Many different techniques have been used for parameter estimation in cardiac electrophysiology models, from optimization algorithms to heuristic and frequentist statistical methods. However, the fixed parameter values obtained from such approaches cannot provide a complete description of variability within an individual or across a population. To overcome this shortcoming, in this work we adopt a Bayesian approach by applying the Hamiltonian Monte Carlo (HMC) algorithm to cardiac electrophysiology models and data for the first time through three studies. (i) Using HMC, we fit synthetic and experimental cardiac voltage data from different pacing rates and find the probability distributions of the parameters of two relatively low-dimensional models, the Mitchell-Schaeffer (MS) and Fenton-Karma (FK) models. We successfully fit synthetic and experimental voltage traces and build populations of action potentials with the posterior probability distributions of the parameters. (ii) We compare the performance of HMC with that of the main Bayesian approach used previously for similar applications, the Approximate Bayesian Computation Sequential Monte Carlo (ABC SMC) algorithm. Both techniques are able to describe the dynamics of synthetic and experimental voltage data using the MS and FK models, with HMC more consistent and ABC SMC more versatile and easier to implement. (iii) We study the variability of cardiac action potentials in space within an individual. We use HMC and a novel approach employing a Gaussian process prior for one spatially varying MS model parameter along with a hierarchical model for the remaining parameters, considered spatially invariant. Using this approach, we do inference and prediction on synthetic cardiac voltage data, exploiting the spatial correlations in cardiac tissue that arise from cellular coupling to use voltage information from a small number of sites to predict parameter value distributions and families of voltage data in other locations. Together these three studies show the potential of Bayesian inference and prediction in providing a framework to represent variability within cardiac electrophysiology modeling
