82,436 research outputs found
On Hidden States in Quantum Random Walks
It was recently pointed out that identifiability of quantum random walks and
hidden Markov processes underlie the same principles. This analogy immediately
raises questions on the existence of hidden states also in quantum random walks
and their relationship with earlier debates on hidden states in quantum
mechanics. The overarching insight was that not only hidden Markov processes,
but also quantum random walks are finitary processes. Since finitary processes
enjoy nice asymptotic properties, this also encourages to further investigate
the asymptotic properties of quantum random walks. Here, answers to all these
questions are given. Quantum random walks, hidden Markov processes and finitary
processes are put into a unifying model context. In this context, quantum
random walks are seen to not only enjoy nice ergodic properties in general, but
also intuitive quantum-style asymptotic properties. It is also pointed out how
hidden states arising from our framework relate to hidden states in earlier,
prominent treatments on topics such as the EPR paradoxon or Bell's
inequalities.Comment: 26 page
Entropy rate calculations of algebraic measures
Let . We use a special class of translation invariant
measures on called algebraic measures to study the entropy rate
of a hidden Markov processes. Under some irreducibility assumptions of the
Markov transition matrix we derive exact formulas for the entropy rate of a
general state hidden Markov process derived from a Markov source corrupted
by a specific noise model. We obtain upper bounds on the error when using an
approximation to the formulas and numerically compute the entropy rates of two
and three state hidden Markov models
Structure and Randomness of Continuous-Time Discrete-Event Processes
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.Comment: 10 pages, 2 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/ctdep.ht
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
On Hidden Markov Processes with Infinite Excess Entropy
We investigate stationary hidden Markov processes for which mutual
information between the past and the future is infinite. It is assumed that the
number of observable states is finite and the number of hidden states is
countably infinite. Under this assumption, we show that the block mutual
information of a hidden Markov process is upper bounded by a power law
determined by the tail index of the hidden state distribution. Moreover, we
exhibit three examples of processes. The first example, considered previously,
is nonergodic and the mutual information between the blocks is bounded by the
logarithm of the block length. The second example is also nonergodic but the
mutual information between the blocks obeys a power law. The third example
obeys the power law and is ergodic.Comment: 12 page
Novel Lower Bounds on the Entropy Rate of Binary Hidden Markov Processes
Recently, Samorodnitsky proved a strengthened version of Mrs. Gerber's Lemma,
where the output entropy of a binary symmetric channel is bounded in terms of
the average entropy of the input projected on a random subset of coordinates.
Here, this result is applied for deriving novel lower bounds on the entropy
rate of binary hidden Markov processes. For symmetric underlying Markov
processes, our bound improves upon the best known bound in the very noisy
regime. The nonsymmetric case is also considered, and explicit bounds are
derived for Markov processes that satisfy the -RLL constraint
Logic and model checking for hidden Markov models
The branching-time temporal logic PCTL* has been introduced to specify quantitative properties over probability systems, such as discrete-time Markov chains. Until now, however, no logics have been defined to specify properties over hidden Markov models (HMMs). In HMMs the states are hidden, and the hidden processes produce a sequence of observations. In this paper we extend the logic PCTL* to POCTL*. With our logic one can state properties such as "there is at least a 90 percent probability that the model produces a given sequence of observations" over HMMs. Subsequently, we give model checking algorithms for POCTL* over HMMs
Hidden processes and hidden Markov processes: classical and quantum
This paper consists of parts. The first part only considers classical
processes and introduces two different extensions of the notion of hidden
Markov process. In the second part, the notion of quantum hidden process is
introduced. In the third part it is proven that, by restricting various types
of quantum Markov chains to appropriate commutative sub--algebras (diagonal
sub--algebras) one recovers all the classical hidden process and, in addition,
one obtains families of processes which are not usual hidden Markov process,
but are included in the above mentioned extensions of these processes. In this
paper we only deal with processes with an at most countable state space
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