5,051 research outputs found
Pathwise Sensitivity Analysis in Transient Regimes
The instantaneous relative entropy (IRE) and the corresponding instanta-
neous Fisher information matrix (IFIM) for transient stochastic processes are
pre- sented in this paper. These novel tools for sensitivity analysis of
stochastic models serve as an extension of the well known relative entropy rate
(RER) and the corre- sponding Fisher information matrix (FIM) that apply to
stationary processes. Three cases are studied here, discrete-time Markov
chains, continuous-time Markov chains and stochastic differential equations. A
biological reaction network is presented as a demonstration numerical example
Stochastic time-evolution, information geometry and the Cramer-Rao Bound
We investigate the connection between the time-evolution of averages of
stochastic quantities and the Fisher information and its induced statistical
length. As a consequence of the Cramer-Rao bound, we find that the rate of
change of the average of any observable is bounded from above by its variance
times the temporal Fisher information. As a consequence of this bound, we
obtain a speed limit on the evolution of stochastic observables: Changing the
average of an observable requires a minimum amount of time given by the change
in the average squared, divided by the fluctuations of the observable times the
thermodynamic cost of the transformation. In particular for relaxation
dynamics, which do not depend on time explicitly, we show that the Fisher
information is a monotonically decreasing function of time and that this
minimal required time is determined by the initial preparation of the system.
We further show that the monotonicity of the Fisher information can be used to
detect hidden variables in the system and demonstrate our findings for simple
examples of continuous and discrete random processes.Comment: 25 pages, 4 figure
Path mutual information for a class of biochemical reaction networks
Living cells encode and transmit information in the temporal dynamics of
biochemical components. Gaining a detailed understanding of the input-output
relationship in biological systems therefore requires quantitative measures
that capture the interdependence between complete time trajectories of
biochemical components. Mutual information provides such a measure but its
calculation in the context of stochastic reaction networks is associated with
mathematical challenges. Here we show how to estimate the mutual information
between complete paths of two molecular species that interact with each other
through biochemical reactions. We demonstrate our approach using three simple
case studies.Comment: 6 pages, 2 figure
Estimation of Continuous Time Models in Economics: an Overview
The dynamics of economic behaviour is often developed in theory as a continuous time system. Rigorous estimation and testing of such systems, and the analysis of some aspects of their properties, is of particular importance in distinguishing between competing hypotheses and the resulting models. The consequences for the international economy during the past eighteen months of failures in the financial sector, and particularly the banking sector, make it essential that the dynamics of financial and commodity markets and of macro-economic policy are well understood. The nonlinearity of the economic system means that itâs properties are heavily dependent on itâs parameter values. The estimators discussed here are tools to provide those parameter estimates.
Variational characterization of free energy: theory and algorithms
The article surveys and extends variational formulations of the thermodynamic free energy and discusses their information-theoretic content from the perspective of mathematical statistics. We revisit the well-known Jarzynski equality for nonequilibrium free energy sampling within the framework of importance sampling and Girsanov change-of-measure transformations.
The implications of the different variational formulations for designing efficient stochastic optimization and nonequilibrium simulation algorithms for computing free energies are discussed and illustrated
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
In this article we consider one-dimensional random systems of hyperbolic
conservation laws. We first establish existence and uniqueness of random
entropy admissible solutions for initial value problems of conservation laws
which involve random initial data and random flux functions. Based on these
results we present an a posteriori error analysis for a numerical approximation
of the random entropy admissible solution. For the stochastic discretization,
we consider a non-intrusive approach, the Stochastic Collocation method. The
spatio-temporal discretization relies on the Runge--Kutta Discontinuous
Galerkin method. We derive the a posteriori estimator using continuous
reconstructions of the discrete solution. Combined with the relative entropy
stability framework this yields computable error bounds for the entire
space-stochastic discretization error. The estimator admits a splitting into a
stochastic and a deterministic (space-time) part, allowing for a novel
residual-based space-stochastic adaptive mesh refinement algorithm. We conclude
with various numerical examples investigating the scaling properties of the
residuals and illustrating the efficiency of the proposed adaptive algorithm
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