In this paper 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 perform this task, which almost dates back to the origin of the Kalman filter, is to replace the static parameter $\theta$ by a Markov chain $(\theta_t)_{t\geq 0}$ on the parameter space and then to apply a standard filtering algorithm to the extended, or self-organized 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 the Markov chain $(\theta_t)_{t\geq 0}$ that ensure the validity of the self-organized SSM (SO-SSM) for joint parameter and state inference. Notably, we show that theoretically valid SO-SSMs can be defined even if $\|\mathrm{Var}(\theta_{t}|\theta_{t-1})\|$ converges to 0 slowly as $t\rightarrow\infty$. This result is important since, as illustrated in our numerical experiments, such models can be efficiently approximated using standard particle filter algorithms. While the idea studied in this work was first introduced for online inference in SSMs, it has also been proved to be useful for computing the maximum likelihood estimator (MLE) of a given SSM, since iterated filtering algorithms can be seen as particle filters applied to SO-SSMs for which the target parameter value is the MLE of interest. Based on this observation, we also derive constructions of $(\theta_t)_{t\geq 0}$ and theoretical results tailored to these specific applications of SO-SSMs, and as a result, we introduce new iterated filtering algorithms. From a practical point of view, the algorithms introduced in this work have the merit of being simple to implement and only requiring minimal tuning to perform well.

翻译：本文考虑由参数 $\theta$ 参数化的状态空间模型（SSM），我们的目标是进行参数与状态的联合推断。执行此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 $\theta$ 替换为参数空间上的马尔可夫链 $(\theta_t)_{t\geq 0}$，然后将标准滤波算法应用于扩展的或自组织的 SSM。然而，以理论合理的方式实现这一思路在实践中一直是一个未解决的问题。本文通过引入多种可能的马尔可夫链 $(\theta_t)_{t\geq 0}$ 构造来填补这一空白，这些构造确保了自组织 SSM（SO-SSM）在参数与状态联合推断中的有效性。值得注意的是，我们证明即使 $\|\mathrm{Var}(\theta_{t}|\theta_{t-1})\|$ 随 $t\rightarrow\infty$ 缓慢收敛至 0，理论上有效的 SO-SSM 仍然可以被定义。这一结果非常重要，因为正如我们的数值实验所示，此类模型可以使用标准粒子滤波算法进行高效近似。尽管本文研究的思想最初是针对 SSM 的在线推断提出的，但它也被证明对计算给定 SSM 的最大似然估计量（MLE）是有用的，因为迭代滤波算法可被视为应用于 SO-SSM 的粒子滤波器，其目标参数值正是我们感兴趣的 MLE。基于这一观察，我们还推导了针对 SO-SSM 这些特定应用的 $(\theta_t)_{t\geq 0}$ 构造及相关理论结果，并由此引入了新的迭代滤波算法。从实践角度来看，本文提出的算法具有易于实现、仅需极少调优即可获得良好性能的优点。