270 research outputs found
Prediction, Retrodiction, and The Amount of Information Stored in the Present
We introduce an ambidextrous view of stochastic dynamical systems, comparing
their forward-time and reverse-time representations and then integrating them
into a single time-symmetric representation. The perspective is useful
theoretically, computationally, and conceptually. Mathematically, we prove that
the excess entropy--a familiar measure of organization in complex systems--is
the mutual information not only between the past and future, but also between
the predictive and retrodictive causal states. Practically, we exploit the
connection between prediction and retrodiction to directly calculate the excess
entropy. Conceptually, these lead one to discover new system invariants for
stochastic dynamical systems: crypticity (information accessibility) and causal
irreversibility. Ultimately, we introduce a time-symmetric representation that
unifies all these quantities, compressing the two directional representations
into one. The resulting compression offers a new conception of the amount of
information stored in the present.Comment: 17 pages, 7 figures, 1 table;
http://users.cse.ucdavis.edu/~cmg/compmech/pubs/pratisp.ht
A note on retrodiction and machine evolution
Biomolecular communication demands that interactions between parts of a
molecular system act as scaffolds for message transmission. It also requires an
evolving and organized system of signs - a communicative agency - for creating
and transmitting meaning. Here I explore the need to dissect biomolecular
communication with retrodiction approaches that make claims about the past
given information that is available in the present. While the passage of time
restricts the explanatory power of retrodiction, the use of molecular structure
in biology offsets information erosion. This allows description of the gradual
evolutionary rise of structural and functional innovations in RNA and proteins.
The resulting chronologies can also describe the gradual rise of molecular
machines of increasing complexity and computation capabilities. For example,
the accretion of rRNA substructures and ribosomal proteins can be traced in
time and placed within a geological timescale. Phylogenetic, algorithmic and
theoretical-inspired accretion models can be reconciled into a congruent
evolutionary model. Remarkably, the time of origin of enzymes, functional RNA,
non-ribosomal peptide synthetase (NRPS) complexes, and ribosomes suggest they
gradually climbed Chomsky's hierarchy of formal grammars, supporting the
gradual complexification of machines and communication in molecular biology.
Future retrodiction approaches and in-depth exploration of theoretical models
of computation will need to confirm such evolutionary progression.Comment: 7 pages, 1 figur
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
Quantifying Self-Organization with Optimal Wavelets
The optimal wavelet basis is used to develop quantitative, experimentally
applicable criteria for self-organization. The choice of the optimal wavelet is
based on the model of self-organization in the wavelet tree. The framework of
the model is founded on the wavelet-domain hidden Markov model and the optimal
wavelet basis criterion for self-organization which assumes inherent increase
in statistical complexity, the information content necessary for maximally
accurate prediction of the system's dynamics. At the same time the method,
presented here for the one-dimensional data of any type, performs superior
denoising and may be easily generalized to higher dimensions.Comment: 12 pages, 3 figure
Adaptive Optical Phase Estimation Using Time-Symmetric Quantum Smoothing
Quantum parameter estimation has many applications, from gravitational wave
detection to quantum key distribution. We present the first experimental
demonstration of the time-symmetric technique of quantum smoothing. We consider
both adaptive and non-adaptive quantum smoothing, and show that both are better
than their well-known time-asymmetric counterparts (quantum filtering). For the
problem of estimating a stochastically varying phase shift on a coherent beam,
our theory predicts that adaptive quantum smoothing (the best scheme) gives an
estimate with a mean-square error up to times smaller than that
from non-adaptive quantum filtering (the standard quantum limit). The
experimentally measured improvement is
The Weak Reality that Makes Quantum Phenomena more Natural: Novel Insights and Experiments
While quantum reality can be probed through measurements, the
Two-State-Vector formalism (TSVF) reveals a subtler reality prevailing between
measurements. Under special pre- and post-selections, odd physical values
emerge. This unusual picture calls for a deeper study. Instead of the common,
wave-based picture of quantum mechanics, we suggest a new, particle-based
perspective: Each particle possesses a definite location throughout its
evolution, while some of its physical variables (characterized by deterministic
operators, some of which obey nonlocal equations of motion) are carried by
"mirage particles" accounting for its unique behavior. Within the time-interval
between pre- and post-selection, the particle gives rise to a horde of such
mirage particles, of which some can be negative. What appears to be
"no-particle," known to give rise to Interaction-Free Measurement, is in fact a
self-canceling pair of positive and negative mirage particles, which can be
momentarily split and cancel out again. Feasible experiments can give empirical
evidence for these fleeting phenomena. In this respect, the Heisenberg ontology
is shown to be conceptually advantageous compared to the Schr\"odinger picture.
We review several recent advances, discuss their foundational significance and
point out possible directions for future research.Comment: An updated version was accepted to Entrop
How Hidden are Hidden Processes? A Primer on Crypticity and Entropy Convergence
We investigate a stationary process's crypticity---a measure of the
difference between its hidden state information and its observed
information---using the causal states of computational mechanics. Here, we
motivate crypticity and cryptic order as physically meaningful quantities that
monitor how hidden a hidden process is. This is done by recasting previous
results on the convergence of block entropy and block-state entropy in a
geometric setting, one that is more intuitive and that leads to a number of new
results. For example, we connect crypticity to how an observer synchronizes to
a process. We show that the block-causal-state entropy is a convex function of
block length. We give a complete analysis of spin chains. We present a
classification scheme that surveys stationary processes in terms of their
possible cryptic and Markov orders. We illustrate related entropy convergence
behaviors using a new form of foliated information diagram. Finally, along the
way, we provide a variety of interpretations of crypticity and cryptic order to
establish their naturalness and pervasiveness. Hopefully, these will inspire
new applications in spatially extended and network dynamical systems.Comment: 18 pages, 18 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/iacp2.ht
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
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