Reclaiming the"frequentist"role of marginal likelihood in Bayesian belief revision
TL;DR
This work shows that the marginal probability possesses a critical dual layer of information: while the posterior probability determines the local magnitude of a belief update upon a solitary trial, the marginal denominator governs the physical, long-run frequentist cadence of that update across a historical horizon.
Abstract
In modern Bayesian computation and parametric estimation, the marginal likelihood, serving as the denominator P(D) in Bayes'Theorem, is routinely bypassed via unnormalized proportionality relations. Even within specialized model-selection frameworks where it is explicitly evaluated to compute Bayes Factors, the denominator is treated purely as a static constant. This note evaluates a subtle analytical oversight resulting from this computational convenience. Through the analysis of a simplified, sequential partial-information system, we show that the marginal probability possesses a critical dual layer of information: while the posterior probability determines the local magnitude of a belief update upon a solitary trial, the marginal denominator governs the physical, long-run frequentist cadence of that update across a historical horizon. Discarding the denominator computes what an observer ought to believe once a specific dataset manifests, but erases the data-generating reality that governs how frequently that inferential state occurs in nature. We propose reclaiming the marginal likelihood as an active, real-time regularizer. We introduce three diagnostic measures to regulate recursive online estimation gains, and construct valid mixture probabilities that blend the prior and posterior to function as surprise-activated or conservative regulators. This paired probability framework may offer a robust regularizing mechanism for sequential estimation architectures and quantitative risk management scenarios under non-stationary distribution shifts.