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On a graph, a binary configuration has probability proportional to $\exp (\alpha {\sum }_v{x}_v+\beta {\sum }_{\{v,w\}}{x}_v{x}_w)$, with each edge counted once. The full conditional probability of a one is logistic with linear predictor $\alpha +\beta {\sum }_{w\sim v}{x}_w$. Local neighbor sums eliminate the need to evaluate the normalizing constant.

A Bayesian network consists of a Directed acyclic graph $G$ whose nodes are random variables, together with conditional distributions satisfying $p(x)={\prod }_{j}p(x_j\mid x_{{\mathrm{pa}}_G(j)})$. Its Directed acyclic graph encodes conditional independence constraints. For unrestricted binary variables it has ${\sum }_j{2}^{|{\mathrm{pa}}_G(j)|}$ free conditional probabilities. A Directed acyclic graph used this way need not have a causal interpretation.

Classification in statistical learning with two possible labels. Given the conditional probability $\eta (x)=\mathbb{P}(Y=1\mid X=x)$, a Bayes classifier under zero-one loss predicts one exactly when $\eta (x)\ge 1/2$, with arbitrary decisions at ties.

When all conditioning events have positive probability, repeatedly apply the definition of conditional probability to factor a joint event into successive conditional factors. For a sequence of discrete responses, this yields the product of conditional masses along the actual outcome path. It requires no unconditional independence assumption.

Independent unit-variance Gaussian observations with mean $x_v$ add $y_v-1/2$ to the local binary field. The posterior distribution conditional probability is $\mathrm{logistic}(\alpha +\beta z_v+y_v-1/2)$. The neighbor interaction is unchanged, so checkerboard conditional independence and the same Gibbs schemes remain valid.

When a response $Y$ is missing at random conditional on a fully observed category $X$, estimate its conditional probabilities using respondents within each category and average them with the category frequencies from all patients. For a binary response, $\hat{P}(Y=1)={\sum }_x\hat{P}(X=x)\hat{P}(Y=1\mid X=x,R=1)$. The weighting uses the full cohort rather than only complete cases.