {X}})=H({\boldsymbol {Z}}|{\boldsymbol {X}})=\operatorname {E} _{\boldsymbol {Z}}[-\log P({\boldsymbol {Z}}|{\boldsymbol {X}})]=\sum _{i=1}^{K}-X_{i}\log X_{i}}$

This function of ${\displaystyle {\boldsymbol {X}}}$ is a scalar random variable. If ${\displaystyle {\boldsymbol {X}}}$ has a symmetric Dirichlet distribution with all ${\displaystyle \alpha _{i}=\alpha }$, the expected value of the entropy (in nat units) is^[12]

${\displaystyle \operatorname {E} [S({\boldsymbol {X}})]=\sum _{i=1}^{K}\operatorname {E} [-X_{i}\ln X_{i}]=\psi (K\alpha +1)-\psi (\alpha +1)}$

Aggregation

If

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

then, if the random variables with subscripts i and j are dropped from the vector and replaced by their sum,

${\displaystyle X'=(X_{1},\ldots ,X_{i}+X_{j},\ldots ,X_{K})\sim \operatorname {Dir} (\alpha _{1},\ldots ,\alpha _{i}+\alpha _{j},\ldots ,\alpha _{K}).}$

This aggregation property may be used to derive the marginal distribution of ${\displaystyle X_{i}}$ mentioned above.

Neutrality

If ${\displaystyle X=(X_{1},\ldots ,X_{K})\sim \operatorname {Dir} ({\boldsymbol {\alpha }})}$, then the vector X is said to be neutral^[13] in the sense that X_K is independent of ${\displaystyle X^{(-K)}}$^[3] where

${\displaystyle X^{(-K)}=\left({\frac {X_{1}}{1-X_{K}}},{\frac {X_{2}}{1-X_{K}}},\ldots ,{\frac {X_{K-1}}{1-X_{K}}}\right),}$

and similarly for removing any of ${\displaystyle X_{2},\ldots ,X_{K-1}}$. Observe that any permutation of X is also neutral (a property not possessed by samples drawn from a generalized Dirichlet distribution).^[14]

Combining this with the property of aggregation it follows that X_j + ... + X_K is independent of ${\displaystyle \left({\frac {X_{1}}{X_{1}+\cdots +X_{j-1}}},{\frac {X_{2}}{X_{1}+\cdots +X_{j-1}}},\ldots ,{\frac {X_{j-1}}{X_{1}+\cdots +X_{j-1}}}\right)}$. In fact it is true, further, for the Dirichlet distribution, that for ${\displaystyle 3\leq j\leq K-1}$, the pair ${\displaystyle \left(X_{1}+\cdots +X_{j-1},X_{j}+\cdots +X_{K}\right)}$, and the two vectors ${\displaystyle \left({\frac {X_{1}}{X_{1}+\cdots +X_{j-1}}},{\frac {X_{2}}{X_{1}+\cdots +X_{j-1}}},\ldots ,{\frac {X_{j-1}}{X_{1}+\cdots +X_{j-1}}}\right)}$ and ${\displaystyle \left({\frac {X_{j}}{X_{j}+\cdots +X_{K}}},{\frac {X_{j+1}}{X_{j}+\cdots +X_{K}}},\ldots ,{\frac {X_{K}}{X_{j}+\cdots +X_{K}}}\right)}$, viewed as triple of normalised random vectors, are mutually independent. The analogous result is true for partition of the indices {1,2,...,K} into any other pair of non-singleton subsets.

Characteristic function

The characteristic function of the Dirichlet distribution is a confluent form of the Lauricella hypergeometric series. It is given by Phillips as^[15]

${\displaystyle CF\left(s_{1},\ldots ,s_{K-1}\right)=\operatorname {E} \left(e^{i\left(s_{1}X_{1}+\cdots +s_{K-1}X_{K-1}\right)}\right)=\Psi ^{\left[K-1\right]}(\alpha _{1},\ldots ,\alpha _{K-1};\alpha _{0};is_{1},\ldots ,is_{K-1})}$

where

${\displaystyle \Psi ^{[m]}(a_{1},\ldots ,a_{m};c;z_{1},\ldots z_{m})=\sum {\frac {(a_{1})_{k_{1}}\cdots (a_{m})_{k_{m}}\,z_{1}^{k_{1}}\cdots z_{m}^{k_{m}}}{(c)_{k}\,k_{1}!\cdots k_{m}!}}.}$

The sum is over non-negative integers ${\displaystyle k_{1},\ldots ,k_{m}}$ and ${\displaystyle k=k_{1}+\cdots +k_{m}}$. Phillips goes on to state that this form is "inconvenient for numerical calculation" and gives an alternative in terms of a complex path integral:

${\displaystyle \Psi ^{[m]}={\frac {\Gamma (c)}{2\pi i}}\int _{L}e^{t}\,t^{a_{1}+\cdots +a_{m}-c}\,\prod _{j=1}^{m}(t-z_{j})^{-a_{j}}\,dt}$

where L denotes any path in the complex plane originating at ${\displaystyle -\infty }$, encircling in the positive direction all the singularities of the integrand and returning to ${\displaystyle -\infty }$.

Inequality

Probability density function ${\displaystyle f\left(x_{1},\ldots ,x_{K-1};\alpha _{1},\ldots ,\alpha _{K}\right)}$ plays a key role in a multifunctional inequality which implies various bounds for the Dirichlet distribution.^[16]

Conjugate prior of the Dirichlet distribution

Because the Dirichlet distribution is an exponential family distribution it has a conjugate prior. The conjugate prior is of the form:^[18]

${\displaystyle \operatorname {CD} ({\boldsymbol {\alpha }}\mid {\boldsymbol {v}},\eta )\propto \left({\frac {1}{\operatorname {B} ({\boldsymbol {\alpha }})}}\right)^{\eta }\exp \left(-\sum _{k}v_{k}\alpha _{k}\right).}$

Here ${\displaystyle {\boldsymbol {v}}}$ is a K-dimensional real vector and ${\displaystyle \eta }$ is a scalar parameter. The domain of ${\displaystyle ({\boldsymbol {v}},\eta )}$ is restricted to the set of parameters for which the above unnormalized density function can be normalized. The (necessary and sufficient) condition is:^[19]

${\displaystyle \forall k\;\;v_{k}>0\;\;\;\;{\text{ and }}\;\;\;\;\eta >-1\;\;\;\;{\text{ and }}\;\;\;\;(\eta \leq 0\;\;\;\;{\text{ or }}\;\;\;\;\sum _{k}\exp -{\frac {v_{k}}{\eta }}<1)}$

The conjugation property can be expressed as

if [prior: ${\displaystyle {\boldsymbol {\alpha }}\sim \operatorname {CD} (\cdot \mid {\boldsymbol {v}},\eta )}$] and [observation: ${\displaystyle {\boldsymbol {x}}\mid {\boldsymbol {\alpha }}\sim \operatorname {Dirichlet} (\cdot \mid {\boldsymbol {\alpha }})}$] then [posterior: ${\displaystyle {\boldsymbol {\alpha }}\mid {\boldsymbol {x}}\sim \operatorname {CD} (\cdot \mid {\boldsymbol {v}}-\log {\boldsymbol {x}},\eta +1)}$].

In the published literature there is no practical algorithm to efficiently generate samples from ${\displaystyle \operatorname {CD} ({\boldsymbol {\alpha }}\mid {\boldsymbol {v}},\eta )}$.

Bayesian models

Dirichlet distributions are most commonly used as the prior distribution of categorical variables or multinomial variables in Bayesian mixture models and other hierarchical Bayesian models. (In many fields, such as in natural language processing, categorical variables are often imprecisely called "multinomial variables". Such a usage is unlikely to cause confusion, just as when Bernoulli distributions and binomial distributions are commonly conflated.)

Inference over hierarchical Bayesian models is often done using Gibbs sampling, and in such a case, instances of the Dirichlet distribution are typically marginalized out of the model by integrating out the Dirichlet random variable. This causes the various categorical variables drawn from the same Dirichlet random variable to become correlated, and the joint distribution over them assumes a Dirichlet-multinomial distribution, conditioned on the hyperparameters of the Dirichlet distribution (the concentration parameters). One of the reasons for doing this is that Gibbs sampling of the Dirichlet-multinomial distribution is extremely easy; see that article for more information.

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. Recall that ${\displaystyle \alpha _{0}=\sum _{i=1}^{K}\alpha _{i}.}$ The ${\displaystyle \alpha _{i}/\alpha _{0}}$ values specify the mean lengths of the cut pieces of string resulting from the distribution. The variance around this mean varies inversely with ${\displaystyle \alpha _{0}}$.

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

From gamma distribution

With a source of Gamma-distributed random variates, one can easily sample a random vector ${\displaystyle x=(x_{1},\ldots ,x_{K})}$ from the K-dimensional Dirichlet distribution with parameters ${\displaystyle (\alpha _{1},\ldots ,\alpha _{K})}$ . First, draw K independent random samples ${\displaystyle y_{1},\ldots ,y_{K}}$ from Gamma distributions each with density

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

and then set

${\displaystyle x_{i}={\frac {y_{i}}{\sum _{j=1}^{K}y_{j}}}.}$

Below is example Python code to draw the sample:

params = [a1, a2, ..., ak]
sample = [random.gammavariate(a, 1) for a in params]
sample = [v / sum(sample) for v in sample]

This formulation is correct regardless of how the Gamma distributions are parameterized (shape/scale vs. shape/rate) because they are equivalent when scale and rate equal 1.0.

From marginal beta distributions

A less efficient algorithm^[21] relies on the univariate marginal and conditional distributions being beta and proceeds as follows. Simulate ${\displaystyle x_{1}}$ from

${\displaystyle {\textrm {Beta}}\left(\alpha _{1},\sum _{i=2}^{K}\alpha _{i}\right)}$

Then simulate ${\displaystyle x_{2},\ldots ,x_{K-1}}$ in order, as follows. For ${\displaystyle j=2,\ldots ,K-1}$, simulate ${\displaystyle \phi _{j}}$ from

${\displaystyle {\textrm {Beta}}\left(\alpha _{j},\sum _{i=j+1}^{K}\alpha _{i}\right),}$

and let

${\displaystyle x_{j}=\left(1-\sum _{i=1}^{j-1}x_{i}\right)\phi _{j}.}$

Finally, set

${\displaystyle x_{K}=1-\sum _{i=1}^{K-1}x_{i}.}$

This iterative procedure corresponds closely to the "string cutting" intuition described above.

Below is example Python code to draw the sample:

params = [a1, a2, ..., ak]
xs = [random.betavariate(params[0], sum(params[1:]))]
for j in range(1, len(params) - 1):
    phi = random.betavariate(params[j], sum(params[j + 1 :]))
    xs.append((1 - sum(xs)) * phi)
xs.append(1 - sum(xs))

When each alpha is 1

When α₁ = ... = α_K = 1, a sample from the distribution can be found by randomly drawing a set of K − 1 values independently and uniformly from the interval [0, 1], adding the values 0 and 1 to the set to make it have K + 1 values, sorting the set, and computing the difference between each pair of order-adjacent values, to give x₁, ..., x_K.

When each alpha is 1/2 and relationship to the hypersphere

When α₁ = ... = α_K = 1/2, a sample from the distribution can be found by randomly drawing K values independently from the standard normal distribution, squaring these values, and normalizing them by dividing by their sum, to give x₁, ..., x_K.

A point (x₁, ..., x_K) can be drawn uniformly at random from the (K−1)-dimensional hypersphere (which is the surface of a K-dimensional hyperball) via a similar procedure. Randomly draw K values independently from the standard normal distribution and normalize these coordinate values by dividing each by the constant that is the square root of the sum of their squares.