Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Generalized Dirichlet distribution
open-in-new
<p>In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the generalized Dirichlet distribution (GD) is a generalization of the <a href="/facts/Dirichlet_distribution/Qq13f5g2">Dirichlet distribution</a> with a more general covariance structure and almost twice the number of parameters. Random vectors with a GD distribution are completely <a href="/facts/Neutral_vector/Mpa7aEzJ">neutral</a>. </p><p>The density function of p 1 , … , p k − 1 {\displaystyle p_{1},\ldots ,p_{k-1}} is </p> [ ∏ i = 1 k − 1 B ( a i , b i ) ] − 1 p k b k − 1 − 1 ∏ i = 1 k − 1 [ p i a i − 1 ( ∑ j = i k p j ) b i − 1 − ( a i + b i ) ] {\displaystyle \left[\prod _{i=1}^{k-1}B(a_{i},b_{i})\right]^{-1}p_{k}^{b_{k-1}-1}\prod _{i=1}^{k-1}\left[p_{i}^{a_{i}-1}\left(\sum _{j=i}^{k}p_{j}\right)^{b_{i-1}-(a_{i}+b_{i})}\right]} <p>where we define p k = 1 − ∑ i = 1 k − 1 p i {\textstyle p_{k}=1-\sum _{i=1}^{k-1}p_{i}} . Here B ( x , y ) {\displaystyle B(x,y)} denotes the <a href="/facts/Beta_function/Dcm1JAxb">Beta function</a>. This reduces to the standard Dirichlet distribution if b i − 1 = a i + b i {\displaystyle b_{i-1}=a_{i}+b_{i}} for 2 ⩽ i ⩽ k − 1 {\displaystyle 2\leqslant i\leqslant k-1} ( b 0 {\displaystyle b_{0}} is arbitrary). </p><p>For example, if <i>k=4</i>, then the density function of p 1 , p 2 , p 3 {\displaystyle p_{1},p_{2},p_{3}} is </p> [ ∏ i = 1 3 B ( a i , b i ) ] − 1 p 1 a 1 − 1 p 2 a 2 − 1 p 3 a 3 − 1 p 4 b 3 − 1 ( p 2 + p 3 + p 4 ) b 1 − ( a 2 + b 2 ) ( p 3 + p 4 ) b 2 − ( a 3 + b 3 ) {\displaystyle \left[\prod _{i=1}^{3}B(a_{i},b_{i})\right]^{-1}p_{1}^{a_{1}-1}p_{2}^{a_{2}-1}p_{3}^{a_{3}-1}p_{4}^{b_{3}-1}\left(p_{2}+p_{3}+p_{4}\right)^{b_{1}-\left(a_{2}+b_{2}\right)}\left(p_{3}+p_{4}\right)^{b_{2}-\left(a_{3}+b_{3}\right)}} <p>where p 1 + p 2 + p 3 < 1 {\displaystyle p_{1}+p_{2}+p_{3}<1} and p 4 = 1 − p 1 − p 2 − p 3 {\displaystyle p_{4}=1-p_{1}-p_{2}-p_{3}} . </p><p>Connor and Mosimann define the PDF as they did for the following reason. Define random variables z 1 , … , z k − 1 {\displaystyle z_{1},\ldots ,z_{k-1}} with z 1 = p 1 , z 2 = p 2 / ( 1 − p 1 ) , z 3 = p 3 / ( 1 − ( p 1 + p 2 ) ) , … , z i = p i / ( 1 − ( p 1 + ⋯ + p i − 1 ) ) {\displaystyle z_{1}=p_{1},z_{2}=p_{2}/\left(1-p_{1}\right),z_{3}=p_{3}/\left(1-(p_{1}+p_{2})\right),\ldots ,z_{i}=p_{i}/\left(1-\left(p_{1}+\cdots +p_{i-1}\right)\right)} . Then p 1 , … , p k {\displaystyle p_{1},\ldots ,p_{k}} have the generalized Dirichlet distribution as parametrized above, if the z i {\displaystyle z_{i}} are independent <a href="/facts/Beta_distribution/DbJk4eeV">beta</a> with parameters a i , b i {\displaystyle a_{i},b_{i}} , i = 1 , … , k − 1 {\displaystyle i=1,\ldots ,k-1} . </p>