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Causal vs. Anticausal merging of predictors
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Studies how merging predictors under causal maximum entropy (CMAXENT) differs in causal vs anticausal directions, showing it reduces to logistic regression (causal) and LDA (anticausal) with full moments and analysing out-of-variable generalisation when some covariances are unknown.
Problem
Merging predictors (experts) of a binary target from different covariate subsets is ill-defined, and the causal direction of covariates relative to the target may change the merged predictor; the paper asks how causal vs anticausal assumptions alter CMAXENT-based merging.
Approach
Applies Causal Maximum Entropy with a binary target and two continuous covariates, deriving that causal-direction CMAXENT yields logistic regression and anticausal-direction CMAXENT yields LDA; shows equal decision-boundary slopes under full moment knowledge but asymmetric behaviour when a predictor-target or predictor-predictor covariance is unknown (enabling out-of-variable generalisation).
Key results
- With all first and second moments known, CMAXENT in the causal direction gives logistic regression and in the anticausal direction gives LDA, with decision-boundary slopes that coincide.
- With unknown predictor-target covariance phi_2, an upper bound on phi_2 is derived as q(1-q)s12*phi1/(q(1-q)s1^2 - phi1^2) under anticausal assumption, enabling a model using X2 without joint observations.
- With unknown predictor covariance s12, the anticausal CMAXENT solution has diagonal conditional covariance and the decision boundaries in causal and anticausal directions differ, shown with synthetic data.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use causal rather than anticausal CMAXENT with all first- and second-moment constraints | CMAXENT predictor · Same-distribution prediction with full moments | 0-1 classification performance | No clear effect |
| Use causal rather than anticausal CMAXENT with full moment constraints | CMAXENT predictor · Full-moment prediction | predictor form | Mixed |
| Assume anticausal generation when the covariance between X2 and Y is unknown (ablation) | CMAXENT predictor · Out-of-variable generalization with unknown predictor-target covariance | ability to include X2 in the predictor | Helped |
| Assume causal generation when the covariance between X2 and Y is unknown (ablation) | CMAXENT predictor · Unknown predictor-target covariance | ability to include X2 in the predictor | Hurt |
| Omit the covariance between predictors X1 and X2 (ablation) | CMAXENT predictor · Unknown predictor covariance | decision-boundary slope | Mixed |
| Use class-conditional second-moment constraints instead of marginal second-moment constraints (ablation) | Anticausal CMAXENT · Quadratic discriminant analysis interpretation | decision-boundary form | Helped |
Surprising findings
- With the same moments but different causal assumptions, CMAXENT decision boundaries differ when the predictor covariance is unknown.
- CMAXENT in the anticausal direction need not yield a Markov (diagonal) conditional covariance when all moments are used, contrary to the DAG implication.
Limitations
- Analysis restricted to a binary target and two continuous covariates, first and second moments only.
- Results on other inductive biases than CMAXENT are conjectured rather than proven.
- Empirical validation limited to synthetic data.
Experimental setup
Baselines: logistic regression linear discriminant analysis quadratic discriminant analysis
Builds on: Causal Maximum Entropy (CMAXENT) Maximum Entropy principle Linear Discriminant Analysis Independent Mechanisms principle
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