# Structure and Randomness of Continuous-Time Discrete-Event Processes

## TL;DR

This paper introduces new methods to calculate the entropy rate and statistical complexity of finite unifilar hidden semi-Markov processes, advancing understanding of their randomness and predictability.

## Contribution

It develops novel mathematical tools and information-theoretic techniques to analyze the entropy and complexity of memoryful, state-dependent renewal processes.

## Key findings

- First calculation of entropy rate for hidden semi-Markov models
- Introduction of -machines for these processes
- New methods for quantifying process complexity

## Abstract

Loosely speaking, the Shannon entropy rate is used to gauge a stochastic process' intrinsic randomness; the statistical complexity gives the cost of predicting the process. We calculate, for the first time, the entropy rate and statistical complexity of stochastic processes generated by finite unifilar hidden semi-Markov models---memoryful, state-dependent versions of renewal processes. Calculating these quantities requires introducing novel mathematical objects ({\epsilon}-machines of hidden semi-Markov processes) and new information-theoretic methods to stochastic processes.