Theorems Matrix variate beta distribution

1 theorems

1.1 distribution of matrix inverse
1.2 orthogonal transform
1.3 partitioned matrix results
1.4 wishart results

theorems
distribution of matrix inverse

if



u
∼

b

p


(
a
,
b
)


{\displaystyle u\sim b_{p}(a,b)}

density of



x
=

u

−
1




{\displaystyle x=u^{-1}}

given by

1


β

p



(
a
,
b
)




det
(
x

)

−
(
a
+
b
)


det


(
x
−

i

p


)


b
−
(
p
+
1
)

/

2




{\displaystyle {\frac {1}{\beta _{p}\left(a,b\right)}}\det(x)^{-(a+b)}\det \left(x-i_{p}\right)^{b-(p+1)/2}}

provided



x
>

i

p




{\displaystyle x>i_{p}}

,



a
,
b
>
(
p
−
1
)

/

2


{\displaystyle a,b>(p-1)/2}

.

orthogonal transform

if



u
∼

b

p


(
a
,
b
)


{\displaystyle u\sim b_{p}(a,b)}

,



h


{\displaystyle h}

constant



p
×
p


{\displaystyle p\times p}

orthogonal matrix,



h
u

h

t


∼
b
(
a
,
b
)
.


{\displaystyle huh^{t}\sim b(a,b).}

also, if



h


{\displaystyle h}

random orthogonal



p
×
p


{\displaystyle p\times p}

matrix independent of



u


{\displaystyle u}

,



h
u

h

t


∼

b

p


(
a
,
b
)


{\displaystyle huh^{t}\sim b_{p}(a,b)}

, distributed independently of



h


{\displaystyle h}

.

if



a


{\displaystyle a}

constant



q
×
p


{\displaystyle q\times p}

,



q
≤
p


{\displaystyle q\leq p}

matrix of rank



q


{\displaystyle q}

,



a
u

a

t




{\displaystyle aua^{t}}

has generalized matrix variate beta distribution,



a
u

a

t


∼
g

b

q



(
a
,
b
;
a

a

t


,
0
)



{\displaystyle aua^{t}\sim gb_{q}\left(a,b;aa^{t},0\right)}

.

partitioned matrix results

if



u
∼

b

p



(
a
,
b
)



{\displaystyle u\sim b_{p}\left(a,b\right)}

, partition



u


{\displaystyle u}

as

u
=


[




u

11





u

12







u

21





u

22





]




{\displaystyle u={\begin{bmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\end{bmatrix}}}

where




u

11




{\displaystyle u_{11}}






p

1


×

p

1




{\displaystyle p_{1}\times p_{1}}

,




u

22




{\displaystyle u_{22}}






p

2


×

p

2




{\displaystyle p_{2}\times p_{2}}

, defining schur complement




u

22
⋅
1




{\displaystyle u_{22\cdot 1}}






u

22


−

u

21





u

11




−
1



u

12




{\displaystyle u_{22}-u_{21}{u_{11}}^{-1}u_{12}}

gives following results:

u

11




{\displaystyle u_{11}}

independent of




u

22
⋅
1




{\displaystyle u_{22\cdot 1}}







u

11


∼

b


p

1





(
a
,
b
)



{\displaystyle u_{11}\sim b_{p_{1}}\left(a,b\right)}







u

22
⋅
1


∼

b


p

2





(
a
−

p

1



/

2
,
b
)



{\displaystyle u_{22\cdot 1}\sim b_{p_{2}}\left(a-p_{1}/2,b\right)}







u

21


∣

u

11


,

u

22
⋅
1




{\displaystyle u_{21}\mid u_{11},u_{22\cdot 1}}

has inverted matrix variate t distribution,




u

21


∣

u

11


,

u

22
⋅
1


∼
i

t


p

2


,

p

1





(
2
b
−
p
+
1
,
0
,

i


p

2




−

u

22
⋅
1


,

u

11


(

i


p

1




−

u

11


)
)

.


{\displaystyle u_{21}\mid u_{11},u_{22\cdot 1}\sim it_{p_{2},p_{1}}\left(2b-p+1,0,i_{p_{2}}-u_{22\cdot 1},u_{11}(i_{p_{1}}-u_{11})\right).}



wishart results

mitra proves following theorem illustrates useful property of matrix variate beta distribution. suppose




s

1


,

s

2




{\displaystyle s_{1},s_{2}}

independent wishart



p
×
p


{\displaystyle p\times p}

matrices




s

1


∼

w

p


(

n

1


,
Σ
)
,

s

2


∼

w

p


(

n

2


,
Σ
)


{\displaystyle s_{1}\sim w_{p}(n_{1},\sigma ),s_{2}\sim w_{p}(n_{2},\sigma )}

. assume



Σ


{\displaystyle \sigma }

positive definite ,




n

1


+

n

2


≥
p


{\displaystyle n_{1}+n_{2}\geq p}

. if

u
=

s

−
1

/

2



s

1




(

s

−
1

/

2


)


t


,


{\displaystyle u=s^{-1/2}s_{1}\left(s^{-1/2}\right)^{t},}

where



s
=

s

1


+

s

2




{\displaystyle s=s_{1}+s_{2}}

,



u


{\displaystyle u}

has matrix variate beta distribution




b

p


(

n

1



/

2
,

n

2



/

2
)


{\displaystyle b_{p}(n_{1}/2,n_{2}/2)}

. in particular,



u


{\displaystyle u}

independent of



Σ


{\displaystyle \sigma }

.