Preprint / Version 2

Bayesian inference of the impulse-response model of athlete training and performance


  • Kangyi Peng Department of Statistics and Actuarial Science, Simon Fraser University
  • Ryan T. Brodie
  • Tim B. Swartz
  • David C. Clarke Department of Biomedical Physiology and Kinesiology, Simon Fraser University



athletic performance, mathematical modeling, Bayes Theorem, Banister impulse-response model, computer simulation, Bayesian data analysis


The Banister impulse-response (IR) model quantitatively relates athletic performance to training. Despite its long history, the model usefulness remains limited due to difficulties in obtaining precise parameter estimates and performance predictions. To address these challenges, we developed a Bayesian implementation of the IR model, which formalizes the combined use of prior knowledge and data. We report the following methodological contributions: 1) we reformulated the model to facilitate the specification of informative priors, 2) we derived the IR model in Bayesian terms, and 3) we developed a method that enabled the JAGS software to be used while enforcing parameter constraints. We applied the model to the training and performance data of a national-class middle-distance runner. We specified the priors from published values of IR model parameters, followed by estimating the posterior distributions from the priors and the athlete’s data. The Bayesian approach led to more precise and plausible parameter estimates than nonlinear least squares. We then drew inferences from the Bayesian model regarding the athlete’s performance and showed how the method can be applied in perpetuity as new data are collected. We conclude that the Bayesian implementation of the IR model overcomes the foremost challenges to its usefulness for athlete monitoring.


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2023-01-23 — Updated on 2023-01-23