Direct and approximately valid probabilistic inference on a class of statistical functionals

TL;DR

This paper introduces a new generalized inferential model framework for direct probabilistic inference on statistical functionals, providing approximately valid uncertainty quantification without relying on a statistical model.

Contribution

It develops a bootstrap-based possibility measure for direct inference on functionals, extending probabilistic inference methods beyond traditional model-based approaches.

Findings

Provides asymptotically well-calibrated plausibility values

Constructs approximately valid confidence regions for functionals

Demonstrates effectiveness in quantile regression and personalized medicine

Abstract

Existing frameworks for probabilistic inference assume the quantity of interest is the parameter of a posited statistical model. In machine learning applications, however, often there is no statistical model/parameter; the quantity of interest is a statistical functional, a feature of the underlying distribution. Model-based methods can only handle such problems indirectly, via marginalization from a model parameter to the real quantity of interest. Here we develop a generalized inferential model (IM) framework for direct probabilistic uncertainty quantification on the quantity of interest. In particular, we construct a data-dependent, bootstrap-based possibility measure for uncertainty quantification and inference. We then prove that this new approach provides approximately valid inference in the sense that the plausibility values assigned to hypotheses about the unknowns are asymptotically well-calibrated in a frequentist sense. Among other things, this implies that confidence regions for the underlying functional derived from our proposed IM are approximately valid. The method is shown to perform well in key examples, including quantile regression, and in a personalized medicine application.