Coarsening at Random: A Small Monograph on Masked-Data Identifiability Across Domains

The coarsening-at-random program is a family of papers showing that several unrelated-looking data problems (reliability with masked failure causes, single-cell and spatial transcriptomics, differential privacy, programmatic weak supervision, electronic-health-record phenotyping, and multiple instance learning) are instances of one statistical structure: a latent quantity observed only through a coarsening, identifiable under the conditions C1, C2, C3 of Heitjan-Rubin and Gill-van der Laan-Robins. This monograph states the shared idea once, at the generality that explains the recurrence, and places each existing paper inside that frame: a reliability foundation, two framework-tier pillars (the C2-holds consistency synthesis and the C2-fails sensitivity theory), and six domain instances. Three results recur in every domain and are stated once here: a consistency identity at the maximum-likelihood fit, an augmented-candidate-set rank condition for identifiability, and a singleton candidate set that restores identifiability when the rank condition fails. The monograph then draws the boundary of the program and names the papers that are missing, stated as a research agenda of gaps whose results would close the framework: a semiparametric-efficiency pillar, estimation of the identifiability geometry from data, optimal and active singleton design, global partial identification, Bayesian and nonparametric treatments, causal and learning-theoretic bridges, and an impossibility and limits paper. It is both an entry point to the program and a plan for its consolidation.