This article develops a methodology allowing application of the complete machinery of particle-based inference methods upon what we call the class of continuous-discrete State Space Models (CD-SSMs). Such models correspond to a latent continuous-time It\^o diffusion process which is observed with noise at discrete time instances. Due to the continuous-time nature of the hidden signal, standard Feynman-Kac formulations and their accompanying particle-based approximations have to overcome several challenges, arising mainly due to the following considerations: (i) finite-time transition densities of the signal are typically intractable; (ii) ancestors of sampled signals are determined w.p.~1, thus cannot be resampled; (iii) diffusivity parameters given a sampled signal yield Dirac distributions. We overcome all above issues by introducing a framework based on carefully designed proposals and transformations thereof. That is, we obtain new expressions for the Feynman-Kac model that accommodate the effects of a continuous-time signal and overcome induced degeneracies. The constructed formulations will enable use of the full range of particle-based algorithms for CD-SSMs: for filtering/smoothing and parameter inference, whether online or offline. Our framework is compatible with guided proposals in the filtering steps that are essential for efficient algorithmic performance in the presence of informative observations or in higher dimensions, and is applicable for a very general class of CD-SSMs, including the case when the signal is modelled as a hypo-elliptic diffusion. Our methods can be immediately incorporated to available software packages for particle-based algorithms.

翻译：本文提出了一种方法论，使得我们能够将基于粒子的推断方法的完整机制应用于我们称之为连续-离散状态空间模型（CD-SSMs）的类别。此类模型对应于一个潜在的连续时间Itô扩散过程，该过程在离散时间点上被带有噪声地观测到。由于隐藏信号的连续时间特性，标准的Feynman-Kac公式及其伴随的基于粒子的近似必须克服几个挑战，这些挑战主要源于以下考虑：(i) 信号的有限时间转移密度通常是难以处理的；(ii) 采样信号的祖先几乎必然确定，因此无法进行重采样；(iii) 给定采样信号的扩散参数会产生狄拉克分布。我们通过引入一个基于精心设计的提议分布及其变换的框架来克服上述所有问题。也就是说，我们获得了Feynman-Kac模型的新表达式，这些表达式适应了连续时间信号的影响并克服了由此引起的退化问题。所构建的公式将使得CD-SSMs能够使用全范围的基于粒子的算法：用于滤波/平滑和参数推断，无论是在线还是离线。我们的框架与滤波步骤中的引导提议兼容，这些引导提议对于在存在信息性观测或高维情况下实现高效的算法性能至关重要，并且适用于非常广泛的CD-SSMs类别，包括信号被建模为亚椭圆扩散的情况。我们的方法可以立即整合到现有的基于粒子算法的软件包中。