Bayes, E-values and Testing

TL;DR

This paper develops a structured framework for understanding E-values and E-processes in sequential testing, clarifying their Bayesian connections, validity, and decision-making aspects, with theoretical and empirical insights.

Contribution

It introduces a typed framework separating evidence representation, validity, and decision, and characterizes likelihood ratios as unique evidence representations under certain conditions.

Findings

Likelihood ratio is the unique evidence representation under log-loss.

Stopping times based on likelihood ratios have predictable expected growth.

Prequential codes can yield valid E-processes, unlike regret-optimal codes.

Abstract

E-values and E-processes (nonnegative supermartingales) provide anytime-valid evidence for sequential testing via Ville's inequality, yet their connection to Bayesian reasoning, representational structure, and computational feasibility are often conflated in the literature. We develop a typed framework that separates sequential evidence into three layers: (i) representation (Radon-Nikodym / likelihood-ratio geometry), (ii) validity (supermartingale certificates under optional stopping), and (iii) decision (boundary design and efficiency calibration). Our main results are: (a) under log-loss and Bayes-risk minimization, the likelihood ratio is the unique evidence representation within the coherent predictive subclass; (b) the likelihood-ratio stopping time satisfies E_1[tau_b] = (log b)/mu + O(sqrt(log b)) under Cramer conditions, while validity-only thresholds admit no such growth-rate guarantee; and (c) regret-optimal codes (e.g., NML/MDL) do not in general yield valid E-processes, while prequential codes do. Monte Carlo experiments confirm the theoretical predictions. The framework applies to online model validation, adaptive experimentation, conformal prediction, and sequential changepoint detection.