Matrix variate Beta distributions"

In matrixsampling: Simulations of Matrix Variate Distributions

$\newcommand{\etr}{\textrm{etr}}$

Two definitions of the matrix variate Beta type I distribution were proposed. We will denote them by $\mathcal{B}I_p^{(1)}(a,b,\Theta_1,\Theta_2)$ and $\mathcal{B}I_p^{(2)}(a,b,\Theta_1,\Theta_2)$, where $\Theta_1$ and $\Theta_2$ are the noncentrality parameters. Take two independent Wishart random matrices $W_1 \sim \mathcal{W}_p(2a, I_p, \Theta_1)$ and $W_2 \sim \mathcal{W}_p(2a, I_p, \Theta_2)$. Then $\mathcal{B}I_p^{(1)}(a,b,\Theta_1,\Theta_2)$ is the distribution of $$ U_1 = {(W_1+W_2)}^{-\frac12}W_1{(W_1+W_2)}^{-\frac12}, $$ while $\mathcal{B}I_p^{(2)}(a,b,\Theta_1,\Theta_2)$ is the distribution of $$ U_2 = W_1^\frac12{(W_1+W_2)}^{-1}W_1^\frac12. $$ In the central case, i.e. when both $\Theta_1$ and $\Theta_2$ are the null matrices, these two distributions are the same.

Similarly, two definitions of the matrix variate Beta type II distribution were proposed. We will denote them by $\mathcal{B}II_p^{(1)}(a,b,\Theta_1,\Theta_2)$ and $\mathcal{B}II_p^{(2)}(a,b,\Theta_1,\Theta_2)$. The first one is the distribution of $$ V_1 = W_2^{-\frac12} W_1 W_2^{-\frac12}, $$ while the second one is the distribution of $$ V_2 = W_1^\frac12 {W_2}^{-1} W_1^\frac12. $$ Similarly to the type I, these two distributions are the same in the central case.

Under the second definition, the Beta type I distribution is related to the Beta type II distribution by $U_2 \sim V_2{(I_p+V_2)}^{-1}$.

$\mathcal{B}I_p^{(1)}(a, b, \Theta_1, \Theta_2)$

$$ U = {(W_1+W_2)}^{-\frac12} W_1 {(W_1+W_2)}^{-\frac12}. $$

$$ I_p - U \sim \mathcal{B}I_p^{(1)}(b, a, \Theta_2, \Theta_1). $$

$$ \begin{aligned} \mathcal{B}I_p^{(1)}(U \mid a, b, \Theta_1, \Theta_2) \propto \, & {\det(U)}^{a-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \ & \int_{S>0} \etr\left(-S\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0!F_1\left(a, \frac{1}{2}\Theta_1S^{\frac12} U S^\frac12\right) {}_0!F_1\left(b, \frac{1}{2}\Theta_2S^{\frac12}(I_p-U)S^\frac12\right) \mathrm{d}S. \end{aligned} $$

• If $\Theta_1$ and $\Theta_2$ are scalar: $$ \begin{aligned} \mathcal{B}I_p^{(1)}(U \mid a, b, \theta_1 I_p, \theta_2 I_p) \propto\, & {\det(U)}^{a-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \ & \int_{S>0} \etr\left(-S\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0!F_1\left(a, \frac{\theta_1}{2}SU\right) {}_0!F_1\left(b, \frac{\theta_2}{2}S(I_p-U)\right) \mathrm{d}S. \end{aligned} $$

$\mathcal{B}I_p^{(2)}(a, b, \Theta_1, \Theta_2)$

$$ U = W_1^\frac12{(W_1+W_2)}^{-1}W_1^\frac12. $$

$$ \begin{aligned} \mathcal{B}I_p^{(2)}(U \mid a, b, \Theta_1, \Theta2) \propto \, & {\det(U)}^{-b-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \ & \int_{S>0} \etr\left(-S U^{-1}\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0!F_1\left(a, \frac{1}{2}\Theta_1S\right) {}_0!F_1\left(b, \frac{1}{2}\Theta_2 S^\frac12 U^{-1}(I_p-U)S^{\frac12}\right)\mathrm{d}S. \end{aligned} $$

If $\Theta_1$ and $\Theta_2$ are scalar, it is equal to $\mathcal{B}I_p^{(1)}(a, b, \Theta_1, \Theta2)$.

$\mathcal{B}II_p^{(1)}(a,b,\Theta_1,\Theta_2)$

$$ V = {(W_2^{-\frac12})}' W_1 W_2^{-\frac12} $$

$$ \begin{aligned} \mathcal{B}II_p^{(1)}(V \mid a, b, \Theta_1, \Theta2) \propto \, & {\det(V)}^{a-\frac12(p+1)} \ & \int_{S>0} \etr\left((I_p+V)S\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0!F_1\left(a, \frac{1}{2}\Theta_1{S^{\frac12}}' V S^{\frac12}\right) {}_0!F_1\left(b, \frac{1}{2}\Theta_2S\right) \mathrm{d}S. \end{aligned} $$

If $\Theta_1$ is scalar, the distribution does not depend on the choice of $W_1^\frac12$.

If $\Theta_1$ and $\Theta_2$ are scalar, $V{(I_p+V)}^{-1} \sim \mathcal{B}I_p^{(1)}(a,b,\Theta_1,\Theta_2)$.

$\mathcal{B}II_p^{(2)}(a,b,\Theta_1,\Theta_2)$

$$ V = W_1^{\frac12} W_2^{-1} {(W_1^{-\frac12})}'. $$

$$ V^{-1} \sim \mathcal{B}II_p^{(1)}(b,a,\Theta_2,\Theta_1). $$

$$ \begin{aligned} \mathcal{B}II_p^{(2)}(V \mid a, b, \Theta_1, \Theta2) \propto \, & {\det(V)}^{-b-\frac12(p+1)} \ & \int_{S >0} \etr\left(-S(I_p+V^{-1})\right) {\det(S)}^{a+b-\frac12(p+1)}
{}_0!F_1\left(a, \frac{1}{2}\Theta_1S\right) {}_0!F_1\left(b, \frac{1}{2}\Theta_2{(S^\frac12)}' V^{-1} S^\frac12\right) \mathrm{d}S. \end{aligned} $$

If $\Theta_2$ is scalar, the distribution does not depend on the choice of $W_1^\frac12$.

If we take $W_1^{\frac12}$ the symmetric square root of $W_1$, then $V{(I_p+V)}^{-1} \sim \mathcal{B}I_2(a,b,\Theta_1,\Theta_2)$.

matrixsampling documentation built on Aug. 25, 2019, 1:03 a.m.