scholarly journals Fixed Precision MCMC Estimation by Median of Products of Averages

2009 ◽  
Vol 46 (2) ◽  
pp. 309-329 ◽  
Author(s):  
Wojciech Niemiro ◽  
Piotr Pokarowski
Keyword(s):  
Markov Chain ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Mean Square ◽  
Relative Precision ◽  
Mcmc Estimation ◽  
Finite State ◽  
The Mean ◽  
Expected Values ◽  
Finite State Space

The standard Markov chain Monte Carlo method of estimating an expected value is to generate a Markov chain which converges to the target distribution and then compute correlated sample averages. In many applications the quantity of interestθis represented as a product of expected values,θ=µ1⋯µk, and a natural estimator is a product of averages. To increase the confidence level, we can compute a median of independent runs. The goal of this paper is to analyze such an estimator, i.e. an estimator which is a ‘median of products of averages’ (MPA). Sufficient conditions are given forto have fixed relative precision at a given level of confidence, that is, to satisfy. Our main tool is a new bound on the mean-square error, valid also for nonreversible Markov chains on a finite state space.

scholarly journals Fixed Precision MCMC Estimation by Median of Products of Averages

2009 ◽  
Vol 46 (02) ◽  
pp. 309-329 ◽  
Author(s):  
Wojciech Niemiro ◽  
Piotr Pokarowski
Keyword(s):  
Markov Chain ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Mean Square ◽  
Relative Precision ◽  
Mcmc Estimation ◽  
Finite State ◽  
The Mean ◽  
Expected Values ◽  
Finite State Space

The standard Markov chain Monte Carlo method of estimating an expected value is to generate a Markov chain which converges to the target distribution and then compute correlated sample averages. In many applications the quantity of interest θ is represented as a product of expected values, θ = µ 1 ⋯ µ k , and a natural estimator is a product of averages. To increase the confidence level, we can compute a median of independent runs. The goal of this paper is to analyze such an estimator , i.e. an estimator which is a ‘median of products of averages’ (MPA). Sufficient conditions are given for to have fixed relative precision at a given level of confidence, that is, to satisfy . Our main tool is a new bound on the mean-square error, valid also for nonreversible Markov chains on a finite state space.

scholarly journals Lumpings of Markov Chains, Entropy Rate Preservation, and Higher-Order Lumpability

2014 ◽  
Vol 51 (4) ◽  
pp. 1114-1132 ◽  
Author(s):  
Bernhard C. Geiger ◽  
Christoph Temmel
Keyword(s):  
Growth Rate ◽  
Markov Chain ◽  
Sufficient Conditions ◽  
Entropy Rate ◽  
Transition Graph ◽  
Order Markov Chain ◽  
Finite State ◽  
Random Growth ◽  
Irreducible Markov Chain ◽  
Finite State Space

A lumping of a Markov chain is a coordinatewise projection of the chain. We characterise the entropy rate preservation of a lumping of an aperiodic and irreducible Markov chain on a finite state space by the random growth rate of the cardinality of the realisable preimage of a finite-length trajectory of the lumped chain and by the information needed to reconstruct original trajectories from their lumped images. Both are purely combinatorial criteria, depending only on the transition graph of the Markov chain and the lumping function. A lumping is strongly k-lumpable, if and only if the lumped process is a kth-order Markov chain for each starting distribution of the original Markov chain. We characterise strong k-lumpability via tightness of stationary entropic bounds. In the sparse setting, we give sufficient conditions on the lumping to both preserve the entropy rate and be strongly k-lumpable.

scholarly journals Lumpings of Markov Chains, Entropy Rate Preservation, and Higher-Order Lumpability

2014 ◽  
Vol 51 (04) ◽  
pp. 1114-1132 ◽  
Author(s):  
Bernhard C. Geiger ◽  
Christoph Temmel
Keyword(s):  
Growth Rate ◽  
Markov Chain ◽  
Sufficient Conditions ◽  
Entropy Rate ◽  
Transition Graph ◽  
Order Markov Chain ◽  
Finite State ◽  
Random Growth ◽  
Irreducible Markov Chain ◽  
Finite State Space

A lumping of a Markov chain is a coordinatewise projection of the chain. We characterise the entropy rate preservation of a lumping of an aperiodic and irreducible Markov chain on a finite state space by the random growth rate of the cardinality of the realisable preimage of a finite-length trajectory of the lumped chain and by the information needed to reconstruct original trajectories from their lumped images. Both are purely combinatorial criteria, depending only on the transition graph of the Markov chain and the lumping function. A lumping is strongly k-lumpable, if and only if the lumped process is a kth-order Markov chain for each starting distribution of the original Markov chain. We characterise strong k-lumpability via tightness of stationary entropic bounds. In the sparse setting, we give sufficient conditions on the lumping to both preserve the entropy rate and be strongly k-lumpable.

Lattice-valued random walks with Markov chain dependent steps

1979 ◽  
Vol 86 (1) ◽  
pp. 115-126 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Transition Probabilities ◽  
Three Dimensional ◽  
Mean Square Displacement ◽  
Building Model ◽  
Finite State ◽  
The Mean ◽  
Regenerative Phenomenon ◽  
Finite State Space

AbstractThe probability of ever returning to the origin and the mean square displacement after n steps are studied for some lattice-valued random walks, whose successive steps constitute a Markov chain on a finite state space with transition probabilities of a simple kind, and such that the returns to the origin form a regenerative phenomenon. The case of walks on a diamond lattice with no immediate reversals is included: this example is relevant as a polymer chain building model. The numerical evaluation of the return probabilities of some three-dimensional walks is discussed and examples given.

Global dissipativity of stochastic Lotka–Volterra system with feedback controls

2017 ◽  
Vol 10 (02) ◽  
pp. 1750022 ◽  
Author(s):  
Qimin Zhang ◽  
Xinjing Zhang ◽  
Hongfu Yang
Keyword(s):  
Linear Matrix Inequalities ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Jensen Inequality ◽  
Linear Matrix ◽  
Mean Square ◽  
Matrix Inequalities ◽  
Volterra System ◽  
Feedback Controls ◽  
The Mean

In this paper, a class of stochastic Lotka–Volterra system with feedback controls is considered. The purpose is to establish some criteria to ensure the system is globally dissipative in the mean square. By constructing suitable Lyapunov functions as well as combining with Jensen inequality and It[Formula: see text] formula, the sufficient conditions are established and they are expressed in terms of the feasibility to a couple linear matrix inequalities (LMIs). Finally, the main results are illustrated by examples.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (4) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φt)t≥0 be an additive functional defined by φt=∫0tv(Xs)d s. We consider the case in which the process (φt)t≥0 is oscillating and that in which (φt)t≥0 has a negative drift. In each of these cases, we condition the process (Xt,φt)t≥0 on the event that (φt)t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (02) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0,t] of the integral processwhereSis a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

scholarly journals Antiduality and Möbius monotonicity: generalized coupon collector problem

2019 ◽  
Vol 23 ◽  
pp. 739-769
Author(s):  
Paweł Lorek
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Partial Ordering ◽  
Absorption Time ◽  
Absorbing Markov Chain ◽  
Coupon Collector ◽  
Strong Stationary Duality ◽  
Finite State ◽  
A Chain ◽  
Finite State Space

For a given absorbing Markov chain X* on a finite state space, a chain X is a sharp antidual of X* if the fastest strong stationary time (FSST) of X is equal, in distribution, to the absorption time of X*. In this paper, we show a systematic way of finding such an antidual based on some partial ordering of the state space. We use a theory of strong stationary duality developed recently for Möbius monotone Markov chains. We give several sharp antidual chains for Markov chain corresponding to a generalized coupon collector problem. As a consequence – utilizing known results on the limiting distribution of the absorption time – we indicate separation cutoffs (with their window sizes) in several chains. We also present a chain which (under some conditions) has a prescribed stationary distribution and its FSST is distributed as a prescribed mixture of sums of geometric random variables.

scholarly journals Mean-Square Exponential Synchronization of Stochastic Complex Dynamical Networks with Switching Topology by Impulsive Control

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Xuefei Wu ◽  
Chen Xu
Keyword(s):  
Impulsive Control ◽  
Sufficient Conditions ◽  
Exponential Synchronization ◽  
Complex Dynamical Networks ◽  
Switching Topology ◽  
Linear Matrix ◽  
Mean Square ◽  
Dynamical Networks ◽  
The Mean ◽  
Networks With Switching Topology

This paper investigates the mean-square exponential synchronization issues of delayed stochastic complex dynamical networks with switching topology and impulsive control. By using the Lyapunov functional method, impulsive control theory, and linear matrix inequality (LMI) approaches, some sufficient conditions are derived to guarantee the mean-square exponential synchronization of delay complex dynamical network with switch topology, which are independent of the network size and switch topology. Numerical simulations are given to illustrate the effectiveness of the obtained results in the end.

scholarly journals Attractor and Boundedness of Switched Stochastic Cohen-Grossberg Neural Networks

2016 ◽  
Vol 2016 ◽  
pp. 1-19 ◽  
Author(s):  
Chuangxia Huang ◽  
Jie Cao ◽  
Peng Wang
Keyword(s):  
Neural Networks ◽  
Sufficient Conditions ◽  
Average Dwell Time ◽  
Functional Method ◽  
Linear Matrix ◽  
Mean Square ◽  
Ultimate Boundedness ◽  
Mean Square Exponential Stability ◽  
The Mean ◽  
Uniformly Ultimate Boundedness

We address the problem of stochastic attractor and boundedness of a class of switched Cohen-Grossberg neural networks (CGNN) with discrete and infinitely distributed delays. With the help of stochastic analysis technology, the Lyapunov-Krasovskii functional method, linear matrix inequalities technique (LMI), and the average dwell time approach (ADT), some novel sufficient conditions regarding the issues of mean-square uniformly ultimate boundedness, the existence of a stochastic attractor, and the mean-square exponential stability for the switched Cohen-Grossberg neural networks are established. Finally, illustrative examples and their simulations are provided to illustrate the effectiveness of the proposed results.

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