1. Introduction
2. The Moorean Tension
3. Betting, Belief, and the Single Case
4. Can Calibration Replace Belief?
5. Misunderstanding Bayesian Inference
6. What Counts as Inference?
7. Conclusion: Betting, Belief, and Epistemic Accountability

Introduction

In a recent discussion¹, I found myself pressing on a tension in one particular interpretation of confidence intervals - namely, the claim that they can guide betting behavior in the single case, even while one denies any belief that the interval actually contains the parameter of interest.

A persistent misconception I encounter - especially among frequentists who conflate performance guarantees with epistemic justification - is the claim that statistical models are “convenient yet perhaps imperfect representations of objective reality."¹ This framing obscures a deeper conceptual split in how different schools of inference understand what modeling is for. Here’s the simple point I want to press: the notion that models are “just representations of the world” bakes in assumptions that Bayesians do not share - and for good reason.

Online frequentists, at least from what I find, often charge that (subjectivist) Bayesians do little more than inject bias into their inferences through the use of a subjective prior; frequentism, however, very often claims asymptotic unbiasedness as a virtue for many of its estimators. This is a point in favor of frequentism, or so the proponents say.

There are two ways to read this as a complaint about Bayesianism: from the internal perspective of a frequentist or from an external, potentially non-frequentist or even Bayesian perspective. This is because the very definition of bias depends on what one considers to be the truth of the matter, the norm governing the correctness of the inference.