fig_5 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，均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刻画。这类模型往往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）便可成为有效的分析工具。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，这类方法常面临收敛速度缓慢、混合效果不佳的问题。为解决该问题，学界已提出多种基于近似贝叶斯计算（approximate Bayesian computation, ABC）的改进方案，包括马尔可夫链蒙特卡洛近似贝叶斯计算（MCMC ABC）与序贯蒙特卡洛近似贝叶斯计算（SMC ABC）。尽管常微分方程系统可刻画生成观测数据的内在过程，但实际观测得到的测量值必然包含误差。本文指出，诸多主流近似贝叶斯计算方法未能对这类误差进行恰当建模：其接受概率仅由差异函数（discrepancy function）与容差（tolerance）的选取决定，未对误差项（error term）加以考量。我们发现，由这类方法推导得到的所谓后验分布（posterior distributions），无法准确反映参数取值所对应的认知不确定性（epistemic uncertainties）。此外，针对两组场景：一组为搭载模拟数据的单常微分方程流行病学模型，另一组为涉及阿富汗疟疾传播（malaria transmission）的真实观测数据集，我们通过实验证明，相较于精确贝叶斯推断方法，这类近似贝叶斯计算方法几乎未体现出计算效率优势。