Teacher-Centred Martingale Posteriors for Interpretable Binary Regression

We develop a predictive-first framework for interpreting black-box classifiers through an explicitly subjective but coherent Bayesian analysis. The starting point is the martingale-posterior view of Fong et al. [9], itself rooted in the predictive interpretation of Bayesian uncertainty associated with Doob [7]. In that view, posterior uncertainty is induced from a predictive distribution on missing responses rather than from a prior on model parameters. For binary regression, we combine two ideas. First, we use a black-box classifier—for example random forests or BART [3, 5]—as a teacher that initializes the conditional predictive surface. Second, we retain interpretability by defining the target parameter as the coefficient vector of a sparse logistic projection of the completed data. The resulting posterior on logistic coefficients is not a posterior for the internal parameters of the teacher; it is the martingale posterior of a user-chosen interpretable functional of the completed-data law. We distinguish carefully between random-design and fixed-design regimes, with special emphasis on the latter, where the analyst conditions on a deterministic collection of covariate configurations and no distribution on X is required. A practical contribution is a teacher-centred conditional copula recursion for binary outcomes, together with predictive-resampling algorithms and an implementation blueprint for simulation studies with mixed continuous and categorical covariates. Bayesian logistic distillation appears as a limiting or plug-in approximation to the predictive-first construction. The overall message is that interpretable post-hoc explanation of a black box is necessarily subjective; the value of the martingale-posterior formulation is that this subjectivity is made explicit through the chosen predictive law, the teacher-trust parameter, the covariate similarity metric, and the surrogate class.