Nearly Optimal Bayesian Inference for Structural Missingness

Structural missingness breaks 'just impute and train': values can be undefined by causal or logical constraints, and the mask may depend on observed variables, unobserved variables (MNAR), and other missingness indicators. It simultaneously brings (i) a catch-22 situation with causal loop, prediction needs the missing features, yet inferring them depends on the missingness mechanism, (ii) under MNAR, the unseen are different, the missing part can come from a shifted distribution, and (iii) plug-in imputation, a single fill-in can lock in uncertainty and yield overconfident, biased decisions. In the Bayesian view, prediction via the posterior predictive distribution integrates over the full model posterior uncertainty, rather than relying on a single point estimate. This framework decouples (i) learning an in-model missing-value posterior from (ii) label prediction by optimizing the predictive posterior distribution, enabling posterior integration. This decoupling yields an in-model almost-free-lunch: once the posterior is learned, prediction is plug-and-play while preserving uncertainty propagation. It achieves SOTA on 43 classification and 15 imputation benchmarks, with finite-sample near Bayes-optimality guarantees under our SCM prior.