arXiv:2508.15489v2 Announce Type: replace Abstract: Conformal Prediction (CP) provides prediction sets with formal coverage guarantees, but most existing methods assume fully observed training and calibration data. In many applications, however, outcomes, covariates, or both may be missing or coarsened. We develop a general framework for predictive inference with coarsened data by combining semiparametric theory with conformal methods. We introduce two complementary procedures. $(i)$ Robust Split Prediction (RSP) generalizes the doubly robust estimating-equation approach of Yang et al. [2024] beyond the classical missing outcome problem by constructing an efficient influence function based estimating equation and solving its empirical analogue. This yields multiply robust prediction sets under general coarsening patterns. $(ii)$ Robust Split Conformal Prediction (RSCP) instead combines semiparametric theory with conformal risk control. Because efficient influence functions are generally non-monotone in the threshold, we first estimate the target threshold on an independent sample and freeze the threshold-dependent nuisance functions at this preliminary estimate. This restores monotonicity and allows standard conformal risk control arguments to be applied. The resulting coverage guarantees inherit the higher-order mixed-bias structure of the influence function. We develop the theory for classical, multi-stage monotone, and non-monotone missingness and illustrate the performance of the methods through simulations and a longitudinal rheumatoid-arthritis application.