Invariant Probabilistic Prediction

TL;DR

This paper explores the challenges of achieving invariant and robust probabilistic predictions under distribution shifts, proposing a new method called IPP that demonstrates promising empirical results.

Contribution

It introduces the IPP method for invariant probabilistic prediction and analyzes conditions for invariance under distribution shifts in a causality-inspired framework.

Findings

Arbitrary distribution shifts generally prevent invariant probabilistic predictions.

Invariance can be achieved under restricted distribution shifts in Gaussian heteroscedastic models.

IPP method shows empirical robustness and consistency in simulations and single-cell data.

Abstract

In recent years, there has been a growing interest in statistical methods that exhibit robust performance under distribution changes between training and test data. While most of the related research focuses on point predictions with the squared error loss, this article turns the focus towards probabilistic predictions, which aim to comprehensively quantify the uncertainty of an outcome variable given covariates. Within a causality-inspired framework, we investigate the invariance and robustness of probabilistic predictions with respect to proper scoring rules. We show that arbitrary distribution shifts do not, in general, admit invariant and robust probabilistic predictions, in contrast to the setting of point prediction. We illustrate how to choose evaluation metrics and restrict the class of distribution shifts to allow for identifiability and invariance in the prototypical Gaussian heteroscedastic linear model. Motivated by these findings, we propose a method to yield invariant probabilistic predictions, called IPP, and study the consistency of the underlying parameters. Finally, we demonstrate the empirical performance of our proposed procedure on simulated as well as on single-cell data.