Towards an Approximation Theory of Observable Operator Models
Wojciech Anyszka
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unpublished tutorial [Jae98], an inner product structure on the space of future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.