Electronic supplementary material discretion file from A comparison of approximate versus exact techniques for Bayesian parameter inference in nonlinear ordinary differential equation models

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 one ODE epidemiological models with simulated data and one with real data concerning malaria transmission in Afghanistan.

科学与工程领域诸多过程的行为，可通过由一组常微分方程（ordinary differential equations, ODEs）构成的动力学系统模型精准描述。这类模型通常包含若干难以通过实验数据估计的未知参数，此时贝叶斯推断（Bayesian inference）便成为一项实用工具。原则上，借助马尔可夫链蒙特卡洛（Markov chain Monte Carlo, MCMC）技术可实现精确贝叶斯推断；但在实际应用中，这类方法往往存在收敛缓慢、混合性不佳的问题。为解决这一问题，学界已提出若干基于近似贝叶斯计算（approximate Bayesian computation, ABC）的方法，包括马尔可夫链蒙特卡洛ABC（MCMC ABC）与序贯蒙特卡洛ABC（SMC ABC）。尽管ODE系统可刻画生成数据的内在过程，但实测数据无一例外均包含误差。本文指出，若干主流ABC方法未能对这类误差进行恰当建模，究其原因在于其接受概率仅取决于差异函数的选取与容差设定，完全未考虑误差项的影响。我们发现，通过这类方法得到的所谓后验分布，无法准确反映参数取值的认知不确定性。此外，本文通过两组实验验证了这一点：一组采用模拟数据的ODE流行病学模型，另一组采用阿富汗疟疾传播的实测数据，结果显示这类方法相较精确贝叶斯方法几乎未展现出计算优势。