Bayesian Nonparametrics for Normative Modelling in Multiple Sclerosis via Modularised Inference

Normative models produce per-subject deviation scores that feed directly into downstream analyses, but typical pipelines (i) treat confounders with ad-hoc or purely linear adjustments, and (ii) pass point estimates of deviation scores directly to the downstream model, ignoring uncertainty. We propose an integrated, two-module Bayesian framework that aims to address both limitations. A normative module based on Bayesian Additive Regression Trees (BART) flexibly captures non-linear effects and higher-order interactions while marginalising over image-quality variables via counterfactual averaging. Crucially, we define individual deviation as di = E[Y|Xi,Zi] - (Zi) with (Z) the feature-conditional population mean, not as a residual. A SoftBART survival model then ingests the full posterior distribution of deviation scores via a cut-posterior construction, propagating upstream uncertainty while blocking feedback from the outcome model. Across challenging simulations and a large clinical data set of multiple sclerosis patients (N>8k), the integrated approach yields better calibration, prediction accuracy and time-varying hazard separation between groups than a two-step plug-in Cox regression model. Modularised inference with BART-based normative deviations improves both flexibility and uncertainty quantification, and extends naturally to other outcomes beyond survival.