I encounter this problem in one of my research projects in economic/applied statistics decision making.
Suppose there is a random variable $\omega \in \{1,0\}$ with $\Pr (\omega =1)=q\in (0,1).$ There are $n$ agents each of whom receives a noisy signal $% s_{i}\in \{1,0\}$ about $\omega $. The precision of each signal is $\Pr (s_{i}=\omega |\omega )=\gamma >\frac{1}{2}$. The pairwise conditional correlation between any two signals is $Corr(s_{i}$,$s_{j}|\omega )=\rho \in \lbrack 0,1]$.
My questions are how do I calculate (i) \begin{equation*} \Pr (s_{1}=1,s_{2}=1,...s_{i-1}=1,s_{i+1}=1,...s_{n}=1|s_{i}=0,\omega =1), \end{equation*} i.e., what is the probability that $n-1$ other agents receive $s=1$ conditional on the fact that agent $i$ receives $s_{i}=0$ and the state is $% \omega =1?$
(ii) More generally, what is the probability that $n_{1}$ agents receive $s=1 $ and $n_{2}$ agents receive $s=0$ conditional on the fact that $n_{3}$ agents receive $s=1$ and $n_{4}$ agents receive $s=0$ and the state is $% \omega =1$ such that $n_{1}+n_{2}+n_{3}+n_{4}=n?$
I can solve the problem with $n=2$, that is, for example, $Pr(s_{2}=1|s_{1}=0,w=1)=\gamma(1-\rho)$, but find difficulty from $n=3$ and more. I have also consulted the document "Multivariate Bernoulli Distribution" by Dai et al. However, not much luck yet.
I will appreciate some help.