\(\newcommand{\etr}{\textrm{etr}}\)
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 \(U_2 \sim V_2{(I_p+V_2)}^{-1}\).
\[ U = {(W_1+W_2)}^{-\frac12} W_1 {(W_1+W_2)}^{-\frac12}. \]
\[ I_p - U \sim \mathcal{B}I_p^{(1)}(b, a, \Theta_2, \Theta_1). \]
\[ \begin{aligned} \mathcal{B}I_p^{(1)}(U \mid a, b, \Theta_1, \Theta_2) \propto \, & {\det(U)}^{a-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \\ & \int_{S>0} \etr\left(-S\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1S^{\frac12} U S^\frac12\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2S^{\frac12}(I_p-U)S^\frac12\right) \mathrm{d}S. \end{aligned} \]
\[ U = W_1^\frac12{(W_1+W_2)}^{-1}W_1^\frac12. \]
\[ \begin{aligned} \mathcal{B}I_p^{(2)}(U \mid a, b, \Theta_1, \Theta2) \propto \, & {\det(U)}^{-b-\frac12(p+1)} {\det(I_p-U)}^{b-\frac12(p+1)} \\ & \int_{S>0} \etr\left(-S U^{-1}\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1S\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2 S^\frac12 U^{-1}(I_p-U)S^{\frac12}\right)\mathrm{d}S. \end{aligned} \]
If \(\Theta_1\) and \(\Theta_2\) are scalar, it is equal to \(\mathcal{B}I_p^{(1)}(a, b, \Theta_1, \Theta2)\).
\[ V = {(W_2^{-\frac12})}' W_1 W_2^{-\frac12} \]
\[ \begin{aligned} \mathcal{B}II_p^{(1)}(V \mid a, b, \Theta_1, \Theta2) \propto \, & {\det(V)}^{a-\frac12(p+1)} \\ & \int_{S>0} \etr\left((I_p+V)S\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1{S^{\frac12}}' V S^{\frac12}\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2S\right) \mathrm{d}S. \end{aligned} \]
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, \(V{(I_p+V)}^{-1} \sim \mathcal{B}I_p^{(1)}(a,b,\Theta_1,\Theta_2)\).
\[ V = W_1^{\frac12} W_2^{-1} {(W_1^{-\frac12})}'. \]
\[ V^{-1} \sim \mathcal{B}II_p^{(1)}(b,a,\Theta_2,\Theta_1). \]
\[ \begin{aligned} \mathcal{B}II_p^{(2)}(V \mid a, b, \Theta_1, \Theta2) \propto \, & {\det(V)}^{-b-\frac12(p+1)} \\ & \int_{S >0} \etr\left(-S(I_p+V^{-1})\right) {\det(S)}^{a+b-\frac12(p+1)} {}_0\!F_1\left(a, \frac{1}{2}\Theta_1S\right) {}_0\!F_1\left(b, \frac{1}{2}\Theta_2{(S^\frac12)}' V^{-1} S^\frac12\right) \mathrm{d}S. \end{aligned} \]
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 \(V{(I_p+V)}^{-1} \sim \mathcal{B}I_2(a,b,\Theta_1,\Theta_2)\).