Finite-volume Gibbs pseudo-likelihood with fixed boundary

Let $d\ge 1$, let $\mathrm{\Lambda }$ be a finite nonempty subset of ${\mathbb{Z}}^d$, let $S$ be a finite nonempty spin set, let $(\mathrm{\Theta },\mathcal{T})$ be a nonempty measurable parameter space, and fix a boundary configuration $\eta \in S^{{\mathbb{Z}}^d\setminus \mathrm{\Lambda }}$. Let $\mathcal{C}$ be a finite collection of finite subsets $C\subseteq {\mathbb{Z}}^d$ with $C\cap \mathrm{\Lambda }\mathrm{\ne }\mathrm{\varnothing }$. For each $C\in \mathcal{C}$, let ${\mathrm{\Phi }}_C:S^C\times \mathrm{\Theta }\to \mathbb{R}$ be finite-valued and measurable in $\theta$ for each spin assignment. For $\sigma \in S^{\mathrm{\Lambda }}$, let $\bar{\sigma }$ equal $\sigma$ on $\mathrm{\Lambda }$ and $\eta$ outside $\mathrm{\Lambda }$, and define the finite-volume energy $H_{\theta }^{\eta }(\sigma ):={\sum }_{C\in \mathcal{C}}{\mathrm{\Phi }}_C(\bar{\sigma }{\mathrm{\mid }}_C;\theta )\mathrm{.}$ Its normalized Gibbs law is ${\pi }_{\theta }^{\eta }(\sigma ):=\frac{e^{-H_{\theta }^{\eta }(\sigma )}}{{\sum }_{\tau \in S^{\mathrm{\Lambda }}}{e}^{-H_{\theta }^{\eta }(\tau )}}\mathrm{.}$ For site $i\in \mathrm{\Lambda }$, define its finite interaction neighborhood $N_i:=\left({\bigcup }_{\begin{array}{c}C\in \mathcal{C}\\ i\in C\end{array}}C\right)\setminus \{i\}\mathrm{.}$ For $\sigma \in S^{\mathrm{\Lambda }}$ and $s\in S$, write ${\sigma }^{i=s}$ for the configuration obtained by replacing only its $i$th spin with $s$. The one-site conditional and finite-volume pseudo-likelihood are $q_{i,\theta }^{\eta }(s\mid {\sigma }_{\mathrm{\Lambda }\setminus \{i\}}):=\frac{e^{-H_{\theta }^{\eta }({\sigma }^{i=s})}}{{\sum }_{a\in S}{e}^{-H_{\theta }^{\eta }({\sigma }^{i=a})}},P{L}_{\theta }^{\eta }(\sigma ):={\prod }_{i\in \mathrm{\Lambda }}{q}_{i,\theta }^{\eta }({\sigma }_i\mid {\sigma }_{\mathrm{\Lambda }\setminus \{i\}})\mathrm{.}$ Every conditional uses the same fixed exterior boundary and is determined by the specified finite neighborhood $N_i$. The product is a finite conditional objective; it is not defined as a probability law on $S^{\mathrm{\Lambda }}$.

The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$

The exponential function is strictly increasing

The real exponential function and the number $e$ by a power series

The product in this Definition is a pseudo-likelihood in the sense of Besag: it multiplies one-site conditional probabilities with the boundary and neighborhood fixed. It generally differs from the joint Gibbs likelihood and does not define an exact posterior for the joint Gibbs observation model. The B-page counterexample Two-spin pseudo-likelihood is not the exact joint posterior input ↗ verifies this distinction in a two-spin model. A consistency or concentration theorem for pseudo-likelihood would require additional, explicitly proved identifiability and spatial mixing or ergodicity inputs; none is asserted here.