It has recently been established that the numerical solution of ordinary
differential equations can be posed as a nonlinear Bayesian inference problem,
which can be approximately solved via Gaussian filtering and smoothing,
whenever a Gauss--Markov prior is used. In this paper the class of $\nu$ times
differentiable linear time invariant Gauss--Markov priors is considered. A
taxonomy of Gaussian estimators is established, with the maximum a posteriori
estimate at the top of the hierarchy, which can be computed with the iterated
extended Kalman smoother. The remaining three classes are termed explicit,
semi-implicit, and implicit, which are in similarity with the classical notions
corresponding to conditions on the vector field, under which the filter update
produces a local maximum a posteriori estimate. The maximum a posteriori
estimate corresponds to an optimal interpolant in the reproducing Hilbert space
associated with the prior, which in the present case is equivalent to a Sobolev
space of smoothness $\nu+1$. Consequently, using methods from scattered data
approximation and nonlinear analysis in Sobolev spaces, it is shown that the
maximum a posteriori estimate converges to the true solution at a polynomial
rate in the fill-distance (maximum step size) subject to mild conditions on the
vector field. The methodology developed provides a novel and more natural
approach to study the convergence of these estimators than classical methods of
convergence analysis. The methods and theoretical results are demonstrated in
numerical examples.

本文讨论使用高斯 - 马尔科夫先验将常微分方程的数值解作为非线性贝叶斯推断问题来解决的方法，并针对一些高斯马尔科夫先验阐述了高斯估计器的分类方法，其中最大后验估计处于分类层次结构的顶部。

Bayesian ODE 求解器：最大后验估计

Bayesian ODE Solvers: The Maximum A Posteriori Estimate

We consider approximate maximum likelihood parameter estimation in nonlinear
state-space models. We discuss both direct optimization of the likelihood and
expectation--maximization (EM). For EM, we also give closed-form expressions
for the maximization step in a class of models that are linear in parameters
and have additive noise. To obtain approximations to the filtering and
smoothing distributions needed in the likelihood-maximization methods, we focus
on using Gaussian filtering and smoothing algorithms that employ sigma-points
to approximate the required integrals. We discuss different sigma-point schemes
based on the third, fifth, seventh, and ninth order unscented transforms and
the Gauss--Hermite quadrature rule. We compare the performance of the methods
in two simulated experiments: a univariate nonlinear growth model as well as
tracking of a maneuvering target. In the experiments, we also compare against
approximate likelihood estimates obtained by particle filtering and extended
Kalman filtering based methods. The experiments suggest that the higher-order
unscented transforms may in some cases provide more accurate estimates

研究用于非线性状态空间模型的近似极大似然参数估计，讨论了直接最大化似然和 EM 算法，使用 sigma 点来近似所需积分的高斯滤波和平滑算法，比较了不同阶的 unscented 变换以及高斯 - 埃尔米特积分规则得出的方法在两个模拟实验中的表现