Meta-Statistical Learning: Supervised Learning of Statistical Estimators

Abstract

Statistical inference, a central tool of science, revolves around the study and the usage of statistical estimators: functions that map finite samples to predictions about unknown distribution parameters. In the frequentist framework, estimators are evaluated based on properties such as bias, variance (for parameter estimation), accuracy, power, and calibration (for hypothesis testing). However, crafting estimators with desirable properties is often analytically challenging, and sometimes impossible, e.g., there exists no universally unbiased estimator for the standard deviation. In this work, we introduce meta-statistical learning, an amortized learning framework that recasts estimator design as an optimization problem via supervised learning. This takes a fully empirical approach to discovering statistical estimators; entire datasets are input to permutation-invariant neural networks, such as Set Transformers, trained to predict the target statistical property. The trained model is the estimator, and can be analyzed through the classical frequentist lens. We demonstrate the approach on two tasks: learning a normality test (classification) and estimating mutual information (regression), achieving strong results even with small models. Looking ahead, this paradigm opens a path to automate the discovery of generalizable and flexible statistical estimators.