Missing at random: a stochastic process perspective

We offer a natural and extensible measure-theoretic treatment of missingness at random. Within the standard missing data framework, we give a novel characterisation of the observed data as a stopping-set sigma algebra. We demonstrate that the usual missingness at random conditions are equivalent to requiring particular stochastic processes to be adapted to a set-indexed filtration of the complete data: measurability conditions that suffice to ensure the likelihood factorisation necessary for ignorability. Our rigorous statement of the missing at random conditions also clarifies a common confusion: what is fixed, and what is random?