Abstract
The behaviour of many processes in science and engineering can be accurately described by dynamical system models consisting of a set of ordinary differential equations (ODEs). Often these models have several unknown parameters that are difficult to estimate from experimental data, in which case Bayesian inference can be a useful tool. In principle, exact Bayesian inference using Markov chain Monte Carlo (MCMC) techniques is possible; however, in practice, such methods may suffer from slow convergence and poor mixing. To address this problem, several approaches based on approximate Bayesian computation (ABC) have been introduced, including Markov chain Monte Carlo ABC (MCMC ABC) and sequential Monte Carlo ABC (SMC ABC). While the system of ODEs describes the underlying process that generates the data, the observed measurements invariably include errors. In this paper, we argue that several popular ABC approaches fail to adequately model these errors because the acceptance probability depends on the choice of the discrepancy function and the tolerance without any consideration of the error term. We observe that the so-called posterior distributions derived from such methods do not accurately reflect the epistemic uncertainties in parameter values. Moreover, we demonstrate that these methods provide minimal computational advantages over exact Bayesian methods when applied to two ODE epidemiological models with simulated data and one with real data concerning malaria transmission in Afghanistan.
Highlights
Models of dynamical systems consisting of sets of ordinary differential equations (ODEs) are an essential tool to describe many processes in science and engineering
In order to improve the low acceptance rate in the basic approximate Bayesian computation (ABC) algorithm, an sequential Monte Carlo ABC (SMC ABC) algorithm was proposed in Sisson et al [14], based on the sequential Monte Carlo (SMC) sampler methodology developed by Del Moral et al [27]
Estimation of the noise parameter is standard using exact Bayesian inference (MCMC), but not with the current practice with ABC-based approaches when applying to a system of ODEs that we investigated here
Summary
Models of dynamical systems consisting of sets of ordinary differential equations (ODEs) are an essential tool to describe many processes in science and engineering. An SMC ABC approach was developed by Toni et al [16], with application to dynamical systems Their algorithm is theoretically sound, but we question the validity of the Bayesian posteriors they produce when they apply ABC to several examples involving ODE models. We show in this paper that an exact Bayesian approach is more computationally efficient than this ‘correct’ ABC implementation, questioning the need for considering ABC in the first place when attempting to estimate the posterior distribution for ODE models. Does this make the comparison invalid, and the resulting approximate posterior distribution produced by SMC ABC does not represent the uncertainty around the parameter values In another example, Silk et al [18] present applications to molecular dynamical systems in which they ‘have focused on the sequential ABC algorithm proposed by Toni et al [16]’.
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