Matrix variate beta distribution

In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the matrix variate beta distribution is a generalization of the <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a>. If 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is a 
 
 
 
 p
 ×
 p
 
 
 {\displaystyle p\times p}
 
 <a href="/facts/Positive_definite_matrix/Hbr6AuPS">positive definite matrix</a> with a matrix variate beta distribution, and 
 
 
 
 a
 ,
 b
 >
 (
 p
 −
 1
 )
 
 /
 
 2
 
 
 {\displaystyle a,b>(p-1)/2}
 
 are real parameters, we write 
 
 
 
 U
 ∼
 
 B
 
 p
 
 
 
 (
 
 a
 ,
 b
 
 )
 
 
 
 {\displaystyle U\sim B_{p}\left(a,b\right)}
 
 (sometimes 
 
 
 
 
 B
 
 p
 
 
 I
 
 
 
 (
 
 a
 ,
 b
 
 )
 
 
 
 {\displaystyle B_{p}^{I}\left(a,b\right)}
 
). The <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> for 
 
 
 
 U
 
 
 {\displaystyle U}
 
 is:

{
            
              
                β
                
                  p
                
              
              
                (
                
                  a
                  ,
                  b
                
                )
              
            
            }
          
          
            −
            1
          
        
        det
        
          
            (
            U
            )
          
          
            a
            −
            (
            p
            +
            1
            )
            
              /
            
            2
          
        
        det
        
          
            (
            
              
                I
                
                  p
                
              
              −
              U
            
            )
          
          
            b
            −
            (
            p
            +
            1
            )
            
              /
            
            2
          
        
        .
      
    
    {\displaystyle \left\{\beta _{p}\left(a,b\right)\right\}^{-1}\det \left(U\right)^{a-(p+1)/2}\det \left(I_{p}-U\right)^{b-(p+1)/2}.}

Here 
 
 
 
 
 β
 
 p
 
 
 
 (
 
 a
 ,
 b
 
 )
 
 
 
 {\displaystyle \beta _{p}\left(a,b\right)}
 
 is the multivariate beta function:

β
          
            p
          
        
        
          (
          
            a
            ,
            b
          
          )
        
        =
        
          
            
              
                Γ
                
                  p
                
              
              
                (
                a
                )
              
              
                Γ
                
                  p
                
              
              
                (
                b
                )
              
            
            
              
                Γ
                
                  p
                
              
              
                (
                
                  a
                  +
                  b
                
                )
              
            
          
        
      
    
    {\displaystyle \beta _{p}\left(a,b\right)={\frac {\Gamma _{p}\left(a\right)\Gamma _{p}\left(b\right)}{\Gamma _{p}\left(a+b\right)}}}

where 
 
 
 
 
 Γ
 
 p
 
 
 
 (
 a
 )
 
 
 
 {\displaystyle \Gamma _{p}\left(a\right)}
 
 is the <a href="/facts/Multivariate_gamma_function/G8ATeYBq">multivariate gamma function</a> given by

Γ
          
            p
          
        
        
          (
          a
          )
        
        =
        
          π
          
            p
            (
            p
            −
            1
            )
            
              /
            
            4
          
        
        
          ∏
          
            i
            =
            1
          
          
            p
          
        
        Γ
        
          (
          
            a
            −
            (
            i
            −
            1
            )
            
              /
            
            2
          
          )
        
        .
      
    
    {\displaystyle \Gamma _{p}\left(a\right)=\pi ^{p(p-1)/4}\prod _{i=1}^{p}\Gamma \left(a-(i-1)/2\right).}

Matrix variate beta distribution open-in-new

Matrix variate beta distribution