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12
THE MARKOV CHAIN MONTE CARLO REVOLUTION
"... Abstract. The use of simulation for highdimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through microlocal analysis. 1. ..."
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Abstract. The use of simulation for highdimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through microlocal analysis. 1.
POWER DIAGRAMS AND INTERACTION PROCESSES FOR UNIONS OF DISCS
, 2007
"... We study a flexible class of finite disc process models with interaction between the discs. We let U denote the random set given by the union of discs, and use for the disc process an exponential family density with the canonical sufficient statistic only depending on geometric properties of U such ..."
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Cited by 8 (4 self)
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We study a flexible class of finite disc process models with interaction between the discs. We let U denote the random set given by the union of discs, and use for the disc process an exponential family density with the canonical sufficient statistic only depending on geometric properties of U such as the area, perimeter, EulerPoincaré characteristic, and number of holes. This includes the quarmassinteraction process and the continuum random cluster model as special cases. Viewing our model as a connected component Markov point process, and thereby establish local and spatial Markov properties, becomes useful for handling the problem of edge effects when only U is observed within a bounded observation window. The power tessellation and its dual graph become major tools when establishing inclusionexclusion formulae, formulae for computing geometric characteristics of U, and stability properties of the underlying disc process density. Algorithms for constructing the power tessellation of U and for simulating the disc process are discussed, and the software is made public available.
Perfect Simulation for a Class of Positive Recurrent Markov Chains
, 2006
"... Abstract This paper generalises the work of [11], which showed that perfect simulation, in the form of dominated coupling from the past, is always possible (though not necessarily practical) for geometrically ergodic Markov chains. Here we consider the more general situation of positive recurrent ch ..."
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Cited by 7 (3 self)
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Abstract This paper generalises the work of [11], which showed that perfect simulation, in the form of dominated coupling from the past, is always possible (though not necessarily practical) for geometrically ergodic Markov chains. Here we consider the more general situation of positive recurrent chains, and explore when it is possible to produce such a simulation algorithm for these chains. We introduce a class of chains which we name tame, for which we show that perfect simulation is possible.
Geometric Ergodicity of Two–dimensional Hamiltonian systems with a Lennard–Jones–like Repulsive Potential. ArXiv eprints
, 2011
"... Molecular dynamics simulation is among the most important and widely used tools in the study of molecular systems, providing fundamental insights into molecular mechanisms at a level of detail unattainable by experimental methods [AT87, Lea96, FS96, Sch02]. Usage of molecular dynamics spans a diver ..."
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Cited by 1 (1 self)
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Molecular dynamics simulation is among the most important and widely used tools in the study of molecular systems, providing fundamental insights into molecular mechanisms at a level of detail unattainable by experimental methods [AT87, Lea96, FS96, Sch02]. Usage of molecular dynamics spans a diverse array of fields,
Perfect sampling for infinite server and loss systems. arXiv 1312.4088
, 2013
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Positive Recurrent Markov Chains
, 2006
"... Abstract This paper generalises the work of [13], which showed that perfect simulation, in the form of dominated coupling from the past, is always possible (though not necessarily practical) for geometrically ergodic Markov chains. Here we consider the more general situation of positive recurrent ch ..."
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Abstract This paper generalises the work of [13], which showed that perfect simulation, in the form of dominated coupling from the past, is always possible (though not necessarily practical) for geometrically ergodic Markov chains. Here we consider the more general situation of positive recurrent chains, and explore when it is possible to produce such a simulation algorithm for these chains. We introduce a class of chains which we name tame, for which we show that perfect simulation is possible.
Comment: Gibbs Sampling, Exponential Families, and Orthogonal Polynomials
, 808
"... It is our pleasure to congratulate the authors (hereafter DKSC) on an interesting paper that was a delight to read. While DKSC provide a remarkable collection of connections between different representations of the Markov chains in their paper, we will focus on the “running time analysis ” portion. ..."
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It is our pleasure to congratulate the authors (hereafter DKSC) on an interesting paper that was a delight to read. While DKSC provide a remarkable collection of connections between different representations of the Markov chains in their paper, we will focus on the “running time analysis ” portion. This is a familiar problem to statisticians; given a target population, how can we obtain a representative sample? In the context of Markov chain Monte Carlo (MCMC) the problem can be stated as follows. Let Φ = {X0,X1,X2,...} be an irreducible aperiodic Markov chain with invariant probability distribution π having support X and let P n denote the distribution of Xn  X0 for n ≥ 1, that is, P n (x,A) = Pr(Xn ∈ A  X0 = x). Then, given ω> 0, can we find a positive integer n ∗ such that (1) ‖P n∗ (x, ·) − π(·) ‖ ≤ ω where ‖ · ‖ is the total variation norm? If we can find such an n ∗ , then, since ‖P n − π ‖ is nonincreasing in n, every draw past n ∗ will also be within ω of π, thus providing a representative sample if we keep only the draws after n ∗. There is an enormous amount of research (too much to list here!) on this problem for a wide variety of Markov chains. Unfortunately, there is apparently little that can be said generally about this problem so that we are forced to analyze each Markov chain individually or at most within a limited class of models or situations. This is somewhat reflected in the current
Perfect Simulation of M/G/c Queues
, 2014
"... In this paper we describe a perfect simulation algorithm for the stableM/G/c queue. Sigman (2011: Exact Simulation of the Stationary Distribution of the FIFO M/G/c Queue. Journal of Applied Probability, 48A, 209213) showed how to build a dominated CFTP algorithm for perfect simulation of the supers ..."
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In this paper we describe a perfect simulation algorithm for the stableM/G/c queue. Sigman (2011: Exact Simulation of the Stationary Distribution of the FIFO M/G/c Queue. Journal of Applied Probability, 48A, 209213) showed how to build a dominated CFTP algorithm for perfect simulation of the superstable M/G/c queue operating under First Come First Served discipline, with dominating process provided by the corresponding M/G/1 queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service), and exploiting the fact that the workload process for theM/G/1 queue remains the same under different queueing disciplines, in particular under the Processor Sharing discipline, for which a dynamic reversibility property holds. We generalize Sigman's construction to the stable case by comparing the M/G/c queue to a copy run under Random Assignment. This allows us to produce a naïve perfect simulation algorithm based on running the dominating process back to the time it first empties. We also construct a more efficient algorithm that uses sandwiching by lower and upper processes constructed as coupledM/G/c queues started respectively from the empty state and the state of the M/G/c queue under