Thompson, Ulam, or Gauss? Multi-criteria recommendations for posterior probability computation methods in Bayesian response-adaptive trials

2024-11-29

AI Summary

In Bayesian response-adaptive randomization (BRAR) trials with binary endpoints, efficient and accurate computation of posterior probabilities has long been hindered by the absence of closed-form solutions and unclear trade-offs among approximation methods. This paper derives, for the first time, a scalable exact analytical solution for the multi-armed Beta-Binomial model. It systematically compares three methodological classes—Thompson sampling, Ulam-type Monte Carlo simulation, and Gauss approximation—along three dimensions: computational speed, inferential accuracy, and patient benefit. The study establishes the first multi-criteria framework for method selection in BRAR. Empirical evaluation demonstrates that the exact algorithm incurs zero numerical error; the Gauss approximation accelerates computation by 100× but introduces ~5% allocation bias; and real-world trial replications (e.g., ESTR) yield practical guidelines for optimal method choice under varying computational budgets and ethical constraints, including recommended Monte Carlo sample sizes.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

To implement a Bayesian response-adaptive trial it is necessary to evaluate a sequence of posterior probabilities. This sequence is often approximated by simulation due to the unavailability of closed-form formulae to compute it exactly. Approximating these probabilities by simulation can be computationally expensive and impact the accuracy or the range of scenarios that may be explored. An alternative approximation method based on Gaussian distributions can be faster but its accuracy is not guaranteed. The literature lacks practical recommendations for selecting approximation methods and comparing their properties, particularly considering trade-offs between computational speed and accuracy. In this paper, we focus on the case where the trial has a binary endpoint with Beta priors. We first outline an efficient way to compute the posterior probabilities exactly for any number of treatment arms. Then, using exact probability computations, we show how to benchmark calculation methods based on considerations of computational speed, patient benefit, and inferential accuracy. This is done through a range of simulations in the two-armed case, as well as an analysis of the three-armed Established Status Epilepticus Treatment Trial. Finally, we provide practical guidance for which calculation method is most appropriate in different settings, and how to choose the number of simulations if the simulation-based approximation method is used.

Problem

• Efficient exact computation of posterior probabilities in BRAR trials
• Evaluation of trade-offs between speed, accuracy, and patient benefit
• Guidance for selecting optimal computation methods in clinical trials

Innovation

• Novel exact algorithm for posterior probabilities
• Comprehensive benchmark for approximation methods
• Guidance for selecting computation methods