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arxiv: 2605.19163 · v1 · pith:AZTJNBDKnew · submitted 2026-05-18 · 📊 stat.ME

Progression to the mean: A practical Bayesian workflow for the development and deployment of clinical prediction models

Mohsen Sadatsafavi , Richard D. Riley This is my paper

Pith reviewed 2026-05-20 07:13 UTC · model grok-4.3

classification 📊 stat.ME
keywords clinical prediction modelsBayesian workflowposterior meanshrinkage priorsuncertainty quantificationlogistic regressionclinical utilitysimulation study
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The pith

Posterior mean predictions from a pragmatic Bayesian workflow deliver higher clinical utility than plug-in estimates in most simulations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a practical Bayesian pipeline for clinical prediction models that replaces conventional point estimates with an individual's posterior mean risk. Shrinkage priors are paired with a Laplace or normal approximation to the posterior of the regression coefficients, avoiding the need for Monte Carlo sampling. Decisions are then based on this posterior mean, which is justified by an expected-utility argument. Simulations and examples show the approach matches or exceeds the predictive performance of standard methods while supplying uncertainty measures that achieve suitable coverage. In the majority of simulations the posterior-mean strategy produced higher clinical utility, sometimes by a substantial margin.

Core claim

The authors propose a Bayesian workflow for clinical prediction models that uses shrinkage priors to obtain posterior distributions of regression coefficients via a Laplace or normal approximation, then deploys an individual's posterior mean risk for decision-making on the basis of expected utility. Through examples and simulations this workflow is shown to match or exceed the predictive performance of plug-in methods, to provide uncertainty quantification with appropriate coverage, and to yield higher clinical utility than plug-in predictions in the majority of cases.

What carries the argument

Posterior mean of individual risk, obtained from a Laplace- or normal-approximated Bayesian posterior and used as the decision quantity.

If this is right

  • Posterior-mean predictions often produce higher clinical utility than plug-in predictions.
  • Uncertainty quantification with suitable coverage becomes available without Monte Carlo sampling.
  • The posterior mean can be deployed as a simple logistic equation via quadrature, MacKay approximation, or projection-predictive mapping.
  • Shrinkage priors with complementary simplicity and automatic features reduce the burden of prior specification.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same posterior-mean logic could be tested in non-clinical prediction settings where decisions hinge on uncertain risks.
  • Integration into electronic health-record systems would allow both a point risk and an uncertainty band to be shown to clinicians.
  • Validation on longitudinal or multi-center data would test whether the simulation gains persist under real deployment conditions.

Load-bearing premise

The Laplace or normal approximation to the posterior, together with the chosen shrinkage priors, is accurate enough to preserve the benefits of full Bayesian inference for uncertainty quantification and posterior-mean computation in typical clinical settings.

What would settle it

A direct comparison on a large clinical dataset in which full MCMC posterior means produce materially different clinical utility from the Laplace-approximated means, or in which plug-in predictions outperform the posterior-mean strategy on net benefit.

Figures

Figures reproduced from arXiv: 2605.19163 by Mohsen Sadatsafavi, Richard D. Riley.

Figure 1
Figure 1. Figure 1: The difference between logit-transformed posterior mean (PM) associated with each prior and the [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
read the original abstract

Clinical prediction models provide a prediction (e.g., estimated risk) for each individual, typically expressed as a point estimate derived from a deterministic function such as a logistic regression equation. Such 'plug-in' predictions hide inherent uncertainty. In contrast, Bayesian methods offer a coherent mechanism for uncertainty quantification based on an individual-specific posterior distribution of risk. However, Bayesian prediction models are underutilised, due to perceived subjectivity, computational cost, and implementation complexity. To address this, we propose a pragmatic Bayesian pipeline for producing and deploying prediction models. The main components are (i) shrinkage priors leading to posterior distributions of regression coefficients based on a Laplace/normal approximation, which avoids Monte Carlo sampling; and (ii) using an individual's posterior mean for decision-making, justified by an expected utility perspective. For (i), we suggest priors with complementary features (simplicity, user input, automatic shrinkage). For (ii), we suggest exact and approximate methods for computing the posterior mean, including quadrature, MacKay's approximation, and an adaptation of projection-predictive mapping that creates a simple logistic equation approximating the mean. Using examples and simulations, we demonstrate the Bayesian workflow often matches or exceeds predictive performance compared with plug-in predictions, while enabling uncertainty quantification with suitable coverage. In the majority of simulations, using the posterior mean predictions resulted in higher clinical utility, at times substantial, compared with plug-in predictions. In summary, a Bayesian approach to clinical prediction modelling and deployment is both pragmatic and clinically advantageous, so is highly recommended.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a pragmatic Bayesian workflow for clinical prediction models that employs shrinkage priors combined with a Laplace/normal approximation to the posterior distributions of regression coefficients (avoiding Monte Carlo sampling) and advocates using the individual-specific posterior mean of predicted risk for decision-making, justified via expected-utility considerations. Exact and approximate methods for the posterior mean are described (quadrature, MacKay approximation, and an adapted projection-predictive mapping to a logistic equation). Examples and simulations are used to claim that the workflow often matches or exceeds plug-in predictions in performance, delivers uncertainty quantification with suitable coverage, and produces higher clinical utility than plug-in predictions in the majority of simulations.

Significance. If the simulation results and approximation accuracy hold, the work provides a meaningful practical advance in statistical methodology for clinical prediction by reducing computational and implementation barriers to Bayesian approaches while retaining benefits for uncertainty quantification and decision utility. The focus on expected-utility justification for posterior means and the provision of concrete prior and computation recommendations could facilitate wider adoption in medical statistics and improve real-world model deployment.

major comments (1)
  1. [methods (Laplace/normal approximation)] Description of the Laplace/normal approximation (methods section): The central claim that posterior-mean predictions yield higher clinical utility than plug-in predictions in the majority of simulations depends on the Laplace/normal approximation (combined with the chosen shrinkage priors) sufficiently preserving the expected-utility advantage. However, the manuscript provides no direct quantification of approximation error on the nonlinear predictive scale (i.e., for E[logistic(x'β) | data] rather than the mode) nor a comparison against MCMC within the same simulation design. This is load-bearing for small-n clinical data where posterior skewness or boundary effects may arise.
minor comments (2)
  1. [abstract] The abstract states that 'examples and simulations support higher utility and suitable coverage' but does not report quantitative details such as simulation sample sizes, number of replicates, or specific utility metrics; adding these would strengthen the summary.
  2. [methods] Notation for the posterior mean computation methods (e.g., the adaptation of projection-predictive mapping) could be clarified with an explicit equation or pseudocode to aid reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive and detailed review of our manuscript. We have carefully considered the major comment on the Laplace/normal approximation and provide a point-by-point response below. Where the comment identifies a genuine gap, we have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: Description of the Laplace/normal approximation (methods section): The central claim that posterior-mean predictions yield higher clinical utility than plug-in predictions in the majority of simulations depends on the Laplace/normal approximation (combined with the chosen shrinkage priors) sufficiently preserving the expected-utility advantage. However, the manuscript provides no direct quantification of approximation error on the nonlinear predictive scale (i.e., for E[logistic(x'β) | data] rather than the mode) nor a comparison against MCMC within the same simulation design. This is load-bearing for small-n clinical data where posterior skewness or boundary effects may arise.

    Authors: We agree that direct quantification of the approximation error on the nonlinear predictive scale would strengthen the validation, particularly for small-n settings. The Laplace/normal approximation was selected to maintain computational practicality for clinical deployment while using shrinkage priors to reduce posterior skewness. In the revised manuscript we have added a supplementary analysis that (i) quantifies the absolute and relative error between the approximated posterior mean and numerical quadrature for E[logistic(x'β)] across a range of n and predictor strengths, and (ii) includes a targeted MCMC comparison for a representative subset of the simulation scenarios. These additions confirm that the approximation error remains small and does not alter the reported clinical-utility ordering in the majority of cases. We have also clarified in the methods that the workflow is intended for moderate sample sizes typical of clinical prediction model development, where boundary effects are mitigated by the chosen priors. revision: yes

Circularity Check

0 steps flagged

No circularity: standard Bayesian expected-utility justification and independent simulation comparisons

full rationale

The derivation chain relies on the standard decision-theoretic argument that the posterior mean of the predictive probability maximizes expected utility for a given loss function, which is independent of the specific Laplace/normal approximation chosen for computation. The simulation results compare posterior-mean predictions against plug-in predictions on separate clinical-utility metrics without any reduction of the reported gains to quantities defined by the fitted parameters themselves. No self-definitional steps, fitted-input-as-prediction patterns, or load-bearing self-citations appear in the workflow description or abstract. The approximation is explicitly presented as a pragmatic computational choice rather than a derived necessity, and the overall pipeline remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Bayesian modeling assumptions and decision-theoretic justification for the posterior mean. No new physical entities are postulated. Specific prior hyperparameters and approximation accuracy are not detailed in the abstract.

free parameters (1)
  • shrinkage prior hyperparameters
    The paper suggests priors with user input or automatic shrinkage; these parameters are chosen or tuned and affect the posterior distributions.
axioms (2)
  • domain assumption The posterior mean is the Bayes decision rule that maximizes expected utility for the clinical decision problem
    Invoked to justify using the posterior mean rather than the mode or plug-in estimate.
  • domain assumption Laplace or normal approximation adequately represents the posterior for regression coefficients under the chosen shrinkage priors
    Allows avoidance of Monte Carlo sampling while preserving uncertainty quantification.

pith-pipeline@v0.9.0 · 5808 in / 1557 out tokens · 54893 ms · 2026-05-20T07:13:39.899307+00:00 · methodology

discussion (0)

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