Likelihood function ($L(\theta ;x)$)

For observed data $x$, the likelihood function is the joint probability mass or density $p_{\theta }(x)$ regarded as a function of the model parameter $\theta$.

An observed-data likelihood integrates the full-data statistical probability density over unobserved values, rather than inserting arbitrary values or discarding incomplete records. When the observation pattern is random, its mechanism belongs to the joint observation model unless the conditions for an ignorable missingness mechanism permit its factor to be omitted.

A Bayesian deviance is minus twice the log-likelihood, with any chosen additive data-only constant held consistent across the models being compared. Its Bayesian posterior expectation measures average fit. In the deviance information criterion, evaluating it at a parameter's posterior mean also enters the effective complexity penalty. This likelihood-based convention differs by a data-only constant from a saturated-model exponential-family deviance when a common saturated model exists.

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.

For a parameter of interest $\psi$ and nuisance parameter $\lambda$, the profile likelihood isIt compares values of $\psi$ after fitting the nuisance parameter as favourably as possible at each value.