I encountered this question while reading about MCMC methods to solve image reconstruction problems.
Consider a black and white image where $-1$ corresponds to white and $+1$ to black. $X_{i,j}$ represents the observed image with $X_{i,j} \in \{-1, +1\}$ representing a pixel at row $i$ and column $j$. The true/unknown image, $Y_{i,j}$ with $Y_{i,j} \in \{-1, +1\}$. $Y_{i,j}$ has four immediate neighbors. E is the corresponding set of edges of the neighbors
Given the following information, how would one derive the conditional probability that a pixel $(i, j)$ is black given its Markov blanket, $P(Y_{i,j}=1|Y_{M(i,j)})$? $Y_{M(i, j)}$ represents the variables in the blanket.
Joint: $$P(y, x) = \frac{1}{Z}\{ \prod_{i=1}^N \prod_{j=1}^M exp^{\eta y_{i,j} x_{i,j}} \}{\prod_{((i,j),(i',j')) \in E} \{ exp^{\beta y_{i,j} y_{i',j'}} \} }$$
Normalizing Factor: $$z = \sum_{y,x} exp \{E(y,x) \}$$ where $E(y,x)$ is the Boltzmann distribution.