25,879 research outputs found
Linear Distances between Markov Chains
We introduce a general class of distances (metrics) between Markov chains,
which are based on linear behaviour. This class encompasses distances given
topologically (such as the total variation distance or trace distance) as well
as by temporal logics or automata. We investigate which of the distances can be
approximated by observing the systems, i.e. by black-box testing or simulation,
and we provide both negative and positive results
Perturbation bounds and degree of imprecision for uniquely convergent imprecise Markov chains
The effect of perturbations of parameters for uniquely convergent imprecise
Markov chains is studied. We provide the maximal distance between the
distributions of original and perturbed chain and maximal degree of
imprecision, given the imprecision of the initial distribution. The bounds on
the errors and degrees of imprecision are found for the distributions at finite
time steps, and for the stationary distributions as well.Comment: 20 pages, 2 figure
Competition between Two Kinds of Correlations in Literary Texts
A theory of additive Markov chains with long-range memory is used for
description of correlation properties of coarse-grained literary texts. The
complex structure of the correlations in texts is revealed. Antipersistent
correlations at small distances, L 300 define
this nontrivial structure. For some concrete examples of literary texts, the
memory functions are obtained and their power-law behavior at long distances is
disclosed. This property is shown to be a cause of self-similarity of texts
with respect to the decimation procedure.Comment: 7 pages, 7 figures, Submitted to Physica
Information-geometric Markov Chain Monte Carlo methods using Diffusions
Recent work incorporating geometric ideas in Markov chain Monte Carlo is
reviewed in order to highlight these advances and their possible application in
a range of domains beyond Statistics. A full exposition of Markov chains and
their use in Monte Carlo simulation for Statistical inference and molecular
dynamics is provided, with particular emphasis on methods based on Langevin
diffusions. After this geometric concepts in Markov chain Monte Carlo are
introduced. A full derivation of the Langevin diffusion on a Riemannian
manifold is given, together with a discussion of appropriate Riemannian metric
choice for different problems. A survey of applications is provided, and some
open questions are discussed.Comment: 22 pages, 2 figure
Computing Probabilistic Bisimilarity Distances for Probabilistic Automata
The probabilistic bisimilarity distance of Deng et al. has been proposed as a
robust quantitative generalization of Segala and Lynch's probabilistic
bisimilarity for probabilistic automata. In this paper, we present a
characterization of the bisimilarity distance as the solution of a simple
stochastic game. The characterization gives us an algorithm to compute the
distances by applying Condon's simple policy iteration on these games. The
correctness of Condon's approach, however, relies on the assumption that the
games are stopping. Our games may be non-stopping in general, yet we are able
to prove termination for this extended class of games. Already other algorithms
have been proposed in the literature to compute these distances, with
complexity in and \textbf{PPAD}. Despite the
theoretical relevance, these algorithms are inefficient in practice. To the
best of our knowledge, our algorithm is the first practical solution.
The characterization of the probabilistic bisimilarity distance mentioned
above crucially uses a dual presentation of the Hausdorff distance due to
M\'emoli. As an additional contribution, in this paper we show that M\'emoli's
result can be used also to prove that the bisimilarity distance bounds the
difference in the maximal (or minimal) probability of two states to satisfying
arbitrary -regular properties, expressed, eg., as LTL formulas
Additive N-Step Markov Chains as Prototype Model of Symbolic Stochastic Dynamical Systems with Long-Range Correlations
A theory of symbolic dynamic systems with long-range correlations based on
the consideration of the binary N-step Markov chains developed earlier in Phys.
Rev. Lett. 90, 110601 (2003) is generalized to the biased case (non equal
numbers of zeros and unities in the chain). In the model, the conditional
probability that the i-th symbol in the chain equals zero (or unity) is a
linear function of the number of unities (zeros) among the preceding N symbols.
The correlation and distribution functions as well as the variance of number of
symbols in the words of arbitrary length L are obtained analytically and
verified by numerical simulations. A self-similarity of the studied stochastic
process is revealed and the similarity group transformation of the chain
parameters is presented. The diffusion Fokker-Planck equation governing the
distribution function of the L-words is explored. If the persistent
correlations are not extremely strong, the distribution function is shown to be
the Gaussian with the variance being nonlinearly dependent on L. An equation
connecting the memory and correlation function of the additive Markov chain is
presented. This equation allows reconstructing a memory function using a
correlation function of the system. Effectiveness and robustness of the
proposed method is demonstrated by simple model examples. Memory functions of
concrete coarse-grained literary texts are found and their universal power-law
behavior at long distances is revealed.Comment: 19 pages, 8 figure
- âŚ