Ignorable missingness mechanism

A missingness mechanism is ignorable for likelihood or Bayesian inference about outcome-model parameters when the missingness-model factor can be omitted without changing that inference. missing at random together with distinct outcome and missingness parameters is a standard sufficient condition for likelihood inference; Bayesian inference also requires the corresponding prior factorization, so outcome and missingness parameters are independent a priori.

If a regression response $Y$ is observed but covariate $X$ is sometimes missing, likelihood inference requires a joint model: for example $f_{\beta }(y\mid x)g_{\eta }(x)$. An incomplete record contributes $\int f_{\beta }(y\mid x)g_{\eta }(x)dx$ to the observed-data likelihood. Missing at random with Distinct parameters allows the missingness mechanism to be omitted; it does not remove the need to integrate over $X$.

Outcome and missingness parameters are distinct when their joint parameter space is the Cartesian product of their separate parameter spaces, so variation-independent choices can be made for the two models.