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

This paper proposes a pragmatic Bayesian workflow for developing and deploying clinical prediction models that utilizes shrinkage priors and posterior mean approximations to overcome traditional barriers of subjectivity and complexity, thereby delivering superior predictive performance and essential uncertainty quantification compared to standard deterministic methods.

Original authors: Mohsen Sadatsafavi, Richard D. Riley

Published 2026-05-20

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohsen Sadatsafavi, Richard D. Riley

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: From "One Guess" to "A Range of Possibilities"

Imagine you are a doctor trying to predict a patient's risk of a heart attack in the next 30 days.

The Old Way (The "Plug-in" Method):
Currently, most doctors use a standard formula (like a logistic regression equation). You plug the patient's numbers (age, blood pressure, etc.) into the machine, and it spits out one single number.

Example: "Your risk is 4.7%."
The Problem: This number looks very precise, but it hides the fact that the formula itself is an estimate based on a limited sample of people. It's like looking at a blurry photo and being told, "This is exactly what the person looks like," without admitting the photo is fuzzy. It ignores the uncertainty.

The New Way (The "Bayesian" Method):
The authors propose a smarter way to do this. Instead of giving just one number, they treat the prediction as a cloud of possibilities.

The Analogy: Imagine throwing a dart at a board. The "Old Way" says, "The dart landed exactly here." The "New Way" says, "The dart landed somewhere in this circle, and here is the most likely spot in the middle of that circle."
This approach gives you a range (a "Credible Interval") to show how sure the model is, and it uses a specific type of average (the "Posterior Mean") to make the final decision.

The Three Main Problems with the Old Way

It's too confident: It acts like the model knows the truth perfectly, even though it was built on a finite amount of data.
It's hard to interpret: Standard statistics use "Confidence Intervals," which are confusing. They say, "If we did this experiment 100 times, the result would fall in this range 95% of the time." Patients and doctors don't care about 100 hypothetical experiments; they want to know, "What is the chance my risk is in this range?"
It's computationally heavy: Traditional "Bayesian" methods used to require super-computers to run millions of simulations (like rolling dice billions of times) to get the answer. This made it too slow and difficult for everyday use.

The Authors' Solution: A "Pragmatic" Workflow

The authors say, "We don't need to reinvent the wheel or use super-computers. We can make Bayesian thinking practical and fast." They propose a workflow with two main tricks:

Trick 1: The "Smart Shrinkage" (The Prior)

In the old Bayesian world, you had to guess a "prior" (a starting belief) about the data, which some people thought was too subjective.

The Analogy: Imagine you are guessing the average height of people in a new city.
- Flat Prior: You assume every height is equally likely (very wild guess).
- Smart Prior: You assume heights are likely to be average, with extreme giants or dwarfs being rare.
What they do: They suggest using specific, simple "priors" (like the Jeffreys prior or Log-F prior) that act like a gentle hand on the data. They "shrink" extreme predictions toward the middle.
Why? If a model predicts a 99% risk for a patient, it's often an overreaction to noise in the data. The "shrinkage" pulls that 99% down to something more realistic (like 85%), preventing the model from being too extreme.

Trick 2: The "Fast Average" (The Posterior Mean)

Usually, to get the best average from a cloud of possibilities, you have to do heavy math.

The Analogy: Imagine you have a bag of marbles of different colors representing different risk levels. To find the "average" color, you used to have to pull them all out one by one (slow).
What they do: They use a mathematical shortcut (called Laplace approximation) that assumes the cloud of possibilities looks like a smooth, symmetrical hill (a bell curve).
The Result: You can calculate the "Posterior Mean" (the best single number to use for decision-making) instantly, without needing the super-computer simulations. They even offer a "self-projection" method that turns this complex math back into a simple equation that fits on a piece of paper or a basic calculator.

Why Use the "Posterior Mean"?

The paper argues that for making medical decisions, the Posterior Mean is the best number to use.

The Analogy: Imagine you are betting on a horse race.
- The "Plug-in" method gives you the horse that might win.
- The "Posterior Mean" gives you the horse that, on average, wins the most money over many races.
The Logic: In medicine, we want to maximize the "Net Benefit" (doing good for the patient while avoiding harm). The authors show mathematically that the Posterior Mean is the only number that guarantees the best decision in this "betting" scenario. It balances the risk of treating someone who doesn't need it against the risk of missing someone who does.

What Did They Test?

They tested this new workflow using:

Real Data: A famous dataset of heart attack patients (GUSTO-I).
Fake Data: Simulated scenarios where they knew the "true" answer to see if the model was right.

The Findings:

Accuracy: The new method was just as good (or better) at predicting risks than the old method.
Better Decisions: In many cases, using the "Posterior Mean" led to better clinical decisions (more "Net Benefit") than the old "Plug-in" numbers. Sometimes, the improvement was as big as if they had doubled the size of their study group.
Uncertainty: The new method successfully provided a "range" (Credible Interval) that actually captured the true risk the right amount of time (about 95% of the time, as promised).

The Bottom Line

The authors are saying: Stop using single, overly confident numbers.

Instead, use a workflow that:

Gently pulls extreme predictions toward the middle (Shrinkage).
Calculates the "best average" risk instantly (Posterior Mean).
Gives a clear range to show how uncertain the model is.

They claim this isn't a radical, impossible change. It's a "pragmatic" update that uses tools statisticians already have, making it easy to adopt for better, more honest patient care.

Technical Summary: A Practical Bayesian Workflow for Clinical Prediction Models

Problem Statement
Current clinical prediction models predominantly rely on frequentist frameworks that produce deterministic "plug-in" predictions. These models typically utilize Maximum Likelihood Estimates (MLE) of regression coefficients to generate a single point estimate of risk for an individual. This approach obscures the inherent uncertainty arising from finite sample sizes, model development, and deployment variations. While frequentist methods can quantify uncertainty via confidence intervals (CIs), these are often misinterpreted by clinicians and patients as direct probability statements about an individual's risk. Furthermore, frequentist penalized methods (e.g., Ridge, LASSO) often rely on computationally intensive bootstrapping for uncertainty quantification, which introduces Monte Carlo noise and convergence issues. There is a recognized need for a framework that provides coherent uncertainty quantification and aligns with decision-making theory, yet Bayesian methods have been underutilized due to perceived subjectivity, computational complexity, and implementation barriers.

Methodology
The authors propose a pragmatic Bayesian pipeline designed to be reproducible, computationally tractable, and immediately applicable to existing statistical workflows. The workflow consists of two primary components:

Shrinkage Priors with Laplace Approximation:
Instead of relying on Markov Chain Monte Carlo (MCMC) sampling, the authors utilize priors that allow the posterior distribution of regression coefficients to be approximated by a multivariate normal distribution (Laplace approximation). This enables the model to be packaged for deployment as a vector of point estimates and a covariance matrix. The paper evaluates four specific prior strategies:
- Flat Prior: An improper prior offering a Bayesian interpretation of unpenalized regression.
- Jeffreys Prior: A tuning-free prior that reduces small-sample bias and provides mild shrinkage.
- Log-F Prior: A user-specified prior (e.g., log-F(2,2)) allowing for selective shrinkage of specific predictors via data augmentation, without shrinking the intercept.
- Gaussian Prior (Bayesian Ridge): A hierarchical prior where the shrinkage strength is determined by the data, offering an alternative to cross-validated frequentist Ridge regression.
Posterior Mean as the Decision Metric:
The authors argue that the posterior mean of the predicted probability, $\mathbb{E}[\pi(x; \beta)]$ , is the optimal metric for clinical decision-making based on expected utility theory (specifically the threshold argument by Pauker and Kassirer). Unlike the plug-in estimate (which uses the mode/MAP), the posterior mean integrates over the full parameter uncertainty. To compute this mean without MCMC, the paper proposes three deployment-time calculation methods:
- Gauss-Hermite Quadrature: A high-precision numerical integration method.
- MacKay's Approximation: A simplified analytical approximation requiring only a single inverse-logit evaluation.
- Self-Projection: A two-step regression approach that minimizes Kullback-Leibler (KL) divergence to approximate the posterior mean with a simple logistic equation, facilitating the creation of nomograms or simplified equations.

Key Results
The proposed workflow was evaluated through simulation studies using both the GUSTO-I dataset (real-world data) and fully synthetic data with varying model complexities, event prevalences, and sample sizes.

Predictive Performance: In the majority of simulations, using the posterior mean resulted in higher clinical utility (measured by Standardized Net Benefit, sNB) compared to traditional plug-in predictions. In small samples (e.g., 10 events per variable), the gain in sNB from using the posterior mean was substantial, equivalent to a significant fraction of the gain expected from doubling the sample size.
Calibration: While plug-in estimates (particularly with flat priors) often preserved the Observed-to-Expected (O/E) ratio at 1 in the development sample, they frequently exhibited poor calibration slopes in new data (indicating overfitting). The posterior mean introduced a "shrinkage" effect toward 0.5, trading exact O/E preservation for improved individual-level calibration (slope closer to 1).
Uncertainty Quantification: The posterior distributions generated by the proposed priors demonstrated acceptable coverage probabilities (close to the nominal 95%) for credible intervals. The Bayesian Ridge showed slightly lower coverage than nominal, while other priors performed well.
Computation: The approximation methods (MacKay and Self-Projection) yielded performance nearly identical to the exact quadrature method, confirming that complex numerical integration is not strictly necessary for deployment.

Significance and Claims
The paper claims that a Bayesian approach to clinical prediction modeling is both pragmatic and clinically advantageous. The authors emphasize that their workflow does not require a "revolution" in analytical tools but rather a "minor tweak" to current machinery.

Pragmatism: By utilizing Laplace approximations and avoiding MCMC, the method retains the objectivity and reproducibility valued in the field while enabling coherent uncertainty propagation.
Decision-Theoretic Alignment: The use of the posterior mean is justified not merely as a statistical summary but as the unique minimizer of expected loss (quadratic and information-theoretic) and the optimal metric for maximizing expected utility in clinical decisions.
Shift in Evaluation: The authors suggest a paradigm shift in how model performance is assessed. Rather than prioritizing finite-sample unbiasedness (e.g., forcing O/E = 1), the Bayesian framework prioritizes efficient learning under relevant objective functions, such as clinical utility and individual-level prediction accuracy.
Accessibility: The proposed methods (Jeffreys, Log-F via data augmentation, Bayesian Ridge via existing packages like mgcv) are readily implementable in standard statistical software, lowering the barrier to entry for uncertainty-aware prediction modeling.

The paper concludes that adopting this workflow allows for the communication of uncertainty in a way that directly addresses patient questions ("What is the risk for someone like me?") and supports shared decision-making, all while maintaining or exceeding the predictive performance of conventional plug-in methods.