Matrix variate Dirichlet distribution

In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the matrix variate Dirichlet distribution is a generalization of the <a href="/facts/Matrix_variate_beta_distribution/KovHm7au">matrix variate beta distribution</a> and of the <a href="/facts/Dirichlet_distribution/Qq13f5g2">Dirichlet distribution</a>.
Suppose 
 
 
 
 
 U
 
 1
 
 
 ,
 …
 ,
 
 U
 
 r
 
 
 
 
 {\displaystyle U_{1},\ldots ,U_{r}}
 
 are 
 
 
 
 p
 ×
 p
 
 
 {\displaystyle p\times p}
 
 <a href="/facts/Positive_definite_matrix/Hbr6AuPS">positive definite matrices</a> with 
 
 
 
 
 I
 
 p
 
 
 −
 
 ∑
 
 i
 =
 1
 
 
 r
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle I_{p}-\sum _{i=1}^{r}U_{i}}
 
 also positive-definite, where 
 
 
 
 
 I
 
 p
 
 
 
 
 {\displaystyle I_{p}}
 
 is the 
 
 
 
 p
 ×
 p
 
 
 {\displaystyle p\times p}
 
 <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. Then we say that the 
 
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{i}}
 
 have a matrix variate Dirichlet distribution, 
 
 
 
 
 (
 
 
 U
 
 1
 
 
 ,
 …
 ,
 
 U
 
 r
 
 
 
 )
 
 ∼
 
 D
 
 p
 
 
 
 (
 
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 r
 
 
 ;
 
 a
 
 r
 +
 1
 
 
 
 )
 
 
 
 {\displaystyle \left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r};a_{r+1}\right)}
 
, if their joint <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> is

{
            
              
                β
                
                  p
                
              
              
                (
                
                  
                    a
                    
                      1
                    
                  
                  ,
                  …
                  ,
                  
                    a
                    
                      r
                    
                  
                  ,
                  
                    a
                    
                      r
                      +
                      1
                    
                  
                
                )
              
            
            }
          
          
            −
            1
          
        
        
          ∏
          
            i
            =
            1
          
          
            r
          
        
        det
        
          
            (
            
              U
              
                i
              
            
            )
          
          
            
              a
              
                i
              
            
            −
            (
            p
            +
            1
            )
            
              /
            
            2
          
        
        det
        
          
            (
            
              
                I
                
                  p
                
              
              −
              
                ∑
                
                  i
                  =
                  1
                
                
                  r
                
              
              
                U
                
                  i
                
              
            
            )
          
          
            
              a
              
                r
                +
                1
              
            
            −
            (
            p
            +
            1
            )
            
              /
            
            2
          
        
      
    
    {\displaystyle \left\{\beta _{p}\left(a_{1},\ldots ,a_{r},a_{r+1}\right)\right\}^{-1}\prod _{i=1}^{r}\det \left(U_{i}\right)^{a_{i}-(p+1)/2}\det \left(I_{p}-\sum _{i=1}^{r}U_{i}\right)^{a_{r+1}-(p+1)/2}}

where 
 
 
 
 
 a
 
 i
 
 
 >
 (
 p
 −
 1
 )
 
 /
 
 2
 ,
 i
 =
 1
 ,
 …
 ,
 r
 +
 1
 
 
 {\displaystyle a_{i}>(p-1)/2,i=1,\ldots ,r+1}
 
 and 
 
 
 
 
 β
 
 p
 
 
 
 (
 ⋯
 )
 
 
 
 {\displaystyle \beta _{p}\left(\cdots \right)}
 
 is the multivariate beta function.
If we write 
 
 
 
 
 U
 
 r
 +
 1
 
 
 =
 
 I
 
 p
 
 
 −
 
 ∑
 
 i
 =
 1
 
 
 r
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{r+1}=I_{p}-\sum _{i=1}^{r}U_{i}}
 
 then the PDF takes the simpler form

{
            
              
                β
                
                  p
                
              
              
                (
                
                  
                    a
                    
                      1
                    
                  
                  ,
                  …
                  ,
                  
                    a
                    
                      r
                      +
                      1
                    
                  
                
                )
              
            
            }
          
          
            −
            1
          
        
        
          ∏
          
            i
            =
            1
          
          
            r
            +
            1
          
        
        det
        
          
            (
            
              U
              
                i
              
            
            )
          
          
            
              a
              
                i
              
            
            −
            (
            p
            +
            1
            )
            
              /
            
            2
          
        
        ,
      
    
    {\displaystyle \left\{\beta _{p}\left(a_{1},\ldots ,a_{r+1}\right)\right\}^{-1}\prod _{i=1}^{r+1}\det \left(U_{i}\right)^{a_{i}-(p+1)/2},}

on the understanding that 
 
 
 
 
 ∑
 
 i
 =
 1
 
 
 r
 +
 1
 
 
 
 U
 
 i
 
 
 =
 
 I
 
 p
 
 
 
 
 {\displaystyle \sum _{i=1}^{r+1}U_{i}=I_{p}}
 
.

Matrix variate Dirichlet distribution open-in-new

Matrix variate Dirichlet distribution