Initial_parameters_Malariah_final 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.

科学与工程领域的诸多过程行为，均可通过由一组常微分方程（ordinary differential equations, ODEs）构建的动力学系统模型实现精准描述。此类模型通常包含若干未知参数，且难以通过实验数据完成参数估计，此时贝叶斯推断便是一种极具实用价值的分析工具。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，但实际应用中，此类方法往往存在收敛速度缓慢、混合性差的问题。为解决这一问题，学界已提出多种基于近似贝叶斯计算（approximate Bayesian computation, ABC）的方法，涵盖马尔可夫链蒙特卡洛ABC（MCMC ABC）与序列蒙特卡洛ABC（SMC ABC）等类型。尽管常微分方程系统可刻画生成观测数据的内在过程，但实际获取的测量数据总会附带误差。本文指出，诸多主流ABC方法均未能对这类误差进行恰当建模，究其根源在于其接受概率仅取决于差异函数的选择与容差设定，未对误差项予以考量。我们发现，通过此类方法得到的所谓后验分布，无法准确反映参数取值中的认知不确定性。此外，我们通过实验证实，在分别应用于搭载模拟数据的常微分方程流行病学模型，以及涉及阿富汗疟疾传播的真实数据集对应的常微分方程流行病学模型时，此类方法相较于精确贝叶斯方法仅具备极微弱的计算优势。