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In statistics , the matrix variate beta distribution is a generalization of the beta distribution . If
U
{\displaystyle U}
p
×
p
{\displaystyle p\times p}
positive definite matrix with a matrix variate beta distribution, and
a
,
b
>
(
p
−
1
)
/
2
{\displaystyle a,b>(p-1)/2}
U
∼
B
p
(
a
,
b
)
{\displaystyle U\sim B_{p}\left(a,b\right)}
B
p
I
(
a
,
b
)
{\displaystyle B_{p}^{I}\left(a,b\right)}
probability density function for
U
{\displaystyle U}
{
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}.}
Matrix variate beta distribution Notation
B
p
(
a
,
b
)
{\displaystyle {\rm {B}}_{p}(a,b)}
Parameters
a
,
b
{\displaystyle a,b}
Support
p
×
p
{\displaystyle p\times p}
U
{\displaystyle U}
I
p
−
U
{\displaystyle I_{p}-U}
positive definite PDF
{
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)}
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)}
multivariate gamma function 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).}
See also
References
A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.
