Bayesian change-plane regression

2026-04-26

AI Summary

This study addresses the challenges in traditional change-plane regression—namely, non-differentiability of the objective function, weak boundary identifiability, and unstable large-sample inference caused by hard-threshold boundaries. To overcome these issues, the authors propose a Bayesian inference framework based on a probit-gated smooth likelihood, which employs a controllable smoothing approximation to handle non-smoothness while asymptotically recovering the true hard-threshold boundary as the smoothing scale approaches zero. The work innovatively incorporates a misspecification-robust Bernstein–von Mises theorem to characterize the trade-off between smoothing scale and Gaussian approximation error, and introduces a decision-theoretic mechanism that disentangles evidence for heterogeneity from boundary reporting. Simulation studies and empirical analyses demonstrate that the proposed method outperforms existing frequentist approaches in both estimation accuracy and uncertainty quantification, successfully uncovering heterogeneous treatment effects in lifestyle interventions.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Bayesian Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• change-plane regression
• nonregular inference
• hard-threshold boundary
• weak identification
• likelihood-based inference

Innovation

• Bayesian change-plane regression
• probit-gated likelihood
• misspecified Bernstein–von Mises
• boundary-margin condition
• posterior membership probability