MCAR, MAR, and MNAR relative to an observation pattern

Fix the full-data/observed-data partition of Full and observed data under a missingness pattern. A response mechanism is a specified conditional law $g(r\mid w)$ of the pattern $R$ given full data $W=w$, represented by a probability kernel from the full-data space to the finite pattern space (Conditional law given a random element, Measure kernel and probability kernel). In a parameterized model, write the response kernel as $g_{\psi }$; it must be a probability distribution over patterns for each full-data value.

Relative to a specified full-data law, the mechanism is missing completely at random (MCAR) when, for every pattern $r$, $g(r\mid w)$ is constant in $w$ almost surely. It is missing at random (MAR) when, for every pattern $r$, there is a measurable function $h_r$ on that pattern's observed-data space such that $g(r\mid w)=h_r(W_{\mathrm{o}\mathrm{b}\mathrm{s}}(r))\text{almost surely}\mathrm{.}$ For a family of full-data laws, the condition is required under each law with the same response mechanism and observed-coordinate function. It is missing not at random (MNAR) when MAR fails for at least one pattern, for the declared full-data and observed-data partition. MCAR is a special case of MAR, since a constant is a function of the observed coordinates.

These conditions concern a supplied version of the conditional law; values off the full-data model's support do not affect the classification. This definition does not assert that a response mechanism is empirically true or that every joint law comes with a selected conditional kernel.

When the full-data and pattern spaces are standard Borel and a kernel must be obtained from a joint law rather than supplied as part of the model, the library's RCD existence and factor-through results provide that construction under AC: Existence of regular conditional distributions for standard borel targets and Regular conditional kernels factor through a standard borel conditioning variable. The finite examples on the companion page specify their response probabilities directly and do not use that existence argument.