Lower Bounds on the Error Probability for Invariant Causal Prediction

TL;DR

This paper investigates the limits of invariant causal prediction by linking it to Gaussian channel theory, deriving lower bounds on error probability, and demonstrating these through simulations.

Contribution

It introduces a novel connection between invariant causal prediction and Gaussian channel capacity, providing fundamental lower bounds on error probability.

Findings

Derived lower bounds on error probability using Gaussian channel theory

Connected invariant causal prediction to information-theoretic limits

Validated bounds through simulation experiments

Abstract

It is common practice to collect observations of feature and response pairs from different environments. A natural question is how to identify features that have consistent prediction power across environments. The invariant causal prediction framework proposes to approach this problem through invariance, assuming a linear model that is invariant under different environments. In this work, we make an attempt to shed light on this framework by connecting it to the Gaussian multiple access channel problem. Specifically, we incorporate optimal code constructions and decoding methods to provide lower bounds on the error probability. We illustrate our findings by various simulation settings.