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Dirichlet distribution

In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted Dir ⁡ ( α ) {\displaystyle \operatorname {Dir} ({\boldsymbol {\alpha }})} , is a family of continuous multivariate probability distributions parameterized by a vector α of positive reals. It is a multivariate generalization of the beta…

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Entropy
H ( X ) = log ⁡ B ( α ) {\displaystyle H(X)=\log \mathrm {B} ({\boldsymbol {\alpha }})} + ( α 0 − K ) ψ ( α 0 ) − {\displaystyle +(\alpha _{0}-K)\psi (\alpha _{0})-} ∑ j = 1 K (…
Mean
E ⁡ [ X i ] = α i α 0 {\displaystyle \operatorname {E} [X_{i}]={\frac {\alpha _{i}}{\alpha _{0}}}} E ⁡ [ ln ⁡ X i ] = ψ ( α i ) − ψ ( α 0 ) {\displaystyle \operatorname {E} [\ln…
Method of moments
α i = E [ X i ] ( E [ X j ] ( 1 − E [ X j ] ) V [ X j ] − 1 ) {\displaystyle \alpha _{i}=E[X_{i}]\left({\frac {E[X_{j}](1-E[X_{j}])}{V[X_{j}]}}-1\right)} where j is any index, p…
Mode
x i = α i − 1 α 0 − K , α i > 1. {\displaystyle x_{i}={\frac {\alpha _{i}-1}{\alpha _{0}-K}},\quad \alpha _{i}>1.}
Parameters
K ≥ 2 {\displaystyle K\geq 2} number of categories (integer) α = ( α 1 , … , α K ) {\displaystyle {\boldsymbol {\alpha }}=(\alpha _{1},\ldots ,\alpha _{K})} concentration parame…
PDF
1 B ( α ) ∏ i = 1 K x i α i − 1 {\displaystyle {\frac {1}{\mathrm {B} ({\boldsymbol {\alpha }})}}\prod _{i=1}^{K}x_{i}^{\alpha _{i}-1}} where B ( α ) = ∏ i = 1 K Γ ( α i ) Γ ( α…

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Dirichlet distribution

Nodes100
Edges99
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Avg. degree1.98
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Dirichlet distribution

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related to External links · 17
Dirichlet distribution → Compound Poisson Random Variables, Construction, Dirichlet, Dirichlet DistributionHow, Dirichlet Random Measures, EM, EMS Press, Encyclopedia, Exchangeability Properties, Gamma DistributionSciencesPo, Luc Devroye, Mathematics, Method, Non-Uniform Random Variate Generation, October, Pólya, Retrieved
related to Bayesian models · 10
Dirichlet distribution → Bayesian, Bernoulli, Dirichlet, Dirichlet-multinomial, Gibbs, In, Inference, One, Such, This
related to With specified expectations · 10
Dirichlet distribution → Beta, Dirichlet, For, In, Instead, Jeffreys, K/2, The, This, When
related to Characteristic function · 8
Dirichlet distribution → CF, Dirichlet, It, K-1, Lauricella, Phillips, Psi, The
related to Support · 8
Dirichlet distribution → Another, Dirichlet, For, K-dimensional, K-dimensional Dirichlet, K-way, The, These
is a · 6
Dirichlet distribution → confluent form of the Lauricella hypergeometric series, conjugate prior distribution of the categorical distribution, conjugate prior of the categorical distribution and multinomial distribution.The infinite-dimensional generalization of the Dirichlet distribution is the Dirichlet process, exponential family distribution it has a conjugate prior, open standard, set of K-dimensional vectors x whose entries are real numbers in the interval
related to Conjugate to categorical or multinomial · 6
Dirichlet distribution → Dirichlet, Formally, Given, Intuitively, The Dirichlet, This
related to From gamma distribution · 5
Dirichlet distribution → First, Gamma, Gamma-distributed, K-dimensional Dirichlet, With
related to Inequality · 5
Dirichlet distribution → Another, Dirichlet, K-1, Kullback-Leibler, Probability
related to Kullback–Leibler divergence · 5
Dirichlet distribution → Dir, Dirichlet, KL, Leibler, The Kullback

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displaystyle distribution dirichlet alpha operatorname sum frac boldsymbol ldots gamma prior left right function distributions random concentration k-1 parameter probability

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SubjectPredicateObjectConfidenceSrc
Dirichlet distributionEntropyH ( X ) = log ⁡ B ( α ) {\displaystyle H(X)=\log \mathrm {B} ({\boldsymbol {\alpha }})} + ( α 0 − K ) ψ ( α 0 ) − {\displaystyle +(\alpha _{0}-K)\psi (\alpha _{0})-} ∑ j = 1 K (…1.00infobox
Dirichlet distributionMeanE ⁡ [ X i ] = α i α 0 {\displaystyle \operatorname {E} [X_{i}]={\frac {\alpha _{i}}{\alpha _{0}}}} E ⁡ [ ln ⁡ X i ] = ψ ( α i ) − ψ ( α 0 ) {\displaystyle \operatorname {E} [\ln…1.00infobox
Dirichlet distributionMethod of momentsα i = E [ X i ] ( E [ X j ] ( 1 − E [ X j ] ) V [ X j ] − 1 ) {\displaystyle \alpha _{i}=E[X_{i}]\left({\frac {E[X_{j}](1-E[X_{j}])}{V[X_{j}]}}-1\right)} where j is any index, p…1.00infobox
Dirichlet distributionModex i = α i − 1 α 0 − K , α i > 1. {\displaystyle x_{i}={\frac {\alpha _{i}-1}{\alpha _{0}-K}},\quad \alpha _{i}>1.}1.00infobox
Dirichlet distributionParametersK ≥ 2 {\displaystyle K\geq 2} number of categories (integer) α = ( α 1 , … , α K ) {\displaystyle {\boldsymbol {\alpha }}=(\alpha _{1},\ldots ,\alpha _{K})} concentration parame…1.00infobox
Dirichlet distributionPDF1 B ( α ) ∏ i = 1 K x i α i − 1 {\displaystyle {\frac {1}{\mathrm {B} ({\boldsymbol {\alpha }})}}\prod _{i=1}^{K}x_{i}^{\alpha _{i}-1}} where B ( α ) = ∏ i = 1 K Γ ( α i ) Γ ( α…1.00infobox
Dirichlet distributionSupportx 1 , … , x K {\displaystyle x_{1},\ldots ,x_{K}} where x i ∈ [ 0 , 1 ] {\displaystyle x_{i}\in [0,1]} and ∑ i = 1 K x i = 1 {\displaystyle \sum _{i=1}^{K}x_{i}=1} (i.e. a K − 1…1.00infobox
Dirichlet distributionVarianceVar ⁡ [ X i ] = α ~ i ( 1 − α ~ i ) α 0 + 1 , {\displaystyle \operatorname {Var} [X_{i}]={\frac {{\tilde {\alpha }}_{i}(1-{\tilde {\alpha }}_{i})}{\alpha _{0}+1}},} Cov ⁡ [ X i…1.00infobox
Dirichlet distributionis aconjugate prior of the categorical distribution and multinomial distribution.The infinite-dimensional generalization of the Dirichlet distribution is the Dirichlet process0.90text
Dirichlet distributionis aset of K-dimensional vectors x whose entries are real numbers in the interval0.90text
Dirichlet distributionis aopen standard0.90text
Dirichlet distributionis aconjugate prior distribution of the categorical distribution0.90text
Dirichlet distributionis aconfluent form of the Lauricella hypergeometric series0.90text
Dirichlet distributionis aexponential family distribution it has a conjugate prior0.90text

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