An ordinary differential equation (ODE) model, whose regression curves are a set of solution curves for some ODEs, poses a challenge in parameter estimation. The challenge, due to the frequent absence of analytic solutions and the complicated likelihood surface, tends to be more severe especially for larger models with many parameters and variables. Yang and Lee (2021) proposed a state-space model with variational Bayes (SSVB) for ODE, capable of fast and stable estimation in somewhat large ODE models. The method has shown excellent performance in parameter estimation but has a weakness of underestimation of the posterior covariance, which originates from the mean-field variational method. This paper proposes a way to overcome the weakness by using the Laplace approximation. In numerical experiments, the covariance modified by the Laplace approximation showed a high degree of improvement when checked against the covariances obtained by a standard Markov chain Monte Carlo method. With the improved covariance estimation, the SSVB renders fairly accurate posterior approximations.
翻译:普通的差分方程(ODE)模型,其回归曲线是某些ODE的一套解决方案曲线,对参数估计构成挑战。由于经常缺乏分析解决方案和复杂的可能性表面,挑战往往更为严峻,特别是对于具有许多参数和变量的较大模型而言。杨和李(2021年)提议了一个国家空间模型,对ODE采用可变贝(SSVB),能够在一些较大的ODE模型中快速和稳定地估算。这种方法在参数估计方面表现优异,但对于源自平均场变异法的后方变量估计不足。本文建议了一种方法,通过使用Laplace近比来克服弱点。在数字实验中,由Laplace近比修改的共差值显示在与标准Markov 链 Monte Carlo 方法获得的共差值进行校验时有很大的改进。随着参数估计的改进,SSSVBB提供了相当准确的后方位近似值。