Self-Normalizing Denominators in Rational Causal Estimation

What the math gives to ML

The paper identifies covariance-polynomial denominators whose sampling noise is exactly multiplicative: their relative uncertainty is constant rather than increasing when the denominator becomes small. For the front-door covariance minor, this yields the exact Gaussian variance relation \(\operatorname{avar}(\sqrt{n}(\hat D-D))=10D^2\), so an apparent denominator-strength diagnostic is identically uninformative. The transferable asset is a scale-free treatment of positive covariance determinants and ratios: neural causal or covariance-prediction modules should model log-denominators and relative error, not penalize small raw denominators. This is most useful as a stability and uncertainty-calibration layer for neural estimators that compute ratios of learned covariance statistics.