Beyond Coordinates: Meta-Equivariance in Statistical Inference
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909600634109952 |
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| author | Cook, William |
| author_facet | Cook, William |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10667 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Beyond Coordinates: Meta-Equivariance in Statistical Inference Cook, William Other Computer Science 62F12, 90C25, 53B20 G.1.6; G.3 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. |
| title | Beyond Coordinates: Meta-Equivariance in Statistical Inference |
| topic | Other Computer Science 62F12, 90C25, 53B20 G.1.6; G.3 |
| url | https://arxiv.org/abs/2504.10667 |