Ordinary differential equation (ODE) models are widely used to describe systems in many areas of science. To ensure these models provide accurate and interpretable representations of real-world dynamics, it is often necessary to infer parameters from data, which involves specifying the form of the ODE system as well as a statistical model describing the observational process. A popular and convenient choice for the error model is a Gaussian distribution with constant variance. However, the choice may not be realistic in many systems, since the variance of the observational error may vary over time or have some dependence on the system state (heteroscedastic), reflecting changes in measurement conditions, environmental fluctuations, or intrinsic system variability. Misspecification of the error model can lead to substantial inaccuracies of the posterior estimates of the ODE model parameters and predictions. More elaborate parametric error models could be specified, but this would increase computational cost because additional parameters would need to be estimated within the MCMC procedure and may still be misspecified. In this work we propose a two-step semi-parametric framework for Bayesian parameter estimation of ODE model parameters when there exists heteroscedasticity in the error process. The first step applies a heteroscedastic Gaussian process to estimate the time-dependent error, and the second step performs Bayesian inference for the ODE model parameters using the estimated time-dependent error estimated from step one in the likelihood function. Through a simulation study and two real-world applications, we demonstrate that the proposed approach yields more reliable posterior inference and predictive uncertainty compared to the standard homoscedastic models. Although our focus is on heteroscedasticity, the framework could be applied to handle more complex error processes.

翻译：常微分方程（ODE）模型广泛应用于科学诸多领域的系统描述中。为确保这些模型能够准确且可解释地反映真实世界动态，通常需要从数据中推断参数，这涉及指定ODE系统的形式以及描述观测过程的统计模型。误差模型的一个常用且便捷的选择是恒定方差的高斯分布。然而，这一选择在许多系统中可能不切实际，因为观测误差的方差可能随时间变化或与系统状态存在某种依赖关系（异方差性），反映测量条件的变化、环境波动或系统内在变异性。误差模型的错误设定可能导致ODE模型参数后验估计及预测出现显著偏差。虽然可以指定更精细的参数化误差模型，但这会增加计算成本，因为需要在MCMC过程中额外估计参数，且其本身仍可能被错误设定。本文针对误差过程中存在异方差性的情况，提出了一种用于ODE模型参数贝叶斯估计的两步半参数框架。第一步运用异方差高斯过程估计时变误差，第二步利用第一步估计得到的时变误差构建似然函数，对ODE模型参数进行贝叶斯推断。通过仿真研究及两项实际应用，我们证明了与标准同方差模型相比，所提方法能产生更可靠的后验推断和预测不确定性。尽管本文聚焦于异方差性，但该框架可推广应用于处理更复杂的误差过程。