Finite-window point-process pseudolikelihood

Let $W$ and $\mathcal{C}(W)$ be as in Finite-window spatial observations and configurations. Fix an observed configuration $\eta \in \mathcal{C}(W)$ and a boundary convention $b$ (a specified exterior configuration held fixed, or no exterior configuration). When a model supplies a Papangelou conditional intensity ${\lambda }_{\theta }(u;\zeta ,b)$, define its finite-window pseudolikelihood criterion by ${\mathrm{PL}}_{W,b}(\theta ;\eta ):=({\prod }_{x\in \eta }{\lambda }_{\theta }(x;\eta -{\delta }_x,b))\exp \{-{\int }_W{\lambda }_{\theta }(u;\eta ,b){\mathrm{\ell }}_W(du)\}\mathrm{.}$ The finite product counts multiplicities, the empty product is $1$, and one occurrence is deleted in each event factor. The criterion is used only when the displayed intensity values and integral are finite; a zero event factor makes the criterion zero. The exterior pattern $b$ is fixed throughout the calculation. This criterion is not asserted to be a joint likelihood, and its definition supplies no consistency or asymptotic guarantee.