The inference of models, prediction of future symbols, and entropy rate estimation of discrete-time, discrete-event processes is well-worn ground. However, many time series are better conceptualized as continuous-time, discrete-event processes. Here, we provide new methods for inferring models, predicting future symbols, and estimating the entropy rate of continuous-time, discrete-event processes. The methods rely on an extension of Bayesian structural inference that takes advantage of neural network’s universal approximation power. Based on experiments with simple synthetic data, these new methods seem to be competitive with state-of- the-art methods for prediction and entropy rate estimation as long as the correct model is inferred. Much scientific data is dynamic, meaning that we see not a static image of a system but its time evolution. The additional richness of dynamic data should allow us to better understand the system, but we may not know how to process the richer data in a way that will yield new insight into the system in question. For example, we have records of when earthquakes have occurred, but still lack the ability to predict earthquakes well or estimate their intrinsic randomness (Geller, 1997); we know which neurons have spiked when, but lack an understanding of the neural code (Rieke et al., 1999) ; and finally, we can observe organisms, but have difficulty modeling their behavior (Berman et al., 2016; Cavagna et al., 2014) . Such examples are not only continuous-time, but also discreteevent, meaning that the observations belong to a finite set (e.g, neuron spikes or is silent) and are not better-described as a collection of real numbers. These disparate scientific problems are begging for a unified framework for inferring expressive continuous-time, discrete-event models and for using those models to make predictions and, potentially, estimate the intrinsic randomness of the system. In this paper, we present a step towards such a unified framework that takes advantage of: the inference and the predictive advantages of unifilarity-meaning that the hidden Markov model's underlying state (the so-called "causal state" (Shalizi & Crutchfield, 2001) or "predictive state representation" (Littman & Sutton, 2002) ) can be uniquely identified from the past data; and the universal approximation power of neural networks (Hornik, 1991) . Indeed, one could view the proposed algorithm for model inference as the continuous-time extension of Bayesian structural inference Strelioff & Crutchfield (2014) . We focus on time series that are discrete-event and inherently stochastic. In particular, we infer the most likely unifilar hidden semi-Markov model (uhsMm) given data using the Bayesian information criterion. This class of models is slightly more powerful than semi-Markov models, in which the future symbol depends only on the prior symbol, but for which the dwell time of the next symbol is drawn from a non-exponential distribution. With unifilar hidden semi-Markov models, the probability of a future symbol depends on arbitrarily long pasts of prior symbols, and the dwell time distribution for that symbol is non-exponential. Beyond just model inference, we can use the inferred model and the closed-form expressions in Ref. (Marzen & Crutchfield, 2017) to estimate the process' entropy rate, and we can use the inferred states of the uhsMm to predict future input via a k-nearest neighbors approach. We compare the latter two algorithms to reasonable extensions of state-of-the-art algorithms. Our new algorithms appear competitive as long as model inference is in-class, meaning that the true model producing the data is equivalent to one of the models in our search. In Sec. 3, we introduce the reader to unifilar hidden semi-Markov models. In Sec. 4, we describe our new algorithms for model inference, entropy rate estimation, and time series prediction and test our algorithms on synthetic data that is memoryful. And in Sec. 5, we discuss potential extensions and applications of this research. We have introduced a new algorithm for inferring causal states (Shalizi & Crutchfield, 2001 ) of a continuous-time, discrete-event process using the groundwork of Ref. (Marzen & Crutchfield, 2017) . We have introduced a new estimator of entropy rate that uses the causal states. And finally, we have shown that a predictor based on causal states is more accurate and less compute-heavy than other predictors. The new inference, estimation, and prediction algorithms could be used to infer a predictive model of complex continuous-time, discrete-event processes, such as animal behavior, and calculate estimates of the intrinsic randomness of such complex processes. Future research could delve into improving estimators of other time series information measures (James et al., 2011) , using something more accurate than BIC to calculate MAP models, or enumerating the topology of all possible uhsMm models for non-binary alphabets (Johnson et al.) .
A new method for inferring a model of, estimating the entropy rate of, and predicting continuous-time, discrete-event processes.
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