Extended Fiducial Inference: Toward an Automated Process of Statistical Inference

TL;DR

This paper introduces extended Fiducial inference (EFI), a scalable statistical inference method that combines advanced computing techniques with neural networks to improve parameter estimation and automate hypothesis testing, especially in big data contexts.

Contribution

The paper develops EFI, integrating stochastic gradient MCMC and sparse DNNs, to enhance fiducial inference with automation and robustness against outliers.

Findings

EFI achieves higher fidelity in parameter estimation.

EFI automates hypothesis testing without reference distributions.

EFI is scalable for big data applications.

Abstract

While fiducial inference was widely considered a big blunder by R.A. Fisher, the goal he initially set --`inferring the uncertainty of model parameters on the basis of observations' -- has been continually pursued by many statisticians. To this end, we develop a new statistical inference method called extended Fiducial inference (EFI). The new method achieves the goal of fiducial inference by leveraging advanced statistical computing techniques while remaining scalable for big data. EFI involves jointly imputing random errors realized in observations using stochastic gradient Markov chain Monte Carlo and estimating the inverse function using a sparse deep neural network (DNN). The consistency of the sparse DNN estimator ensures that the uncertainty embedded in observations is properly propagated to model parameters through the estimated inverse function, thereby validating downstream statistical inference. Compared to frequentist and Bayesian methods, EFI offers significant advantages in parameter estimation and hypothesis testing. Specifically, EFI provides higher fidelity in parameter estimation, especially when outliers are present in the observations; and eliminates the need for theoretical reference distributions in hypothesis testing, thereby automating the statistical inference process. EFI also provides an innovative framework for semi-supervised learning.