Informational and Causal Architecture of
Continuous-Time Renewal Processes

ABSTRACT: We introduce the minimal maximally predictive models (ε-machines) of processes generated by certain hidden semi-Markov models. Their causal states are either discrete, mixed, or continuous random variables and causal-state transitions are described by partial differential equations. As an application, we present a complete analysis of the ε-machines of continuous-time renewal processes. This leads to closed-form expressions for their entropy rate, statistical complexity, excess entropy, and differential information anatomy rates.