Dirichlet distribution

Dirichlet distribution

Probability density function
Parameters	${\displaystyle K\geq 2}$ number of categories (integer) ${\displaystyle {\boldsymbol {\alpha }}=(\alpha _{1},\ldots ,\alpha _{K})}$ concentration parameters, where ${\displaystyle \alpha _{i}>0}$
Support	${\displaystyle x_{1},\ldots ,x_{K}}$ where ${\displaystyle x_{i}\in [0,1]}$ and ${\displaystyle \sum _{i=1}^{K}x_{i}=1}$
PDF	${\displaystyle {\frac {1}{\mathrm {B} ({\boldsymbol {\alpha }})}}\prod _{i=1}^{K}x_{i}^{\alpha _{i}-1}}$ where ${\displaystyle \mathrm {B} ({\boldsymbol {\alpha }})={\frac {\prod _{i=1}^{K}\Gamma (\alpha _{i})}{\Gamma {\bigl (}\alpha _{0}{\bigr )}}}}$ where ${\displaystyle \alpha _{0}=\sum _{i=1}^{K}\alpha _{i}}$
Mean	${\displaystyle \operatorname {E} [X_{i}]={\frac {\alpha _{i}}{\alpha _{0}}}}$ ${\displaystyle \operatorname {E} [\ln X_{i}]=\psi (\alpha _{i})-\psi (\alpha _{0})}$ (where ${\displaystyle \psi }$ is the digamma function)
Mode	${\displaystyle x_{i}={\frac {\alpha _{i}-1}{\alpha _{0}-K}},\quad \alpha _{i}>1.}$
Variance	${\displaystyle \operatorname {Var} [X_{i}]={\frac {{\tilde {\alpha }}_{i}(1-{\tilde {\alpha }}_{i})}{\alpha _{0}+1}},}$ ${\displaystyle \operatorname {Cov} [X_{i},X_{j}]={\frac {\delta _{ij}\,{\tilde {\alpha }}_{i}-{\tilde {\alpha }}_{i}{\tilde {\alpha }}_{j}}{\alpha _{0}+1}}}$ where ${\displaystyle {\tilde {\alpha }}_{i}={\frac {\alpha _{i}}{\alpha _{0}}}}$, and ${\displaystyle \delta _{ij}}$ is the Entropy			${\displaystyle H(X)=\log \mathrm {B} ({\boldsymbol {\alpha }})}$${\displaystyle +(\alpha _{0}-K)\psi (\alpha _{0})-}$${\displaystyle \sum _{j=1}^{K}(\alpha _{j}-1)\psi (\alpha _{j})}$ with ${\displaystyle \alpha _{0}}$ defined as for variance, above; and ${\displaystyle \psi }$ is the digamma function
Method of Moments	${\displaystyle \alpha _{i}=E[X_{i}]\left({\frac {E[X_{j}](1-E[X_{j}])}{V[X_{j}]}}-1\right)}$ where ${\displaystyle j}$ is any index, possibly ${\displaystyle i}$ itself

In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted ${\displaystyle \operatorname {Dir} ({\boldsymbol {\alpha }})}$, is a family of

parameterized by a vector ${\displaystyle {\boldsymbol {\alpha }}}$ of positive

The infinite-dimensional generalization of the Dirichlet distribution is the Dirichlet process.

Definitions

Probability density function

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

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^[1], the symmetric Dirichlet distribution is equivalent to a uniform distribution over the open

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.

Properties

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 in the following way. For ${\displaystyle {\boldsymbol {t}}=(t_{1},\dotsc ,t_{K})\in \mathbb {R} ^{K}}$, denote by ${\displaystyle {\boldsymbol {t}}^{\circ i}=(t_{1}^{i},\dotsc ,t_{K}^{i})}$ its ${\displaystyle i}$-th Hadamard power. Then,^[6]

${\displaystyle \operatorname {E} \left[({\boldsymbol {t}}\cdot {\boldsymbol {X}})^{n}\right]={\frac {n!\,\Gamma (\alpha _{0})}{\Gamma (\alpha _{0}+n)}}\sum {\frac {{t_{1}}^{k_{1}}\cdots {t_{K}}^{k_{K}}}{k_{1}!\cdots k_{K}!}}\prod _{i=1}^{K}{\frac {\Gamma (\alpha _{i}+k_{i})}{\Gamma (\alpha _{i})}}={\frac {n!\,\Gamma (\alpha _{0})}{\Gamma (\alpha _{0}+n)}}Z_{n}({\boldsymbol {t}}^{\circ 1}\cdot {\boldsymbol {\alpha }},\cdots ,{\boldsymbol {t}}^{\circ n}\cdot {\boldsymbol {\alpha }}),}$

where the sum is over non-negative integers ${\displaystyle k_{1},\ldots ,k_{K}}$ with ${\displaystyle n=k_{1}+\cdots +k_{K}}$, and ${\displaystyle Z_{n}}$ is the cycle index polynomial of the Symmetric group of degree ${\displaystyle n}$.

The multivariate analogue ${\textstyle \operatorname {E} \left[({\boldsymbol {t}}_{1}\cdot {\boldsymbol {X}})^{n_{1}}\cdots ({\boldsymbol {t}}_{q}\cdot {\boldsymbol {X}})^{n_{q}}\right]}$ for vectors ${\displaystyle {\boldsymbol {t}}_{1},\dotsc ,{\boldsymbol {t}}_{q}\in \mathbb {R} ^{K}}$ can be expressed^[7] in terms of a color pattern of the exponents ${\displaystyle n_{1},\dotsc ,n_{q}}$ in the sense of Pólya enumeration theorem.

Particular cases include the simple computation^[8]

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

Mode

The mode of the distribution is^[9] 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:^[10]

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

Conjugate to categorical or multinomial

The Dirichlet distribution is the

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

In Bayesian

If X is a ${\displaystyle \operatorname {Dir} ({\boldsymbol {\alpha }})}$ random variable, the differential entropy of X (in nat units) is^[11]

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

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

The following formula for ${\displaystyle \operatorname {E} [\ln(X_{i})]}$ can be used to derive the differential

${\displaystyle \ln(X_{i})}$

${\displaystyle \ln(X_{i})}$

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

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^[13]

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

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

${\displaystyle {\boldsymbol {Z}}}$

${\displaystyle {\boldsymbol {X}}}$

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

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^[14]

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

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^[15] 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).^[16]

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

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.[18]

Related distributions

For K independently distributed Gamma distributions:

${\displaystyle Y_{1}\sim \operatorname {Gamma} (\alpha _{1},\theta ),\ldots ,Y_{K}\sim \operatorname {Gamma} (\alpha _{K},\theta )}$

we have:^[19]: 402

${\displaystyle V=\sum _{i=1}^{K}Y_{i}\sim \operatorname {Gamma} \left(\alpha _{0},\theta \right),}$

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

Although the X_is are not independent from one another, they can be seen to be generated from a set of K independent gamma random variable.^[19]: 594 Unfortunately, since the sum V is lost in forming X (in fact it can be shown that V is stochastically independent of 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.

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:^[20]

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

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:^[21]

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

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 ${\displaystyle \operatorname {CD} ({\boldsymbol {\alpha }}\mid {\boldsymbol {v}},\eta )}$.

Occurrence and applications

Bayesian models

Dirichlet distributions are most commonly used as the

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

The concentration parameter

Dirichlet distributions are very often used as

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).^[22]

For a formal proof, note that the proportions of the different colored balls form a bounded [0,1]^K-valued

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^[23] 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

• Generalized Dirichlet distribution
• Grouped Dirichlet distribution
• Inverted Dirichlet distribution
• Latent Dirichlet allocation
• Dirichlet process
• Matrix variate Dirichlet distribution

References