Informational and Causal Architecture of Discrete-Time Renewal Processes

Renewal processes are broadly used to model stochastic behavior consisting of isolated events separated by periods of quiescence, whose durations are specified by a given probability law. Here, we identify the minimal sufficient statistic for their prediction (the set of causal states), calculate the historical memory capacity required to store those states (statistical complexity), delineate what information is predictable (excess entropy), and decompose the entropy of a single measurement into that shared with the past, future, or both. The causal state equivalence relation defines a new subclass of renewal processes with a finite number of causal states despite having an unbounded interevent count distribution. We use these formulae to analyze the output of the parametrized Simple Nonunifilar Source, generated by a simple two-state hidden Markov model, but with an infinite-state epsilon-machine presentation. All in all, the results lay the groundwork for analyzing processes with infinite statistical complexity and infinite excess entropy.Comment: 18 pages, 9 figures, 1 table; http://csc.ucdavis.edu/~cmg/compmech/pubs/dtrp.ht

Information Anatomy of Stochastic Equilibria

A stochastic nonlinear dynamical system generates information, as measured by its entropy rate. Some---the ephemeral information---is dissipated and some---the bound information---is actively stored and so affects future behavior. We derive analytic expressions for the ephemeral and bound informations in the limit of small-time discretization for two classical systems that exhibit dynamical equilibria: first-order Langevin equations (i) where the drift is the gradient of a potential function and the diffusion matrix is invertible and (ii) with a linear drift term (Ornstein-Uhlenbeck) but a noninvertible diffusion matrix. In both cases, the bound information is sensitive only to the drift, while the ephemeral information is sensitive only to the diffusion matrix and not to the drift. Notably, this information anatomy changes discontinuously as any of the diffusion coefficients vanishes, indicating that it is very sensitive to the noise structure. We then calculate the information anatomy of the stochastic cusp catastrophe and of particles diffusing in a heat bath in the overdamped limit, both examples of stochastic gradient descent on a potential landscape. Finally, we use our methods to calculate and compare approximations for the so-called time-local predictive information for adaptive agents.Comment: 35 pages, 3 figures, 1 table; http://csc.ucdavis.edu/~cmg/compmech/pubs/iase.ht

Signatures of Infinity: Nonergodicity and Resource Scaling in Prediction, Complexity, and Learning

We introduce a simple analysis of the structural complexity of infinite-memory processes built from random samples of stationary, ergodic finite-memory component processes. Such processes are familiar from the well known multi-arm Bandit problem. We contrast our analysis with computation-theoretic and statistical inference approaches to understanding their complexity. The result is an alternative view of the relationship between predictability, complexity, and learning that highlights the distinct ways in which informational and correlational divergences arise in complex ergodic and nonergodic processes. We draw out consequences for the resource divergences that delineate the structural hierarchy of ergodic processes and for processes that are themselves hierarchical.Comment: 8 pages, 1 figure; http://csc.ucdavis.edu/~cmg/compmech/pubs/soi.pd

Child poverty in rural America: new data shows increases in 41 states

A study by the Carsey Institute, based on U.S. Census Bureau data, found that in forty-one states, a higher percentage of rural children live in poverty than did in 2000. While the poverty level in 2006 was relatively stagnant compared to 2005\u27s poverty level, the situation is clearly becoming worse for rural kids

Informational and Causal Architecture of Continuous-time Renewal and Hidden Semi-Markov Processes

We introduce the minimal maximally predictive models ({\epsilon}-machines) of processes generated by certain hidden semi-Markov models. Their causal states are either hybrid discrete-continuous or continuous random variables and causal-state transitions are described by partial differential equations. Closed-form expressions are given for statistical complexities, excess entropies, and differential information anatomy rates. We present a complete analysis of the {\epsilon}-machines of continuous-time renewal processes and, then, extend this to processes generated by unifilar hidden semi-Markov models and semi-Markov models. Our information-theoretic analysis leads to new expressions for the entropy rate and the rates of related information measures for these very general continuous-time process classes.Comment: 16 pages, 7 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/ctrp.ht

Child poverty high in rural America

On August 28, 2007, new data from the U.S. Census Bureau\u27s American Community Survey show that 22 percent of rural children are living in poverty, up from 19 percent in 2000. On average, rates are highest in the nonmetropolitan South (27 percent) and have climbed the most in the nonmetropolitan Midwest (by 3.9 percentage points)

Optimized Bacteria are Environmental Prediction Engines

Experimentalists have observed phenotypic variability in isogenic bacteria populations. We explore the hypothesis that in fluctuating environments this variability is tuned to maximize a bacterium's expected log growth rate, potentially aided by epigenetic markers that store information about past environments. We show that, in a complex, memoryful environment, the maximal expected log growth rate is linear in the instantaneous predictive information---the mutual information between a bacterium's epigenetic markers and future environmental states. Hence, under resource constraints, optimal epigenetic markers are causal states---the minimal sufficient statistics for prediction. This is the minimal amount of information about the past needed to predict the future as well as possible. We suggest new theoretical investigations into and new experiments on bacteria phenotypic bet-hedging in fluctuating complex environments.Comment: 7 pages, 1 figure; http://csc.ucdavis.edu/~cmg/compmech/pubs/obepe.ht

Prediction and Power in Molecular Sensors: Uncertainty and Dissipation When Conditionally Markovian Channels Are Driven by Semi-Markov Environments

Sensors often serve at least two purposes: predicting their input and minimizing dissipated heat. However, determining whether or not a particular sensor is evolved or designed to be accurate and efficient is difficult. This arises partly from the functional constraints being at cross purposes and partly since quantifying the predictive performance of even in silico sensors can require prohibitively long simulations. To circumvent these difficulties, we develop expressions for the predictive accuracy and thermodynamic costs of the broad class of conditionally Markovian sensors subject to unifilar hidden semi-Markov (memoryful) environmental inputs. Predictive metrics include the instantaneous memory and the mutual information between present sensor state and input future, while dissipative metrics include power consumption and the nonpredictive information rate. Success in deriving these formulae relies heavily on identifying the environment's causal states, the input's minimal sufficient statistics for prediction. Using these formulae, we study the simplest nontrivial biological sensor model---that of a Hill molecule, characterized by the number of ligands that bind simultaneously, the sensor's cooperativity. When energetic rewards are proportional to total predictable information, the closest cooperativity that optimizes the total energy budget generally depends on the environment's past hysteretically. In this way, the sensor gains robustness to environmental fluctuations. Given the simplicity of the Hill molecule, such hysteresis will likely be found in more complex predictive sensors as well. That is, adaptations that only locally optimize biochemical parameters for prediction and dissipation can lead to sensors that "remember" the past environment.Comment: 21 pages, 4 figures, http://csc.ucdavis.edu/~cmg/compmech/pubs/piness.ht