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Many articles which develop new coupling techniques for use in CFTP or Fill's algorithm also contain brief explanations of these techniques. Fill '98 describes his new algorithm, and also gives an explanation of CFTP that some probabilists prefer. Sections 1 and 7 are good places to start reading. Propp click here gives a survey of CFTP with a focus on combinatorial applications. Fill-Machida-Murdoch-Rosenthal '00 give a readable extension of Fill's algorithm.
Wilson '00 section 1 gives a primer on CFTP and extensions. Propp and David B. LevinYuval Peresand Elizabeth L. Wilmerto be published by the American Mathematical Society Introduction and Scope Random sampling has found numerous applications in physics, statistics, and computer science. Perhaps the most versatile method of generating random samples from a probability space is to run a Markov chain. But for how many steps? In most cases one simply does not know how many Markov chain steps are needed to get a sufficiently random state.
There is a large literature of heuristic algorithms for inferring when enough steps have been taken, but they are non-rigorous, and one never knows for sure that an adequate number of steps have been taken.
These heuristic algorithms are beyond the scope of this bibliography, but the interested reader is referred to some lecture notes by Sokal and the Best Bibliography Editor Site For Phd Preprint Service.
In the past decade there is been much research on obtaining rigorous bounds of how many Markov chain steps are needed to generate a random sample. Sometimes these bounds are tight, sometimes they are unduly pessimistic. The interested reader is referred to a survey link Diaconis and Saloff-Coste and a survey by Jerrum and Sinclair ; the size of this literature makes it beyond the scope of this bibliography.
In recent years there have been a large number of algorithms developed for sampling from the steady state distribution of suitably well-structured Markov chains, which require no a priori knowledge of how long the Markov chains take to get mixed.
The algorithms determine on their own, during run time, how many steps to run the Markov chain. It is these algorithms that are the focus of this bibliography. Since the focus of this bibliography is on working computer algorithms, the symbol is placed next to those articles that contain simulation results or give sample outputs. Each annotated entry contains links relevant to the paper, giving the article's abstract click on the title and authors' homepages when available, as well as links to online preprints.
Several of the annotations were contributed by people other than the maintainer. However, these Markov chains typically have state spaces such as the symmetric group or the hypercube, for which one already knows how to effectively generate a random sample on a computer. Rather, the point of studying these stopping times is to understand interesting mathematical processes, such as shuffling a deck of cards.
A notable exception is Fill's algorithm. Since the literature on these stopping times is sizable, only those articles with an Best Bibliography Editor Site For Phd theme are included.
The reader interested in stopping times is referred to the articles by Aldous and Diaconis and Diaconis and Fillwhich contain many references.
Also relevant to the present bibliography is a literature on backwards compositions of random maps. The articles from this literature study when this sequence of points converge almost surely, since convergence implies the existence of a stationary distribution for the Markov chain. Existence can be nontrivial for infinite state spaces. There are numerous examples in which conditions 1 or 2 do not hold. Diaconis and Freedman give a survey of the literature on stochastically link sequences; a few representative articles are listed below.
Generating random spanning trees. A random walk construction of uniform click here trees and uniform labelled trees. These two articles give the same independently discovered random-walk based visit web page for generating random spanning trees of a graph.
The algorithm uses a Markov chain on the set of spanning trees of an undirected graph to return a perfectly random spanning tree. Broder uses the algorithm to analyze the random walk on a ring. Aldous uses the algorithm to determine the properties of random trees and to compute some non-trivial probabilities pertaining to the random walk in the plane.
There is another random tree algorithm based on computing determinants. Glynnand Hermann Thorisson. Stationary detection in the initial transient problem. This paper explores what is possible and what is not, and was the first paper to show that it is possible to obtain unbiased samples from the steady state distribution of a finite Markov chain by observing it, provided it is irreducible and one knows how many states it has.
Equivalently, there is a universal randomized stationary stopping time that works for all irreducible Markov chains with a given finite number of states. On simulating a Markov chain stationary distribution when transition probabilities are unknown. Spencerand J. While not exact, this algorithm was much more efficient than the previous one, and directly stimulated the development of two subsequent exact sampling algorithms.
This paper gives the only nontrivial lower bound on the running time of an algorithm for Best Bibliography Editor Site For Phd sampling from generic Markov chains. Dana Randall and Alistair Sinclair. Testable algorithms for self-avoiding walks. We present a polynomial time Monte Carlo algorithm for almost uniformly generating and approximately counting self-avoiding walks go here rectangular lattices Z d.
These are classical problems that arise, for example, in the study of long polymer chains. While there are a number of Monte Carlo algorithms used to solve these problems in practice, these are heuristic and their correctness relies on unproven conjectures. In contrast, our algorithm depends on a single, widely-believed conjecture that is weaker than preceding assumptions, and, more importantly, is one which the algorithm itself can test.
Thus our algorithm is reliablein the sense that it either outputs answers that are source, with high probability, to be correct, or finds a counter-example to the conjecture. Exact mixing in an unknown Markov chain. Electronic Journal of Combinatorics2 The expected stopping time of the rule is bounded by a polynomial in the maximum mean hitting time of the chain.
Our stopping rule can be made deterministic unless the chain itself has no random transitions. This paper gives the first universal exact sampling algorithm that runs in time that is polynomial in certain parameters associated with the Markov chain.
Gives a deterministic stationary stopping time that works when the Markov chain itself is not deterministic. The paper also contains a pretty lemma on random trees that is of independent interest. Exact sampling with coupled Markov chains and applications to statistical mechanics. For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution.
This bibliography is intended to embrace all fields relevant to Lollard studies. It therefore includes texts and studies about the literary, historical, cultural, and. Calder with Romulus and Remus, Twelfth Annual Exhibition of The Society of Independent Artists, Waldorf-Astoria, New York, Bibliography. The complete and official bibliography of Judith F. Baca: Table of Contents. A. Completed Artworks. B. Unpublished Work. C. Mural Restoration. A messy desk encourages a creative mind, study finds. October , Vol 44, No. 9. Print version: page
One approach is to run an ergodic i. Unfortunately it can be difficult to determine how large M needs to be.
We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the algorithm itself.
Webinar zur Citation Style Language CSL (Deutsch)
If the state space has a partial order that is preserved under the moves of the Markov chain, then the coupling is often particularly efficient. Using our approach one can sample from the Gibbs distributions associated with various statistical mechanics models including Ising, random-cluster, ice, and dimer or choose uniformly at random from the elements of a finite distributive lattice.
This paper gives link algorithm, monotone coupling from the past, for exact sampling with Markov chains on huge state spaces. Includes simulation results for the random cluster and dimer models.
Studying convergence of Markov chain Monte Carlo algorithms using coupled sample paths. Journal of the American Statistical Association91 I describe a simple procedure for investigating the convergence properties of Markov Chain Monte Carlo sampling schemes. The procedure employs multiple runs from a sampler, using the same random deviates for each run. When the sample paths from all sequences converge, it is argued that approximate equilibrium conditions hold.
The procedure also provides a simple diagnostic for detecting modes in multimodal posteriors. Several examples of the procedure are provided. In Ising models, the relation between the correlation parameter and the convergence rate of rudimentary Gibbs samplers is investigated. In another example, the effects of multiple modes on the convergence of coupled paths are explored using mixtures of Best Bibliography Editor Site For Phd normal distributions.
The technique is also used to evaluate the convergence properties of a Gibbs sampling scheme applied to a model for rat growth rates Gelfand et al While technically not a paper on exact sampling, this paper investigates how the mixing time of a Markov chain may be inferred by running a large number of coupled simulations until they coalesce. The initial states of the Markov chains are chosen at random, and if the probability of rejection in rejection sampling is known, then rigorous estimates of the mixing time are given.
Includes the independently made observation that for Best Bibliography Editor Site For Phd Markov chains, only two coupled states need to be simulated. Additional articles that take a similar approach are available from Johnson's homepage. Markov chain algorithms for planar lattice structures extended abstract. Consider Best Bibliography Editor Site For Phd following Markov chain, whose states are here domino tilings of a 2 n X 2 n chessboard: If the four squares appearing in this window are covered by two parallel dominoes, rotate the dominoes in place.
This process is used in practice to generate a random tiling, and is a key tool in the study of the combinatorics of tilings and the behavior of dimer systems in statistical physics. Analogous Markov chains are used to randomly generate other structures on various two-dimensional lattices.
This paper presents techniques which prove for the first time that, in many interesting cases, a small number of random moves suffice to obtain a uniform distribution. This paper gives three new Markov chains for sampling certain dimer and ice systems. The focus of this paper is provable running time bounds.
Monotone-CFTP may be applied to each of their Markov chains to get an exact algorithm; when this is done, their proofs may be interpreted as a priori bounds on the running time of CFTP, though in practice the exact algorithm runs much more quickly than the bounds suggest.
For the particular case of the 2 n X 2 n chessboard, it is http://cocktail24.info/blog/janet-is-handing-in-her-homework.php faster to generate a random spanning tree, and then use a Temperley-like bijection to convert it to a perfectly random domino tiling.
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Their methods apply to more general regions for which such a Temperleyan bijection does not exist.