This paper deals with the identification of the stochastic Ornstein-Uhlenbeck (OU) process error model, which is characterized by an inverse time constant, and the unknown variances of the process and observation noises. Although the availability of the explicit expression of the log-likelihood function allows one to obtain the maximum likelihood estimator (MLE), this entails evaluating the nontrivial gradient and also often struggles with local optima. To address these limitations, we put forth a sample-efficient global optimization approach based on the Bayesian optimization (BO) framework, which relies on a Gaussian process (GP) surrogate model for the objective function that effectively balances exploration and exploitation to select the query points. Specifically, each evaluation of the objective is implemented efficiently through the Kalman filter (KF) recursion. Comprehensive experiments on various parameter settings and sampling intervals corroborate that BO-based estimator consistently outperforms MLE implemented by the steady-state KF approximation and the expectation-maximization algorithm (whose derivation is a side contribution) in terms of root mean-square error (RMSE) and statistical consistency, confirming the effectiveness and robustness of the BO for identification of the stochastic OU process. Notably, the RMSE values produced by the BO-based estimator are smaller than the classical Cram\'{e}r-Rao lower bound, especially for the inverse time constant, estimating which has been a long-standing challenge. This seemingly counterintuitive result can be explained by the data-driven prior for the learning parameters indirectly injected by BO through the GP prior over the objective function.

翻译：本文研究随机奥恩斯坦-乌伦贝克（OU）过程误差模型的辨识问题，该模型由逆时间常数以及过程噪声和观测噪声的未知方差所表征。尽管对数似然函数的显式表达式允许获得最大似然估计量（MLE），但这需要计算非平凡梯度且常陷入局部最优。为克服这些局限，我们提出一种基于贝叶斯优化（BO）框架的样本高效全局优化方法，该方法通过高斯过程（GP）代理模型对目标函数进行建模，有效平衡探索与利用以选择查询点。具体而言，每次目标函数评估均通过卡尔曼滤波器（KF）递归高效实现。在不同参数设置与采样间隔下的综合实验表明，基于BO的估计器在均方根误差（RMSE）和统计一致性方面持续优于采用稳态KF近似和期望最大化算法（其推导过程作为附带贡献）实现的MLE，证实了BO在随机OU过程辨识中的有效性与鲁棒性。值得注意的是，基于BO的估计器产生的RMSE值低于经典克拉默-拉奥下界，尤其对于逆时间常数这一长期存在估计挑战的参数。这一看似反直觉的结果可通过BO经目标函数的GP先验间接注入学习参数的数据驱动先验来解释。