For parameters $(\rho_G,\pi_1,\lambda)$ on $0<\rho_G<1$ and $(\pi_1,\lambda)\in\mathcal T$, conditional independence gives
$$
\begin{aligned}
\pi(\rho_G,\pi_1,\lambda\mid y_G,y_{S1},y_{S2})
\propto{}&
(\lambda\rho_G)^{y_G}(1-\lambda\rho_G)^{N-y_G}\\
&\times\pi_1^{y_{S1}}(1-\pi_1)^{n_{S1}-y_{S1}}\\
&\times(\pi_1+\lambda)^{y_{S2}}
(1-\pi_1-\lambda)^{n_{S2}-y_{S2}}\\
&\times\rho_G^{a-1}(1-\rho_G)^{b-1}
\mathbf1_{\mathcal T}(\pi_1,\lambda).
\end{aligned}
$$
This is the joint posterior up to its normalizing constant.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 207 / 1 / h / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-207/1/h/solution