Reclaiming the "frequentist" role of marginal likelihood in Bayesian belief revision

Abdelhakim Aknouche

Home/Archive/arXiv

arXiv · 2607.26259

Reclaiming the "frequentist" role of marginal likelihood in Bayesian belief revision

Abdelhakim Aknouche

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

stat.ME

Keep this discovery

BibTeX RIS

Abdelhakim Aknouche. 2026-07-28. Reclaiming the "frequentist" role of marginal likelihood in Bayesian belief revision. https://arxiv.org/abs/2607.26259

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Surprise Reduction and Nullification in Bayesian and Inverse Bayesian Inference under Ambiguous Prediction-Error Attribution

In non-stationary environments, prediction errors may signal environmental change or transient outliers, and adaptive systems must track such changes without overreacting to outliers. We distinguish surprise reduction, which updates beliefs to fit observations, from surprise nullification, which weakens constraints imposed by the predictive structure, and formalize both within Bayesian and inverse Bayesian (BIB) inference. Belief and likelihood updates are derived from variational objectives sharing a nullification strength, determined endogenously by minimizing surprise under the candidate post-update predictive distribution. In the Gaussian case, nullification expands belief and likelihood variances by a common factor relative to standard Bayesian updating, leaving the ratio unchanged. BIB thus defers attribution of the prediction error, committing to neither latent-state change nor observation-process uncertainty. The nullification strength is carried over as a candidate and is maintained or released according to the predictive surprise of the next observation. In a mean estimation task with outliers and changepoints, no scanned parameter setting of a Sage-Husa-type adaptive Kalman filter, fixed-strength BIB variant, or belief-forgetting-only variant outperforms BIB in both changepoint tracking and post-outlier stability. An oracle-informed reduced Bayesian model tracks changepoints better but is less stable after outliers. Although BIB maintains no explicit hypotheses about changepoints or outliers, it generates event-dependent dynamics. The learning rate increases after changepoints, whereas after outliers, nullification is released, and this increase is suppressed. Deferring attribution and letting subsequent observations differentiate the responses may constitute a principle of adaptive inference in non-stationary environments.

stat.ME↗

arXiv2026

Generalized Ridge Refitting for the Lasso and Prediction Improvement Bounds

We study a class of Lasso based estimators obtained by applying a quadratic correction on the Lasso equicorrelation set. The penalty matrix determines both the magnitude and geometry of the correction and contains, among other cases, the isotropic Lasso--Ridge correction, least squares refitting, Gram proportional interpolation between the Lasso and least squares, and coordinate specific penalties. We first derive a closed form representation and isolate the positive gain component of the resulting prediction improvement. We then control the remaining stochastic linear term in expectation by localizing the random signed equicorrelation model around a deterministic reference support. This yields a finite sample expectation bound that explicitly accounts for the randomness induced by Lasso model selection. The resulting decomposition provides a unified framework for understanding when Lasso based quadratic corrections can improve prediction.

stat.ME↗

arXiv2026

Discretization in covariate-adaptive randomization: gains and losses

Covariate-adaptive randomization(CAR) is widely implemented in clinical trials to balance prognostic covariates across treatment arms. Continuous covariates are often discretized into strata in practice, yet their consequences are not clearly understood. This paper provides a comprehensive study of the impact of discretization on both the CAR design process and the inferential results thereafter. We establish the asymptotic properties of both imbalance measures and treatment effect estimators under discretized and non-discretized settings. Practical recommendations are given on when and how discretization should be employed. We show that discretization in design is generally recommended, as it enhances robustness against model misspecification. However, if the true model is known, the most efficient strategy is to balance covariates according to that model in the design. The theoretical results are corroborated by extensive simulation studies and an empirical application to a diabetes trial dataset. Together, the results clarify the gains and losses of discretization in CAR and pave the way for learning impact of discretization to other designs and beyond.

stat.ME↗