Towards an Approximation Theory of Observable Operator Models
Wojciech Anyszka
TL;DR
The paper tackles the challenge of developing an approximation theory for Observable Operator Models (OOMs) applied to infinite-dimensional stochastic processes. It constructs a rigorous inner-product on the space of future distributions $\mathcal{G}$ by embedding into $L^2(\mu_{\epsilon})$ through Radon-Nikodym densities, and proves continuity of the observable operators with $\|t_a\|\le 1$, enabling a Hilbert-space viewpoint. The central finding is that $(\mathcal{G},\| abla\cdot\|)$ is complete if and only if $\mathcal{G}$ is finite-dimensional, indicating the original infinite-dimensional goal cannot form a Hilbert space in this framework. The work also provides computable formulas for the inner-product structure and density functions, and proposes enlarging the space to the closure of $\phi(\mathcal{G})$ to extend operators to a Hilbert space for finite-rank approximation. This identifies a fundamental obstacle and charts a concrete direction for future research in approximating OOMs of infinite-dimensional processes.
Abstract
Observable operator models (OOMs) offer a powerful framework for modelling stochastic processes, surpassing the traditional hidden Markov models (HMMs) in generality and efficiency. However, using OOMs to model infinite-dimensional processes poses significant theoretical challenges. This article explores a rigorous approach to developing an approximation theory for OOMs of infinite-dimensional processes. Building upon foundational work outlined in an unpublished tutorial [Jae98], an inner product structure on the space of future distributions is rigorously established and the continuity of observable operators with respect to the associated 2-norm is proven. The original theorem proven in this thesis describes a fundamental obstacle in making an infinite-dimensional space of future distributions into a Hilbert space. The presented findings lay the groundwork for future research in approximating observable operators of infinite-dimensional processes, while a remedy to the encountered obstacle is suggested.