Valid Parameters for Predictive State Representations

Predictive state representations (PSRs) represent the state of a dynamical system as a set of predictions about future events. The parameters of a PSR model consist of several matrices and vectors, but not all values for those parameters result in valid PSR models. Our work starts with a general definition of what it means to be a valid PSR model and derives necessary and sufficient constraints for the model parameters to constitute a valid PSR. These same constraints also define the set of valid state vectors for a given PSR model, which we prove to be a convex set. We also derive a set of simplified constraints on the PSR parameters, and we prove that any PSR model has an equivalent arameterization that satisfies those simplified constraints. lastly, we demonstrate one simple application of our constraints: preventing overflow or underflow of the PSR state as it changes over time