The Resolution of Causal Heterogeneity

2026-07-19

AI Summary

This study addresses a fundamental limitation in traditional causal heterogeneity analysis, which relies on pre-specified numbers of subgroups lacking a population-level robust definition. The authors propose the “resolution profile”—a novel, population-identifiable quantity that characterizes the minimal number of subgroups required to explain a given proportion of treatment effect heterogeneity. Their approach dispenses with latent variable assumptions and demonstrates that uncertainty in subgroup count stems from threshold nonregularity rather than model selection. Inference is conducted via a structured moment process combined with a cross-fitted Bayesian bootstrap corrected by influence functions. Theoretically, the estimator is shown to be posteriorly consistent and asymptotically efficient, converging to a Gaussian limit. Simulations confirm favorable finite-sample performance, and an application to the MineThatData email marketing experiment reveals that just two or three subgroups suffice to effectively capture heterogeneity in visitation response.

Abstract

Causal subgroup analyses often report a small number of groups summarizing treatment effect heterogeneity, as if that number were a well-defined estimand. Outside genuinely latent class populations, however, a ``true'' subgroup count is model dependent rather than a population functional. We replace it with a new population estimand, the resolution profile, a functional of the causal feature law giving the fewest groups explaining a prescribed fraction of causal heterogeneity, defined for every population without latent structure. Inference is organized around one cross-fitted Bayesian-bootstrap posterior for a single structured moment process, its scores corrected with influence functions, so that paths, profiles, fixed-resolution summaries, and subgroup effects follow by composition. A uniform conditional Bernstein--von Mises theorem over a loss class containing the nonsmooth quantization losses shows this posterior merges with the efficient Gaussian limit under stated nuisance-rate and margin conditions. Subgroup-number uncertainty is not model selection but threshold nonregularity, the profile being an integer-valued threshold of a continuous path, discontinuous in the law at each knot. At these knots no single-valued selector is locally uniformly consistent over root-$n$ neighborhoods, and the set-valued report obtained by inverting a simultaneous band retains locally uniform validity over exactly the same perturbations. Simulations support the approximations, and an analysis of the MineThatData e-mail experiment illustrates the resolution-indexed report, in which two to three groups summarize the visit response while finer structure falls below a noise-floor diagnostic.

Problem

• causal heterogeneity
• subgroup analysis
• resolution profile
• population estimand
• threshold nonregularity

Innovation

• resolution profile
• causal heterogeneity
• Bayesian bootstrap
• influence function
• nonregular threshold