Invariant Statistical Inference: A Framework Beyond Classical Statistical Inference

Classical statistical inference is traditionally formulated through probabilitymodels, likelihood functions, and estimation procedures. While these methodshave proven extremely successful, many statistical procedures implicitly rely oninvariance properties under transformations of the data. Location estimation, scaleestimation, and invariant hypothesis testing provide well-known examples wherestatistical rules remain structurally stable under admissible transformations.This paper introduces an invariant statistical inference framework in which sta-tistical procedures are characterized through identity-preserving transformations ofdatasets. Let D denote a dataset and let T represent an admissible transformationacting on the data space. The invariant statistical condition is expressed asI(T (D)) = I(D),where I(D) denotes the statistical identity associated with the dataset. Withinthis framework classical statistical inference appears as a special realization of thebroader invariant inference structure.