Imprecise Probability and Valid Statistical Inference

This research project will advance the foundations of statistical, data-driven uncertainty quantification based on imprecise probabilities. The complexity of the problems faced by social, behavioral, and economic scientists makes direct theoretical investigations virtually impossible, so progress relies heavily on data analysis and statistical inference.

However, the recent replication crisis in science has created confusion about and distrust in statistics. Among the statistical factors contributing to the replication crisis is the lack of a solid foundation of statistics. The two dominant schools of thought, frequentist and Bayesian, are very different, but both rely on precise probabilities.

The use of precise probabilities for drawing inference about unknowns based on data, however, has been shown to be invalid in a specific sense that threatens replicability. The shift from precise to imprecise probabilities for statistical inference will have broad positive impacts and create new research opportunities in fields beyond statistics and the social, behavioral, and economic sciences.

The project will use the online Researchers.One platform for dissemination of results. Publicly available software will be created. The project also will provide valuable training and experience to graduate students and an early-career researcher.

A high-level goal of this research project is to create a single theory of statistical inference based on imprecise probabilities. This research project will focus on a framework that uses provably valid, data-dependent, imprecise (or non-additive) probabilities to quantify uncertainty and draw inferences about unknowns. The investigator will demonstrate that this entire framework can be cast in terms of one of the simplest imprecise probability models, namely, possibility measures, and this simplicity benefits practitioners in various ways.

The investigator will prove that, roughly, every exact or conservative frequentist procedure corresponds to a valid imprecise probability, fully establishing the fundamental nature of imprecise probabilities in statistical inference. The investigator also will develop new and powerful imprecise probability-based methods for two general and challenging statistical problems: structure learning in high dimensions and inference without a statistical model.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.