Norm observable operator models

Hidden Markov models (HMMs) are one of the most popular and success-ful statistical models for time series. Observable operator models (OOMs) are generalizations of HMMs which exhibit several attractive advantages. In particular, a variety of highly efficient, constructive and asymptotically cor-rect learning algorithms are available for OOMs. However, the OOM theory suffers from the negative probability problem (NPP): a given, learnt OOM may sometimes predict negative “probabilities ” for certain events. It was recently shown that it is undecidable whether a given OOM will eventually produce such negative values. We propose a novel variant of OOMs, called norm observable operator models (NOOMs), which avoid the NPP by design. Like OOMs, NOOMs use a set of linear operators to update system states. But differing from OOMs, they represent probabilities by the square of the norm of system states, thus precluding negative “probability ” values. While being free of the NPP, NOOMs retain most advantages of OOMs. For example, NOOMs also capture (some) processes that cannot be modelled by HMMs. More importantly, in principle NOOMs can be learnt from data in a constructive way; and the learnt models are asymptotically correct. We also prove that NOOMs capture all Markov chain (MC) describable processes. This contribution presents the mathematical foundations of NOOMs, dis-cusses the expressiveness of the model class, and explains how a NOOM can be estimated from data constructively.