34 research outputs found
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
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
Time Resolution Dependence of Information Measures for Spiking Neurons: Atoms, Scaling, and Universality
The mutual information between stimulus and spike-train response is commonly
used to monitor neural coding efficiency, but neuronal computation broadly
conceived requires more refined and targeted information measures of
input-output joint processes. A first step towards that larger goal is to
develop information measures for individual output processes, including
information generation (entropy rate), stored information (statistical
complexity), predictable information (excess entropy), and active information
accumulation (bound information rate). We calculate these for spike trains
generated by a variety of noise-driven integrate-and-fire neurons as a function
of time resolution and for alternating renewal processes. We show that their
time-resolution dependence reveals coarse-grained structural properties of
interspike interval statistics; e.g., -entropy rates that diverge less
quickly than the firing rate indicate interspike interval correlations. We also
find evidence that the excess entropy and regularized statistical complexity of
different types of integrate-and-fire neurons are universal in the
continuous-time limit in the sense that they do not depend on mechanism
details. This suggests a surprising simplicity in the spike trains generated by
these model neurons. Interestingly, neurons with gamma-distributed ISIs and
neurons whose spike trains are alternating renewal processes do not fall into
the same universality class. These results lead to two conclusions. First, the
dependence of information measures on time resolution reveals mechanistic
details about spike train generation. Second, information measures can be used
as model selection tools for analyzing spike train processes.Comment: 20 pages, 6 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/trdctim.ht
Nearly maximally predictive features and their dimensions
Scientific explanation often requires inferring maximally predictive features from a given data set. Unfortunately, the collection of minimal maximally predictive features for most stochastic processes is uncountably infinite. In such cases, one compromises and instead seeks nearly maximally predictive features. Here, we derive upper bounds on the rates at which the number and the coding cost of nearly maximally predictive features scale with desired predictive power. The rates are determined by the fractal dimensions of a process' mixed-state distribution. These results, in turn, show how widely used finite-order Markov models can fail as predictors and that mixed-state predictive features can offer a substantial improvement.United States. Army Research Office (W911NF-13-1-0390)United States. Army Research Office (W911NF-12-1- 0288
Prediction and Dissipation in Nonequilibrium Molecular Sensors: Conditionally Markovian Channels Driven by Memoryful Environments.
Biological sensors must often predict their input while operating under metabolic constraints. However, determining whether or not a particular sensor is evolved or designed to be accurate and efficient is challenging. This arises partly from the functional constraints being at cross purposes and partly since quantifying the prediction performance of even in silico sensors can require prohibitively long simulations, especially when highly complex environments drive sensors out of equilibrium. To circumvent these difficulties, we develop new expressions for the prediction accuracy and thermodynamic costs of the broad class of conditionally Markovian sensors subject to complex, correlated (unifilar hidden semi-Markov) environmental inputs in nonequilibrium steady state. Predictive metrics include the instantaneous memory and the total predictable information (the mutual information between present sensor state and input future), while dissipation metrics include power extracted from the environment and the nonpredictive information rate. Success in deriving these formulae relies on identifying the environment's causal states, the input's minimal sufficient statistics for prediction. Using these formulae, we study large random channels and the simplest nontrivial biological sensor model-that of a Hill molecule, characterized by the number of ligands that bind simultaneously-the sensor's cooperativity. We find that the seemingly impoverished Hill molecule can capture an order of magnitude more predictable information than large random channels
Statistical Signatures of Structural Organization: The case of long memory in renewal processes
Identifying and quantifying memory are often critical steps in developing a
mechanistic understanding of stochastic processes. These are particularly
challenging and necessary when exploring processes that exhibit long-range
correlations. The most common signatures employed rely on second-order temporal
statistics and lead, for example, to identifying long memory in processes with
power-law autocorrelation function and Hurst exponent greater than .
However, most stochastic processes hide their memory in higher-order temporal
correlations. Information measures---specifically, divergences in the mutual
information between a process' past and future (excess entropy) and minimal
predictive memory stored in a process' causal states (statistical
complexity)---provide a different way to identify long memory in processes with
higher-order temporal correlations. However, there are no ergodic stationary
processes with infinite excess entropy for which information measures have been
compared to autocorrelation functions and Hurst exponents. Here, we show that
fractal renewal processes---those with interevent distribution tails ---exhibit long memory via a phase transition at .
Excess entropy diverges only there and statistical complexity diverges there
and for all . When these processes do have power-law
autocorrelation function and Hurst exponent greater than , they do not
have divergent excess entropy. This analysis breaks the intuitive association
between these different quantifications of memory. We hope that the methods
used here, based on causal states, provide some guide as to how to construct
and analyze other long memory processes.Comment: 13 pages, 2 figures, 3 appendixes;
http://csc.ucdavis.edu/~cmg/compmech/pubs/lrmrp.ht