 Solution

The prior factors involving $s_i$ are $\exp (-J{s}_i{s}_j)$ for $j\in {\mathcal{N}}_i$. A likelihood factor centered at $k$ depends on $s_i$ precisely when $i\in {\mathcal{N}}_k$, equivalently $k\in {\mathcal{N}}_i$. Those factors areOnly incident prior edges and likelihoods centered at neighboring sites are affected. In particular, the likelihood centered at $i$ does not depend on $s_i$: its mean involves only the neighbors of $i$. These are the local likelihood factors in a hidden spin field. The resulting conditional dependence can extend two lattice steps, beyond the nearest-neighbor prior interactions. For $K=1$, both collections are empty.