False confidence, non-additive beliefs, and valid statistical inference

TL;DR

The paper argues that traditional probability-based inference can lead to false confidence and proposes a framework using non-additive beliefs, such as belief functions and random sets, to achieve valid statistical inference and avoid systematic bias.

Contribution

It introduces a novel framework based on non-additive beliefs and validity conditions that ensures reliable inference and avoids false confidence, addressing fundamental issues in statistical theory.

Findings

Traditional probability frameworks suffer from false confidence.

Non-additive beliefs can provide valid, calibrated inference.

Classical procedures like p-values align with the proposed framework.

Abstract

Statistics has made tremendous advances since the times of Fisher, Neyman, Jeffreys, and others, but the fundamental and practically relevant questions about probability and inference that puzzled our founding fathers remain unanswered. To bridge this gap, I propose to look beyond the two dominating schools of thought and ask the following three questions: what do scientists need out of statistics, do the existing frameworks meet these needs, and, if not, how to fill the void? To the first question, I contend that scientists seek to convert their data, posited statistical model, etc., into calibrated degrees of belief about quantities of interest. To the second question, I argue that any framework that returns additive beliefs, i.e., probabilities, necessarily suffers from {\em false confidence}---certain false hypotheses tend to be assigned high probability---and, therefore, risks systematic bias. This reveals the fundamental importance of {\em non-additive beliefs} in the context of statistical inference. But non-additivity alone is not enough so, to the third question, I offer a sufficient condition, called {\em validity}, for avoiding false confidence, and present a framework, based on random sets and belief functions, that provably meets this condition. Finally, I discuss characterizations of p-values and confidence intervals in terms of valid non-additive beliefs, which imply that users of these classical procedures are already following the proposed framework without knowing it.