Self-Organized State-Space Models with Artificial Dynamics

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.