Causal vs. Anticausal merging of predictors

TL;DR

This paper investigates how merging predictors differently in causal and anticausal directions affects model behavior, revealing asymmetries that influence decision boundaries and generalization, using CMAXENT, logistic regression, and LDA.

Contribution

It demonstrates that merging predictors with CMAXENT yields logistic regression in causal and LDA in anticausal directions, highlighting asymmetries in model behavior.

Findings

CMAXENT reduces to logistic regression causally and LDA anticausally when all distributions are observed.

Decision boundaries differ between causal and anticausal merging, affecting out-of-variable generalization.

Asymmetries influence the effectiveness of merging methods in causal inference tasks.

Abstract

We study the differences arising from merging predictors in the causal and anticausal directions using the same data. In particular we study the asymmetries that arise in a simple model where we merge the predictors using one binary variable as target and two continuous variables as predictors. We use Causal Maximum Entropy (CMAXENT) as inductive bias to merge the predictors, however, we expect similar differences to hold also when we use other merging methods that take into account asymmetries between cause and effect. We show that if we observe all bivariate distributions, the CMAXENT solution reduces to a logistic regression in the causal direction and Linear Discriminant Analysis (LDA) in the anticausal direction. Furthermore, we study how the decision boundaries of these two solutions differ whenever we observe only some of the bivariate distributions implications for Out-Of-Variable (OOV) generalisation.