Explain directional reversal under heteroscedastic noise

Derive a closed-form mechanism explaining why least-squares and likelihood objectives systematically favor the wrong causal direction under heteroscedastic noise, and characterize whether covariance matching loses or retains reliable directional information in that regime.

Background

The paper's identifiability guarantees rely on equal-variance exogenous noise. Under heteroscedastic noise, covariance matching's directional advantage weakens, while the least-squares and likelihood objectives tested exhibit consistently negative directional gaps, indicating preference for the wrong direction.

This behavior is reported empirically but is not explained theoretically. The authors describe it as the most important unresolved issue for applying covariance-, least-squares-, or likelihood-based fitting when equal-variance noise cannot be assumed.

References

We report this as an observed pattern, not a proven mechanism: we do not have a closed-form account of it analogous to Lemma~\ref{lem:tie} or Lemma~\ref{lem:separation}, and it is, on the evidence in this paper, the single most important open question for anyone considering covariance-, least-squares-, or likelihood-based fitting under noise that may not be equal-variance in practice.

— Guide, Not Bind: Why Defeasible Priors Fail in Augmented Lagrangian Causal Discovery  (2609.03442 - Sundararaman et al., 3 Sep 2026) in Limitations, item 5; Section 6.2 and Section 6.3