We consider a state-space model (SSM) parametrized by some parameter $\theta$, and our aim is to perform joint parameter and state inference. A simple idea to carry out this task, which almost dates back to the origin of the Kalman filter, is to replace the static parameter $\theta$ by a Markov chain $(\theta_t)_{t\geq 0}$ and then to apply a filtering algorithm to the extended, or self-organized SSM (SO-SSM). However, the practical implementation of this idea in a theoretically justified way has remained an open problem. In this paper we fill this gap by introducing various possible constructions of $(\theta_t)_{t\geq 0}$ that ensure the validity of the SO-SSM for joint parameter and state inference. Notably, we show that such SO-SSMs can be defined even if $\|\mathrm{Var}(\theta_{t}|\theta_{t-1})\|\rightarrow 0$ slowly as $t\rightarrow\infty$. This result is important since, as illustrated in our numerical experiments, these models can be efficiently approximated using particle filter algorithms. While SO-SSMs have been introduced for online inference, the development of iterated filtering (IF) algorithms has shown that they can also serve for computing the maximum likelihood estimator of a given SSM. In this work, we also derive constructions of $(\theta_t)_{t\geq 0}$ and theoretical guarantees tailored to these specific applications of SO-SSMs and, as a result, introduce new IF algorithms. From a practical point of view, the algorithms we develop have the merit of being simple to implement and only requiring minimal tuning to perform well.

翻译：我们考虑一个由参数$\theta$参数化的状态空间模型(SSM)，目标是进行参数与状态的联合推断。执行该任务的一个简单思路——几乎可追溯至卡尔曼滤波的起源——是将静态参数$\theta$替换为马尔可夫链$(\theta_t)_{t\geq 0}$，然后对扩展的或自组织的SSM(SO-SSM)应用滤波算法。然而，以理论合理的方式实现这一思路在实践中始终是未解决的难题。本文通过引入多种确保SO-SSM适用于联合参数与状态推断的$(\theta_t)_{t\geq 0}$构造方案填补了这一空白。值得注意的是，我们证明即使当$\|\mathrm{Var}(\theta_{t}|\theta_{t-1})\|\rightarrow 0$随$t\rightarrow\infty$缓慢趋近于零时，此类SO-SSM仍然可以定义。这一结果具有重要意义，因为如数值实验所示，这些模型可通过粒子滤波算法进行高效近似。虽然SO-SSM最初是为在线推断而提出，但迭代滤波(IF)算法的发展表明它们同样可用于计算给定SSM的最大似然估计量。本工作中，我们还针对SO-SSM的这些特定应用场景推导了$(\theta_t)_{t\geq 0}$的构造方案与理论保证，并由此提出了新的IF算法。从实践角度看，所开发的算法具有实现简单、仅需最小化调参即可获得良好性能的优点。