Introduction
I have been studying the following hierarchical model
$$ X_{i} \sim \begin{cases} \texttt{Po} ( \theta ), & i = 1, \dots , k; \\ \texttt{Po} ( \lambda ), & i = k+1, \dots , n, \end{cases} $$
for the number of disasters in coal mines in the nineteenth century. Furthermore, we assume that
$$\theta \sim \Gamma (a_1 , b_1 ), \ \lambda \sim \Gamma (a_2 , b_2 ), \ k \sim \texttt{U} ( \{ 1 , \dots , n \} )$$
and in turn $b_1 \sim \Gamma ( c_1 , d_1), \ b_2 \sim \Gamma ( c_2 , d_2 )$. We assume the shape and rate Gamma parametrization. The rest of the parameters, i.e., $n, a_1, a_2, c_1, c_2, b_1, b_2$, are known constants. Lastly, we assume that $\theta, \lambda, k$ are independent and that $b_1, b_2$ are independent.
Question
The main interest of this study is to determine the distribution of the following conditional random variable
$$ \theta | \textbf{X}, \lambda , k , b_1 , b_2 $$
and similar random variables for $\lambda, b_1, b_2$ and $k$.
Solution
I have determined that
$$ p_{ \boldsymbol{X} , \lambda , \theta , b_1, b_2, k } ( \boldsymbol{x}, \alpha , \beta, \gamma, \omega, \tau ) = \prod_{i=1}^{\tau} \frac{ \beta^{x_{i}} e^{- \beta} }{x_{i}!} \ \prod_{i= \tau +1}^{n} \frac{ \alpha^{x_{i}} e^{- \alpha} }{x_{i}!} \\ \bigg( \frac{ \omega^{a_2} }{\Gamma (a_2)} \alpha^{a_2 -1} e^{- \alpha \omega} \bigg) \ \bigg( \frac{ \gamma^{a_1} }{\Gamma (a_1)} \beta^{a_1 -1} e^{- \beta \gamma} \bigg) \frac{1}{n} \ \bigg( \frac{ d_1^{c_1} }{\Gamma (c_1)} \gamma^{c_1 -1} e^{- \gamma d_1} \bigg) \ \bigg( \frac{ d_2^{c_2} }{\Gamma (c_2)} \omega^{c_2 -1} e^{- d_2 \omega} \bigg)$$
is the joint probability density function of $( \textbf{X}, \lambda , \theta , k , b_1 , b_2)$. Next, I computed
$$ p_{\theta | \boldsymbol{X}, \lambda , b_1 , b_2 , k} ( \beta | \boldsymbol{x}, \alpha , \gamma , \omega , \tau ) = \frac{ p_{ \boldsymbol{X} , \lambda , \theta , b_1, b_2, k } ( \boldsymbol{x}, \alpha , \beta, \gamma, \omega, \tau ) }{ p_{ \boldsymbol{X}, \lambda , b_1 , b_2 , k } ( \boldsymbol{x}, \alpha , \gamma , \omega , \tau ) }. $$
and then the same for the rest of the parameters. Finally, I have determined that
\begin{equation} \theta | \boldsymbol{X}, \lambda , b_1 , b_2 , k \sim \Gamma \bigg( \sum_{i=1}^{k} X_{i} + a_1 , k + b_1 \bigg) ; \end{equation}
\begin{equation} \lambda | \boldsymbol{X}, \theta , b_1 , b_2 , k \sim \Gamma \bigg( \sum_{i=k+1}^{n} X_{i} + a_2 , n-k+b_2 \bigg) ; \end{equation}
\begin{equation} b_1 | \boldsymbol{X} , \lambda , \theta , b_2 , k \sim \Gamma ( a_1 + c_1, \theta + d_1 ); \end{equation}
\begin{equation} b_2 | \boldsymbol{X} , \lambda , \theta , b_1 , k \sim \Gamma ( a_2 + c_2, \lambda + d_2 ). \end{equation}
However, I do not know whether my approach is correct. I would like for someone to check my computation and perhaps point out my mistakes. I would appreciate any help or advice.