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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.

A conditional likelihood function conditions on an initial observation or another specified statistic. In an autoregressive model, conditioning on the first observed state avoids modeling its initial density. For Gaussian transitions, maximizing this likelihood is a regression problem; treating an autoregressive coefficient as known changes the exact distribution of an estimated mean.

Assume independent individual event times with positive time-constant hazard multipliers and a common baseline hazard whose cumulative hazard tends to infinity. Transforming each event time by that cumulative hazard then gives independent exponential event times with corresponding rates, so all event orders exist. Their complete-order probabilities select each next label proportionally to its remaining multiplier. Marginalizing over the position of a deleted label leaves the order law of the other exponential times, hence the same sequential formula without that label. This proves the deletion identity used when unobserved ranks are summed out. It does not supply a likelihood function for censoring times; ignoring censoring in observed survival analysis still needs independent censoring.

For $Q(\theta \mid {\theta }_0)={\mathbb{E}}_{{\theta }_0}[\log p_{\theta }(Y,Z)\mid Y]$, the Jensen inequality givesThus any expectation-maximization algorithm M-step that increases $Q$ cannot decrease the observed-data likelihood function. Monotonicity does not by itself assert convergence to a global maximum.

For density $r(x-\theta {)}^{r-1}$ on $[\theta ,\theta +1]$, $0<r<1$, the interior likelihood function increases without bound as $\theta$ approaches the sample minimum. The extended-likelihood estimator is that minimum. An arbitrary finite reassignment of the density at the endpoint can eliminate an attained finite maximum, so the endpoint convention must be stated.

The Felsenstein pruning algorithm computes a site likelihood function by sum-product belief propagation from leaves to root. For a finite-state Markov kernel $K$, the subtree likelihood function obeys $L_v(a)={\prod }_{w\text{ child of }v}{\sum }_{b}K(a,b)L_w(b)$, with observed-state indicators at leaves. Summing against the root probability distribution gives the site likelihood function. An outward pass gives edge posterior probabilities and expected transition counts for the expectation-maximization algorithm.

For independent observations $y_s\sim N(\beta +d_s,V_s)$, put $a_s=V_s^{-1}$, $S={\sum }_s{a}_s$, $r_s=y_s-d_s$, $\bar{r}=S^{-1}{\sum }_s{a}_s{r}_s$ and $Q={\sum }_s{a}_s(r_s-\bar{r}{)}^2$. Integrating the likelihood function against a flat improper prior on $\beta$ gives, up to the arbitrary prior constant,Indeed ${\sum }_s{a}_s(r_s-\beta {)}^2=Q+S(\beta -\bar{r}{)}^2$, and the remaining one-dimensional Gaussian integral is $\sqrt{2\pi /S}$. The conditional Bayesian posterior of $\beta$ is $N(\bar{r},S^{-1})$.

A Gaussian likelihood is a likelihood function obtained by modeling the observations conditionally with a normal distribution.

With fixed $X,Z$, compare $N(X\beta ,{\sigma }^{2}I)$ and $N(X\beta ,{\sigma }^{2}I+{\tau }^{2}Z{Z}^T)$ by maximizing the ordinary likelihood function in both models. Their likelihood-ratio test statistic is unchanged by $Y\mapsto Xb+cY$, $c>0$, because both maximized log-likelihoods change by the same $-n\log c$. Its null law can therefore be simulated using independent $N(0,I)$ responses. Comparing the observed statistic with simulated statistics by $(1+\#\{T_b\ge T_{obs}\})/(B+1)$ gives a conservative finite-sample Monte Carlo p-value, subject to correctly maximizing both likelihoods.

For a normal linear model with coefficient vector $\eta$ and residual variance ${\sigma }^2$, a normal-inverse-gamma prior has $\eta \mid {\sigma }^2\sim N(m,{\sigma }^{2}V)$ and ${\sigma }^2\sim \mathrm{IG}(a,b)$, with $V$ a positive-definite matrix and $a,b>0$. The inverse-gamma density is proportional to $({\sigma }^2{)}^{-a-1}{e}^{-b/{\sigma }^2}$. Multiplying by the normal linear model likelihood function and completing the square preserves the family, giving an analytic posterior and evidence integral.

Partial pooling estimates related local parameters jointly through shared hyperparameters. Each local Bayesian posterior balances its own likelihood function against a group prior distribution, with stronger shrinkage when its data are weak or the group variation is small. It lies between forcing all local parameters to be equal and estimating them independently. Random-effects meta-analysis is an important application.