One Probability, Two Roles: The Separation of Coherence and Frequency in Adaptive Regimes
Abstract
Probability plays two distinct roles in modern data-analytic practice: (i) as an internally coherent, filtration-relative language for sequential forecasting and, together with a stated loss or utility, decision making; and (ii) as a foundation for empirical claims such as stabilization, calibration, and repeated-sampling validity. The first role is relative to an assessment law ℚ, whereas the second is evaluated under a governing law ℙ and a declared repetition regime. In classical settings the roles are often aligned by conditional correctness under ℙ together with i.i.d.\ or other stable-law assumptions under a fixed design, or by exchangeability. In adaptive and AI-mediated settings they can diverge because forecasts operate within feedback loops while reliability is evaluated under a potentially different law and regime. We synthesize relevant results from prequential forecasting, calibration, martingale and game-theoretic validity, adaptive data analysis, conformal prediction, and long-context AI reliability into a formal account of \emph{role separation}. The resulting \emph{Role Separation Principle} distinguishes internal coherence from conditional-mean correctness under the evaluation law and from target-specific reference and stability conditions. Two separation results show that a proper coarsening of the limiting-frequency distribution need not identify the governing law, and that even conditionally correct forecasts under that law need not accompany stabilization. We then develop a unified stabilization framework spanning filtration-based, shift-based, policy--environment, and design-based regimes and operationalize it through a regime-conditional audit checklist. Applied illustrations clarify calibration claims, synthetic-data validity, and structural mismatches relevant to some high-confidence unsupported AI outputs.
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BibTeX
@misc{lu2026probability,
title = {{One Probability, Two Roles: The Separation of Coherence and Frequency in Adaptive Regimes}},
author = {Yonggang Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04115},
eprint = {2609.04115},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04115},
}
RIS
TY - UNPB
TI - One Probability, Two Roles: The Separation of Coherence and Frequency in Adaptive Regimes
AU - Yonggang Lu
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04115
UR - https://arxiv.org/abs/2609.04115
AB - Probability plays two distinct roles in modern data-analytic practice: (i) as an internally coherent, filtration-relative language for sequential forecasting and, together with a stated loss or utility, decision making; and (ii) as a foundation for empirical claims such as stabilization, calibration, and repeated-sampling validity. The first role is relative to an assessment law ℚ, whereas the second is evaluated under a governing law ℙ and a declared repetition regime. In classical settings the roles are often aligned by conditional correctness under ℙ together with i.i.d.\ or other stable-law assumptions under a fixed design, or by exchangeability. In adaptive and AI-mediated settings they can diverge because forecasts operate within feedback loops while reliability is evaluated under a potentially different law and regime. We synthesize relevant results from prequential forecasting, calibration, martingale and game-theoretic validity, adaptive data analysis, conformal prediction, and long-context AI reliability into a formal account of \emph{role separation}. The resulting \emph{Role Separation Principle} distinguishes internal coherence from conditional-mean correctness under the evaluation law and from target-specific reference and stability conditions. Two separation results show that a proper coarsening of the limiting-frequency distribution need not identify the governing law, and that even conditionally correct forecasts under that law need not accompany stabilization. We then develop a unified stabilization framework spanning filtration-based, shift-based, policy--environment, and design-based regimes and operationalize it through a regime-conditional audit checklist. Applied illustrations clarify calibration claims, synthetic-data validity, and structural mismatches relevant to some high-confidence unsupported AI outputs.
ER -