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

TL;DR

This paper introduces a practical Bayesian workflow for developing and deploying clinical prediction models that effectively quantify uncertainty and improve clinical utility over traditional plug-in methods.

Contribution

It proposes a pragmatic Bayesian pipeline using shrinkage priors and posterior mean decision-making, avoiding complex sampling and enhancing clinical utility.

Findings

Bayesian workflow matches or exceeds predictive performance of plug-in models.

Using posterior mean predictions often results in higher clinical utility.

The approach is demonstrated to be both pragmatic and advantageous in clinical settings.

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.