Bayesian change-plane regression

TL;DR

This paper introduces a Bayesian framework for change-plane regression that handles the nonregular boundary inference problem by using a smoothed likelihood surrogate, enabling better uncertainty quantification and interpretation.

Contribution

It develops a novel Bayesian approach with a probit-gated likelihood for change-plane models, addressing nonregularity and boundary uncertainty in a computationally regular way.

Findings

Simulations show improved accuracy and uncertainty quantification over frequentist methods.

Application to a lifestyle trial demonstrates practical utility in understanding treatment heterogeneity.

The method effectively separates evidence for heterogeneity from boundary reporting.

Abstract

Change-plane regression identifies subpopulations through an interpretable linear threshold rule, but likelihood-based inference for the hard-threshold boundary is nonregular: objectives are non-smooth, the boundary is weakly identified under no heterogeneity, and standard large-sample approximations are fragile. We develop a new Bayesian inferential framework based on a probit-gated working likelihood -- a computationally regular surrogate that is deliberately misspecified for any fixed smoothing scale. For fixed smoothing, posterior summaries are therefore interpreted for a well-defined smoothed pseudo-true target; inference for the hard-threshold target is recovered only in a vanishing-smoothing regime, where approximation bias is governed by a boundary-margin condition on the covariate distribution. The resulting theory adapts misspecified Bernstein--von Mises arguments to Bayesian change-plane regression and makes explicit the triangular-array trade-off created by sending the smoothing scale to zero: sharper gates worsen the derivative bounds needed for Gaussian approximation, while approximation bias decreases according to the local amount of covariate mass near the boundary. Building on the resulting joint posterior, we further propose a decision-theoretic reporting protocol that separates evidence for clinically meaningful heterogeneity from the reporting of a subgroup boundary, with boundary uncertainty propagated to the covariate level through posterior membership probabilities. Simulations show favorable accuracy and uncertainty quantification of our new methods relative to the frequentist counterpart, and an application to a randomized lifestyle-intervention trial further demonstrates the utility of Bayesian change-plane regression in understanding treatment effect heterogeneity.