Bayes, E-values and Testing
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917327047491584 |
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| author | Polson, Nicholas G. Sokolov, Vadim Zantedeschi, Daniel |
| author_facet | Polson, Nicholas G. Sokolov, Vadim Zantedeschi, Daniel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04146 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bayes, E-values and Testing Polson, Nicholas G. Sokolov, Vadim Zantedeschi, Daniel Statistics Theory 62F15, 62L10, 62F03, 60G46 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. |
| title | Bayes, E-values and Testing |
| topic | Statistics Theory 62F15, 62L10, 62F03, 60G46 |
| url | https://arxiv.org/abs/2602.04146 |