User:Gakera/Lipsum

 $$ \begin{align} \mu&|y \sim t_{22}(\mu | 4.1505, 0.0124)\\ \sigma^2&|y \sim \mbox{Inv-}\chi^2(22, 0.2845) \end{align}\,$$

$$ \mu | \sigma^2, y \sim \mbox{N}\left(\bar y, \frac{\sigma^2}{n}\right)

\,$$

$$ \begin{align} \mbox{p}(\alpha, \lambda | y) &\propto \alpha^n\lambda^{\alpha n} \prod_{i=1}^n {y_i}^{\alpha -1} \mbox{exp}\left 		( -(\lambda y_i)^\alpha \right )\\ &\propto \alpha^n\lambda^{\alpha n} \mbox{exp} \left ( \sum_{i=1}^n -(\lambda y_i)^\alpha \right ) \prod_{i=1}^n 		{y_i}^\alpha \end{align} \,$$ $$T(z)=\frac{\alpha z + \beta}{\gamma z + \delta}$$

=hed= $$ \begin{alignat}{2} &J_{\xi,t}(\xi^*|\xi^{t-1})&\qquad&J_{\xi,t}(\xi^{t-1}|\xi^*)\\ &\quad\beta^t = \frac{\xi^{t-1}}{{\tau_\xi}^2}&&\quad\beta^* = \frac{\xi^*}{{\tau_\xi}^2}\\ &\quad\alpha^t = \beta^t\xi^{t-1} &&\quad\alpha^* = \beta^t\xi^* \end{alignat} \,$$

$$ \begin{align} 1 \quad&\mbox{Draw } \xi^* \mbox{ from Gamma}(\xi^*|\alpha^t,\beta^t)\\ 2 \quad&\mbox{Calculate } log(r)\\ &= log\!\left( p(\xi^*|\xi_{-j}^{t-1}, y) \right) - log\!\left(\mbox{Gamma}(\alpha^t,\beta^t)\right)\\ &-log\!\left( p(\xi^{t-1}|\xi_{-j}^{t-1}, y) \right) + log\!\left(\mbox{Gamma}(\alpha^*,\beta^*)\right)\\ 3 \quad&\mbox{Draw }v^t \sim U(0,1)\mbox{ and set}\\ &\xi^t=\xi^* \ \ \, \mbox{ if } v^t < r\\ &\xi^t=\xi^{t-1} \mbox{ if } v^t \ge r \end{align}\,$$

$$ \alpha(1_{N_1})=\pi\circ i(1_{N_1})=\pi(i(1_{N_1}))=\pi(1_N)=\pi(j(1_{M_1}))=0_{M_2} \,$$