Self-Organizing State-Space Models with Artificial Dynamics

Abstract

We consider the problem of performing parameter and state inference in a state-space model (SSM) parametrized by a static parameter $θ$. A popular idea to address this problem consists of incorporating $θ$ in the state of the system and allowing its time evolution, modelled as a Markov chain $(θ_t)_{t\geq 1}$. This proxy model defines a so-called self-organizing SSM (SO-SSM) to which one may apply standard particle filters. However, the practical implementation of this idea in a theoretically justified manner has remained an open problem until now. In this paper we fill this gap and in particular show that theoretically consistent SO-SSMs can be defined such that $\|\mathrm{Var}(θ_{t+1}|θ_{t})\|\rightarrow 0$ slowly as $t\rightarrow\infty$. This, in turn, leads to particle filter algorithms for online inference in SSMs which we find to be robust in simulation. We also develop constructions of $(θ_t)_{t\geq 1}$ and associated theoretical guarantees tailored to the application of SO-SSMs to maximum likelihood estimation in SSMs, leading to novel iterated filtering algorithms. The algorithms developed in this work have the advantage of being simple to implement and to require minimal tuning to perform well.