Editing Dirichlet distribution (section)

==Definitions==
===Probability density function===
[[Image:LogDirichletDensity-alpha 0.3 to alpha 2.0.gif|thumb|right|250px|Illustrating how the log of the density function changes when {{math|1=''K'' = 3}} as we change the vector {{math|'''α'''}} from {{math|1='''α''' = (0.3, 0.3, 0.3)}} to {{math|(2.0, 2.0, 2.0)}}, keeping all the individual <math>\alpha_i</math>'s equal to each other.]]

The Dirichlet distribution of order {{math|''K'' ≥ 2}} with parameters {{math|''α''{{sub|1}}, ..., ''α''{{sub|''K''}} > 0}} has a [[probability density function]] with respect to [[Lebesgue measure]] on the [[Euclidean space]] {{math|'''R'''{{isup|''K''−1}}}} given by

<math display=block>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}</math>
where <math>\{x_k\}_{k=1}^{k=K}</math> belong to the standard <math>K-1</math> [[simplex]], or in other words:
<math display=block>\sum_{i=1}^{K} x_i = 1 \mbox{ and } x_i \in \left[0,1\right] \mbox{ for all } i \in \{1,\dots,K\}\,.</math>

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

<math display=block>\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).</math>

===Support===
The [[support (mathematics)|support]] of the Dirichlet distribution is the set of {{mvar|K}}-dimensional vectors {{math|'''x'''}} whose entries are real numbers in the interval [0,1] such that <math>\|\boldsymbol x\|_1 = 1</math>, i.e. the sum of the coordinates is equal to 1.  These can be viewed as the probabilities of a {{mvar|K}}-way [[categorical distribution|categorical]] event. Another way to express this is that the domain of the Dirichlet distribution is itself a set of [[probability distribution]]s, specifically the set of {{mvar|K}}-dimensional [[discrete distribution]]s. The technical term for the set of points in the support of a {{mvar|K}}-dimensional Dirichlet distribution is the [[open set|open]] [[standard simplex|standard {{math|(''K'' − 1)}}-simplex]],<ref name=FKG>{{cite web |url=https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2010-0006.pdf |title=Introduction to the Dirichlet Distribution and Related Processes |year=2010 |author1=Bela A. Frigyik |author2=Amol Kapila |author3=Maya R. Gupta |access-date= |format=Technical Report UWEETR-2010-006 |publisher=University of Washington Department of Electrical Engineering |archive-url=https://web.archive.org/web/20150219021331/https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2010-0006.pdf |archive-date=2015-02-19 |url-status=dead }}</ref> which is a generalization of a [[triangle]], embedded in the next-higher dimension.  For example, with {{math|1=''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 {{math|'''α'''}} 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 {{mvar|α}}, called the [[concentration parameter]]. In terms of {{mvar|α}}, the density function has the form

<math display=block>f(x_1,\dots, x_{K}; \alpha) = \frac{\Gamma(\alpha K)}{\Gamma(\alpha)^K} \prod_{i=1}^K x_i^{\alpha - 1}.</math>

When {{math|1=''α'' = 1}},{{ref|concentration-parameter-disambiguation}} the symmetric Dirichlet distribution is equivalent to a uniform distribution over the open [[standard simplex|standard {{math|(''K''−1)}}-simplex]], i.e. it is uniform over all points in its [[support (mathematics)|support]]. This particular distribution is known as the '''flat Dirichlet distribution'''. Values of the concentration parameter above 1 prefer [[random variate|variate]]s 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.

When {{math|1=''α'' = 1/2}}, the distribution is the same as would be obtained by choosing a point uniformly at random from the surface of a {{math|(''K''−1)}}-dimensional [[unit hypersphere]] and squaring each coordinate.  The {{math|1=''α'' = 1/2}} distribution is the [[Jeffreys prior]] for the Dirichlet distribution.

More generally, the parameter vector is sometimes written as the product <math>\alpha \boldsymbol n</math> of a ([[Scalar (mathematics)|scalar]]) [[concentration parameter]] {{mvar|α}} and a ([[Vector (mathematics and physics)|vector]]) [[base measure]] <math>\boldsymbol n=(n_1,\dots,n_K)</math> where {{math|'''n'''}} lies within the {{math|(''K'' − 1)}}-simplex (i.e.: its coordinates <math>n_i</math> sum to one). The concentration parameter in this case is larger by a factor of {{mvar|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 process]]es and is often used in the topic modelling literature.

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:{{note|concentration-parameter-disambiguation}} If we define the concentration parameter as the sum of the Dirichlet parameters for each dimension, the Dirichlet distribution with concentration parameter {{mvar|K}}, the dimension of the distribution, is the uniform distribution on the {{math|(''K'' − 1)}}-simplex.
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