I'm trying to approach the problem here, but I think the explanation I provided there is probably confusing, so I'm simplifying the problem so someone can hopefully provide some insight.
Suppose I have a Poisson prior, $Z \sim \texttt{Poisson} (\lambda)$, for some known $\lambda$. Now, given this observed density:
$ P(Y_i = x | \lambda) = \begin{cases} e^{-\lambda}\lambda^x/x! & \text{if $x \neq 3, x \neq 1$} \\ e^{-\lambda}\lambda^1/1! + e^{-\lambda}\lambda^3/3! & \text{if $x = 1$} \\ 0 & \text{if $x = 3$} \\ \end{cases} $
I'm trying to determine (the expectation of) the posterior distribution $P(Z \mid X)$. Is this tractable?