We consider a state-space model (SSM) parametrized by some parameter $\theta$ and aim at performing joint parameter and state inference. A popular idea to carry out this task is to replace $\theta$ 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 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 these models can be efficiently approximated using a particle filter. While SO-SSMs have been introduced for online inference, the development of iterated filtering (IF) has shown that they can also serve for computing the maximum likelihood estimator of an SSM. We also derive constructions of $(\theta_t)_{t\geq 0}$ and theoretical guarantees tailored to these specific applications of SO-SSMs and introduce new IF algorithms. From a practical point of view, the algorithms we develop are simple to implement and only require 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），旨在进行参数与状态的联合推断。执行该任务的一种流行思路是将$\theta$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仍是一个未解决的问题。本文通过引入确保SO-SSM适用于联合参数与状态推断的$(\theta_t)_{t\geq 0}$构造来填补这一空白。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仍然可以被定义。这一结果具有重要意义，因为这些模型可以通过粒子滤波器进行高效近似。虽然SO-SSM最初是为在线推断提出的，但迭代滤波（IF）的发展表明它们同样可用于计算SSM的最大似然估计量。我们还针对SO-SSM的这些特定应用场景，推导了$(\theta_t)_{t\geq 0}$的构造方法及相关理论保证，并提出了新的IF算法。从实践角度来看，我们所开发的算法实现简单，仅需极少的调参即可获得良好性能。