Beyond Coordinates: Meta-Equivariance in Statistical Inference
Abstract
Optimal statistical decisions should transcend the language used to describe them. Yet, how do we guarantee that the choice of coordinates - the parameterisation of an optimisation problem - does not subtly dictate the solution? This paper reveals a fundamental geometric invariance principle. We first analyse the optimal combination of two asymptotically normal estimators under a strictly convex trace-AMSE risk. While methods for finding optimal weights are known, we prove that the resulting optimal estimator is invariant under direct affine reparameterisations of the weighting scheme. This exemplifies a broader principle we term meta-equivariance: the unique minimiser of any strictly convex, differentiable scalar objective over a matrix space transforms covariantly under any invertible affine reparameterisation of that space. Distinct from classical statistical equivariance tied to data symmetries, meta-equivariance arises from the immutable geometry of convex optimisation itself. It guarantees that optimality, in these settings, is not an artefact of representation but an intrinsic, coordinate-free truth.
Cite
@article{arxiv.2504.10667,
title = {Beyond Coordinates: Meta-Equivariance in Statistical Inference},
author = {William Cook},
journal= {arXiv preprint arXiv:2504.10667},
year = {2025}
}
Comments
21 pages. Includes numerical simulations and visualisations. Developed independently using a live co-discovery process combining human intuition and computational validation. The principle of meta-equivariance generalises convex invariance under affine reparameterisation, with implications for statistical decision theory and information geometry