Hierarchy of Log-Cauchy $LC(0,\pi )$

$$\begin{gathered} {\gamma }_i|w_i～Gamma(p_i,w_i) \\ {w}_i|p_i～Gamma(1?p_i,1) \\ {p}_i～U(0,1) \end{gathered}$$

$$\pi ({\gamma }_i,w_i,p_i)=\frac{w_i^{p_i}{\gamma }_i^{p_i?1}{e}^{?w_i{\gamma }_i}}{\Gamma (p_i)}\frac{w_i^{?p_i}{e}^{?w_i}}{\Gamma (1?p_i)}$$

Recall Euler reflection formula:
$$\Gamma (p_i)\Gamma (1?p_i)=\frac{\pi }{\sin (\pi p_i)}$$

And the joint density becomes
$$\begin{gathered} \pi ({\gamma }_i,w_i,p_i)=\frac{{\gamma }_i^{p_i?1}{e}^{?({\gamma }_i+1)w_i}\sin (\pi p_i)}{\pi } \\ \pi ({\gamma }_i,p_i)=\frac{{\gamma }_i^{p_i?1}\sin (\pi p_i)}{\pi ({\gamma }_i+1)} \end{gathered}$$

Evaluate the integral
$$I={\int }_0^1{\gamma }_i^{p_i}\sin (\pi p_i)d{p}_i=\frac{\pi ({\gamma }_i+1)}{{\pi }^2+{\ln }^2({\gamma }_i)}$$

Above
$$\pi ({\gamma }_i)=\frac{1}{{\gamma }_i[{\ln }^2({\gamma }_i)+{\pi }^2]}$$

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