Self-Organized State-Space Models with Artificial Dynamics

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.