DirichletDistribution

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`DirichletDistribution`

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DirichletDistribution[{α₁,…,α_k+1}]

represents a Dirichlet distribution of dimension k with shape parameters α_i.

Details

DirichletDistribution is also known as multivariate beta distribution.
The probability density for vector in a Dirichlet distribution is proportional to for and .
DirichletDistribution allows α_i to be any positive real number.
DirichletDistribution allows α_i to be a dimensionless quantity.
DirichletDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context

DirichletDistribution[{α₁,…,α_k+1}] represents a continuous multivariate statistical distribution defined over k-tuples of positive real numbers whose sum is less than one and for which . The marginal distributions (MarginalDistribution) of DirichletDistribution[{α₁,…,α_k+1}] are all versions of BetaDistribution parametrized by various linear combinations of the components α_i. As a result, the Dirichlet distribution is parametrized by a -tuple (α₁,…,α_k+1) whose components α_i are all positive real numbers that together determine the overall shape of its probability density function (PDF). The Dirichlet distribution is sometimes referred to as the multivariate beta distribution.
The PDF of a Dirichlet distribution has an absolute maximum at its mean, i.e. at the k-vector whose components are the one-variable means of its univariate marginal PDF. The tails of each of the associated marginal PDFs corresponding to a Dirichlet distribution are "fat," in the sense that the marginal PDF decreases algebraically for large values of rather than decreasing exponentially. (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of these distributions.)
The Dirichlet distribution is named for German mathematician Johann Dirichlet. It has been used as a tool for modeling both real-world and theoretical phenomena since its formal introduction by Thomas Ferguson in the 1970s. The Dirichlet distribution is a finite version of the more general Dirichlet process, an infinite-dimensional stochastic process that, roughly speaking, assigns a probability distribution to each of a collection of probability distributions according to a specific algorithm. Similarly, the Dirichlet distribution often occurs as a prior distribution in Bayesian statistics, including as the conjugate prior for MultinomialDistribution. One interesting application of the Dirichlet distribution is in modeling randomness among accuracies of (fair) dice manufactured over time, noting that dice manufactured more recently tend to be "fairer" because of more precise manufacturing processes. The Dirichlet distribution has also been applied to data mining, machine learning, and computer vision. More recently, the Dirichlet distribution has been used to model the frequencies of words across collections of texts.
RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Dirichlet distribution. Distributed[{x₁,…,x_k},DirichletDistribution[{α₁,…,α_k+1}]], written more concisely as {x₁,…,x_k}DirichletDistribution[{α₁,…,α_k+1}], can be used to assert that a k-tuple (x₁,…,x_k) of random variables is distributed according to a Dirichlet distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
The probability density and cumulative distribution functions may be given using PDF[DirichletDistribution[{α₁,…,α_k+1}],{x₁,…,x_k}] and CDF[DirichletDistribution[{α₁,…,α_k+1}],{x₁,…,x_k}]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively. Moreover, Covariance can be used to compute the covariance matrix corresponding to DirichletDistribution.
DistributionFitTest can be used to test if a given dataset is consistent with a Dirichlet distribution, EstimatedDistribution to estimate a Dirichlet parametric distribution from given data, and FindDistributionParameters to fit data to a Dirichlet distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Dirichlet distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Dirichlet distribution.
TransformedDistribution can be used to represent a transformed Dirichlet distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Dirichlet distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Dirichlet distributions.
DirichletDistribution is closely related to a number of other distributions. As noted previously, DirichletDistribution has close ties with BetaDistribution, in the sense that each of its marginal distributions is a beta distribution. Moreover, any one-dimensional Dirichlet distribution is a beta distribution, in the sense that the PDF of DirichletDistribution[{α₁,α₂}] is precisely that of BetaDistribution[α₁,α₂]. DirichletDistribution is also related to GammaDistribution, in the sense that the random variable is distributed according to DirichletDistribution whenever is the gamma-distributed random variable defined by the sum of k independently distributed gamma variates Y₁,…,Y_k. DirichletDistribution is the conjugate prior of MultinomialDistribution and is also related to BetaPrimeDistribution, CompoundPoissonDistribution, and MultinormalDistribution.

Examples

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Basic Examples (4)Summary of the most common use cases

Probability density function in two dimensions:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-1u74ju

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-d1z75u

Out[2]=2

Cumulative distribution function in two dimensions:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-sx1aea

Out[1]=1

Mean and variance in two dimensions:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-f4s5xv

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-d6xfm7

Out[2]=2

Covariance:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-yzb1ai

Scope (8)Survey of the scope of standard use cases

Generate a sample of pseudorandom vectors from a bivariate Dirichlet distribution:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-ljmvtp

Visualize the sample using a histogram:

In[2]:=2

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-2ycpd9

Out[2]=2

Distribution parameters estimation:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-oc5luj

Estimate the distribution parameters from sample data:

In[2]:=2

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-t4a7qy

Out[2]=2

Goodness-of-fit test:

In[3]:=3

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-drlyn5

Out[3]=3

Skewness:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-8f9y3n

Out[1]=1

Kurtosis:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-1x45sw

Out[1]=1

Correlation:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-1zadqu

Different mixed moments for a Dirichlet distribution:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-slatx7

In[2]:=2

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-2dtdhm

Out[2]=2

Closed form for a symbolic order:

In[3]:=3

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-mtnuyw

Out[3]=3

Mixed central moments:

In[4]:=4

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-s6xlj5

Out[4]=4

Mixed factorial moments:

In[5]:=5

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-igch4a

Out[5]=5

Mixed cumulants:

In[6]:=6

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-oedqsj

Out[6]=6

Hazard function:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-ny3l9w

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-l28ifm

Out[2]=2

Univariate marginals of a Dirichlet distribution follow a BetaDistribution:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-dulj1k

Out[1]=1

Multivariate marginals follow a DirichletDistribution:

In[2]:=2

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-s6veka

Out[2]=2

Applications (5)Sample problems that can be solved with this function

Show a distribution function and its histogram in the same plot:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-ig1qzy

Compare the PDF to its histogram version:

In[2]:=2

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-h0i29c

In[3]:=3

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-ba9s1s

Out[3]=3

Compare the CDF to its histogram version:

In[4]:=4

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-5806k

Out[4]=4

Simulate points on the half-plane with mean :

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-zksto7

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-9y1uk

Out[2]=2

For α3=30:

In[3]:=3

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-v80zz

Out[3]=3

For α3=10:

In[4]:=4

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-zkk3u8

Out[4]=4

The point spread can be controlled by the third parameter:

In[5]:=5

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-rjxyke

Out[5]=5

Use Dirichlet distribution to define a multivariate Pólya distribution as a parameter mixture:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-6xbnwv

Probability density function:

In[2]:=2

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-fxqn5i

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-h9zw6b

Out[3]=3

Find the probability over a Disk for a Dirichlet distribution:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-3077ca

Out[1]=1

Estimate confidence interval for maximum likelihood estimates of distribution parameters:

In[1]:=1

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-fr0igp

In[2]:=2

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https://wolfram.com/xid/0bmuk6a7o0psea8triy-c0fwx

Out[2]=2

Apply fractional random weight bootstrap to estimate confidence interval, by repeating weighted estimation with weights sampled from a DirichletDistribution with unit parameters:

In[3]:=3