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A hidden object lies in box $i$ with posterior probability $p_i$. Searching box $i$ costs $c_i$ and, conditional on the object being there, detects it with probability ${\alpha }_i$. After an unsuccessful search of box $i$, Bayes theorem updates the probabilities to

For observations $\mathcal{D}$ and graph $G$, Bayes theorem gives $p(G\mid \mathcal{D})\propto \pi (G)\int p(\mathcal{D}\mid {\theta }_G,G)\pi ({\theta }_G\mid G)d{\theta }_G$. Here $\pi (G)$ is the graph prior distribution and ${\theta }_G$ the local distribution statistical parameters. The integral is Bayesian model evidence; it averages over nuisance parameters with a proper statistical parameter prior distribution. For independent complete observations, the node-factorized likelihood function and independence of local statistical parameter prior distributions make the evidence factor over nodes; suitable conjugate priors can make the local integrals analytic. With missing node observations, integrating out unobserved values can couple the local parameters, so independence of local prior distributions alone does not guarantee this factorization. Proper priors and coherent hyperparameters matter for comparing different graphs.

For known size $r$ and ${\lambda }_i=e^{x_i^T\beta }$, the expectation-maximization algorithm gives structural-zero responsibility $t_i=0$ for positive counts and $t_i=\pi /[\pi +(1-\pi )(r/(r+{\lambda }_i){)}^r]$ for zero counts. This is Bayes theorem. Maximizing the expected complete-data log-likelihood gives ${\pi }_{\mathrm{new}}=n^{-1}{\sum }_i{t}_i$ and a weighted negative binomial regression with weights $1-t_i$. The objective separates into a Bernoulli mixing term and a weighted count term, which proves the update.

In a proportional frailty model, Bayes theorem tilts the original density $g(u)$ among subjects surviving to $t$ toThis survival selection progressively favors smaller frailties.