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# Topic: Markov Chain Monte Carlo

 Markov Chain Monte Carlo - MLpedia Markov chain Monte Carlo (MCMC) methods, sometimes called random walk Monte Carlo methods, are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its stationary distribution. A Markov chain is constructed in such a way as to have the integrand as its equilibrium distribution. Hybrid Markov chain Monte Carlo: Tries to avoid random walk behaviour by introducing an auxiliary momentum vector and implementing Hamiltonian dynamics where the potential function is the target density. www.mlpedia.org /index.php?title=Markov_Chain_Monte_Carlo   (704 words)

 The World-Wide Web Virtual Library: Random Numbers and Monte Carlo methods MCMC Preprint Service: this link provides a list of all registered papers on Markov chain Monte Carlo methodology currently submitted for publication. The Molecular Monte Carlo Home Page is meant to serve as an information resource for those who use "random walks" (stochastic methods) to simulate and analyze molecular systems throughout the world. The EGS4 computer code system is a general purpose package for the Monte Carlo simulation of the coupled transport of electrons and photons in an arbitrary geometry for particles with energies from a few keV up to several TeV. random.mat.sbg.ac.at /links/monte.html   (1022 words)

 MCMC tutorial It simulates a Markov chain whose invariant states follow a given (target) probability in a very high (say millions) dimensional state space. In computer vision, Monte Carlo integration is used in learning and model estimation, and has also been used in motion tracking etc. C. MCMC Basics--Metropolis, Metropolis-Hastings, and Gibbs Sampling (1) Markov Chains (2) Inference and Estimation via Sampling (3) The Gibbs Sampler (4) Metropolis-Hastings (5) Metropolis and Gibbs (revisited) 3. civs.stat.ucla.edu /MCMC/MCMC_tutorial.htm   (542 words)

 Mathematics of Markov Chain Monte Carlo Mathematics of Markov Chain Monte Carlo was held from Monday, June 12 through Friday, June 16 at the Mathematical Sciences Research Institute in Berkeley, California. Markov chains are a class of stochastic processes which (under mild regularity conditions) converge to a unique stationary distribution. In the spring of 2005, mixing times of finite Markov chains were a major theme of the multidisciplinary research program Probability, Algorithms, and Statistical Physics, held at MSRI. www.oberlin.edu /markov/workshop.html   (496 words)

 CASt R: An application of Markov chain Monte Carlo Monte Carlo methods are a collection of techniques that use pseudo-random (computer simulated) values to estimate solutions to mathematical problems. Monte Carlo methods can also be used for a variety of other purposes, including estimating maxima or minima of functions (as in likelihood-based inference) but we will not discuss these here. Monte Carlo works as follows: Suppose we want to estimate an expectation of a function g(x) with respect to the probability distribution f. www.stat.psu.edu /~mharan/MCMCtut/MCMC.html   (1951 words)

 Theoretical rates of convergence for Markov chain Monte Carlo - Rosenthal (ResearchIndex) We present a general method for proving rigorous, a priori bounds on the number of iterations required to achieve convergence of Markov chain Monte Carlo. ...1 and ae is related to the transition kernel of the chain. The second approach is to monitor the output of the chain, and... citeseer.ist.psu.edu /rosenthal94theoretical.html   (750 words)

 Markov Chain Monte Carlo   (Site not responding. Last check: 2007-11-04) Generally speaking, MCMC provides a mechanism for taking dependent samples in situations where regular sampling is difficult, if not completely intractable. To implement this algorithm, you need a reversible Markov chain to propose new states from any specified given state, the ability to compute the ratio of the posterior densities (probabilities) for any pair of states, and a random number generator. (There are also ways to handle non-reveersible Markov chains.) Notice that if the posterior density at state x, p(x), equals h(x) / C where C is hard to compute, but h(x) is computable, the ratio of posterior densities at x and y equals p(x) / p(y) = h(x) / h(y). www.mathcs.duq.edu /larget/math496/mcmc.html   (351 words)

 Markov chain Monte Carlo - Wikipedia, the free encyclopedia A good chain will have rapid mixingâ€”the stationary distribution is reached quickly starting from an arbitrary positionâ€”described further under Markov chain mixing time. Hybrid Monte Carlo (HMC) (Would be better called `Hamiltonian Monte Carlo'): Tries to avoid random walk behaviour by introducing an auxiliary momentum vector and implementing Hamiltonian dynamics where the potential function is the target density. The Reversible Jump method is a variant of Metropolis-Hastings that allows proposals that change the dimensionality of the space. en.wikipedia.org /wiki/Markov_chain_Monte_Carlo   (756 words)

 Markov Chain Monte Carlo The transition probability q is constructed in such a way that an ergodic Markov process is defined with stationary distribution equal to the desired posterior distribution. Several methods might be used to obtain the sample from the posterior distribution in the MCMC frame work: Metropolis method, Gibbs sampling, stochastic dynamics and Hybrid Monte Carlo. The hybrid Monte Carlo method developed by Duane [2], and used in this paper is a merge of the stochastic dynamics and Metropolis algorithms. www.cs.cmu.edu /People/rafa/docs/acnn97/node3.html   (413 words)

 An Introduction to Markov Chain Monte Carlo Neal's MCMC package is now installed and accessible from any Unix machine with access to the /home part of the tree directory. Lecture I: Introduction, the basics of Monte Carlo Integration, and the elements of statistical physics (part 1). MCMC Application: Neural Networks as a way to specify nonparametric regression and classification models. omega.albany.edu:8008 /cdocs   (270 words)

 Amazon.ca: Markov Chain Monte Carlo in Practice: Books: W. R. Gilks   (Site not responding. Last check: 2007-11-04) Markov Chain Monte Carlo in Practice introduces MCMC methods and their applications while also providing some theoretical background. They offer step-by-step instructions for using the methods presented and show the importance of MCMC in real applications with examples ranging from the simple to the more complex in fields such as archaeology, astronomy, biostatistics, genetics, epidemiology, and image analysis. Monte Carlo experts who want to apply their knowlege to finance should also read: "Options, Futures, and Other Derivatives (5th Edition) by John Hull; and "Credit Derivatives" (2nd Edition) by Janet Tavakoli. www.amazon.ca /Markov-Chain-Monte-Carlo-Practice/dp/0412055511   (655 words)

 MCMC - Markov Chain Monte Carlo Let Q be the transition probability matrix of an irreducible Markov chain on 1,..,m. This is great because for example if we want to generate data from a posterior distribution B is the integral over the marginal distribution, which might be very difficult to find. There are examples where the chain seems to have settled down for very long periods but is not actually at the stationary distribution yet. math.uprm.edu /~wrolke/esma5015/mark2.htm   (861 words)

 Abstract: Markov Chain Monte Carlo The concept of Markov Chains was first introduced by A.A. Markov in 1906, in a paper extending the Law of Large Numbers and Central Limit Theorem to a weakly - Markovianly - dependent sequence of random variables. The importance of the work was recognized early on, and advances were made quickly in the generalization of the Markovian structure to processes with countably infinite and continuum state spaces and continuous time. But over several subsequent decades, the idea introduced there, that one can employ a finite state discrete time Markov chain to describe multiple important combinatoric and physical phenomena, and exercise the chain efficiently to derive limiting behavior, caught fire. www.itl.nist.gov /div898/education/abstmc2.htm   (332 words)

 MCMC Preprint Service Data and Errata for the book "Monte Carlo Methods in Bayesian Computation" by Ming-Hui Chen, Qi-Man Shao, and Joseph G. Ibrahim (Springer, 2000) are available at http://www.stat.uconn.edu/~mhchen/mcbook. It implements the logic of standard MCMC samplers within a framework designed to be easy to use and to extend while allowing integration with other software tools. This a page containing all sorts of Java applets including some illustrative MCMC samplers, such as the independence sampler with exponential target distribution, the Metropolis sampler with exponential target distribution and uniform proposal distributions, and the slice sampler. www.statslab.cam.ac.uk /~mcmc/pages/links.html   (418 words)

 Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo - Rosenthal (ResearchIndex) It is our hope that the methods presented here can be applied quite generally, to many different Markov chain samplers. 40 A new approach to the limit theory of recurrent Markov chain.. 9 A slowly mixing Markov chain with implications for Gibbs sam.. citeseer.ist.psu.edu /8618.html   (819 words)

 MARKOV CHAIN MONTE CARLO Introduction to Markov Chain Monte Carlo Simulations and Their Statistical Analysis (809k) Markov Chain Monte Carlo (MCMC) originated in statistical physics, but has spilled over into various application areas, leading to a corresponding variety of techniques and methods. MCMC in the Analysis of Genetic Data on Pedigrees (E A Thompson) www.worldscibooks.com /mathematics/5904.html   (310 words)

 Amazon.com: Markov Chain Monte Carlo in Practice (Interdisciplinary Statistics): Books: W.R. Gilks,S. Richardson,D.J. ...   (Site not responding. Last check: 2007-11-04) Monte Carlo Statistical Methods (Springer Texts in Statistics) by Christian P. Robert Monte Carlo Hotel — Book low rates for the Monte Carlo Hotel and Casino at VEGAS.com. Markov Chain Monte Carlo in Practice (Interdisciplinary Statistics) by W.R. Gilks www.amazon.com /Markov-Chain-Practice-Interdisciplinary-Statistics/dp/0412055511   (1435 words)

 MARKOV CHAIN MONTE CARLO SIMULATIONS AND THEIR STATISTICAL ANALYSIS The material should be accessible to advanced undergraduate students and is suitable for a course. Readership: Upper-level undergraduates, graduate students, lecturers and researchers in physics, chemistry, biology, computer science, mathematics and statistics who are interested in Markov chain Monte Carlo simulations. The only book on Monte Carlo simulations for which Web-based computer codeallows the reader to verify many numerical examples easily www.worldscibooks.com /physics/5602.html   (352 words)

 Abstract for ``Probabilistic Inference using Markov Chain Monte Carlo Methods''   (Site not responding. Last check: 2007-11-04) Related problems in other fields have been tackled using Monte Carlo methods based on sampling using Markov chains, providing a rich array of techniques that can be applied to problems in artificial intelligence. In computer science, Markov chain sampling is the basis of the heuristic optimization technique of "simulated annealing", and has recently been used in randomized algorithms for approximate counting of large sets. In this review, I outline the role of probabilistic inference in artificial intelligence, present the theory of Markov chains, and describe various Markov chain Monte Carlo algorithms, along with a number of supporting techniques. www.cs.toronto.edu /~radford/review.abstract.html   (295 words)

 Amazon.com: "Markov-chain Monte Carlo": Key Phrase page   (Site not responding. Last check: 2007-11-04) Markov Chain Monte Carlo Simulations And Their Statistical Analysis: With Web-based Fortran Code by Bernd A. Berg Markov Chain Monte Carlo in Practice (Interdisciplinary Statistics) by W.R. Gilks (Editor), S. Richardson (Editor), D.J. Spiegelhalter (Editor) Monte Carlo Hotel -- Book low rates for the Monte Carlo Hotel and Casino at VEGAS.com. www.amazon.com /phrase/Markov_chain-Monte-Carlo   (337 words)

 Main Page - Scythe The Scythe Statistical Library is an open source C++ library for statistical computation written by Andrew D. Martin (Washington University), Kevin M. Quinn (Harvard University), and Daniel Pemstein (University of Illinois). It includes a suite of matrix manipulation functions, a suite of random number generators (useful for Markov chain Monte Carlo methods), and a suite of numerical optimizers (useful for maximum likelihood estimation). Programs written using Scythe are many orders of magnitude faster than those written in commonly used interpreted languages, such as R, GAUSS, Matlab, and Ox, and can be compiled on any system with the GNU GCC compiler (and perhaps with other C++ compilers). scythe.wustl.edu   (590 words)

 MCMC Preprint Service   (Site not responding. Last check: 2007-11-04) This service provides a list of all registered papers on MCMC methodology currently submitted for publication. It is entirely the contributors responsibility to ensure that submitted papers are essentially correct, and that acceptable academic practice is followed at all times. All opinions and ideas expressed in the papers included on the MCMC Preprint Service are those of the authors and do not necessarily reflect the ideas or beliefs of the administrators nor of the University of Cambridge. www.statslab.cam.ac.uk /~mcmc   (165 words)

 The BUGS Project - Bayesian inference Using Gibbs Sampling The BUGS (Bayesian inference Using Gibbs Sampling) project is concerned with flexible software for the Bayesian analysis of complex statistical models using Markov chain Monte Carlo (MCMC) methods. A knowledge of Bayesian statistics is assumed, including recognition of the potential importance of prior distributions, and MCMC is inherently less robust than analytic statistical methods. There is now a page of links to BUGS and MCMC related material. www.mrc-bsu.cam.ac.uk /bugs/welcome.shtml   (547 words)

 R-CODA   (Site not responding. Last check: 2007-11-04) Output from BUGS is stored in R objects of class "mcmc", or "mcmc.list" for multiple chains. Cowles, MK and Carlin, BP (1995) Markov Chain Monte Carlo diagnostics: A comparative review, J Amer Stat Soc, 91, 883-904. Assessing convergence of Markov Chain Monte Carlo algorithms, Statistics and Computing. www-fis.iarc.fr /coda   (399 words)

 Main Page - MCMCpack MCMCpack is a software package designed to allow users to perform Bayesian inference via Markov chain Monte Carlo (MCMC). To maximize computational efficiency, the actual sampling for each model is done in compiled C++ using the Scythe Statistical Library. The posterior samples returned by each function are returned as mcmc objects, which can easily be summarized and manipulated by the coda package. mcmcpack.wustl.edu   (656 words)

 Markov Chain Monte Carlo   (Site not responding. Last check: 2007-11-04) Next: Updates Not `Algorithms' Up: Likelihood Inference for Spatial Previous: General Models Specified by Markov chain Monte Carlo (MCMC) in the form of the Gibbs sampler and the Metropolis, Metropolis-Hastings, and Metropolis-Hastings-Green algorithms, permits the simulation of any distribution on a finite-dimensional state space specified by any unnormalized density. As we shall see, when we can simulate we can also do inference. www.stat.umn.edu /PAPERS/html_prints/points/node11.html   (62 words)

 Analysis of Financial Time Series, Ruey Tsay, Markov Chain Monte Carlo Methods with Applications Analysis of Financial Time Series, Ruey Tsay, Markov Chain Monte Carlo Methods with Applications It starts with the basic idea of Markov Chain simulation and its relationship with Bayesian Inference. The chapter effectively uses examples of practical and theoretical interest, including linear regression with time series errors, univariate and multivariate stochastic volatility models, Markov switching models and outliers detection (for interest rate data). www.garpdigitallibrary.org /display/product.asp?pid=1196   (379 words)

 Markov Chain Monte Carlo   (Site not responding. Last check: 2007-11-04) What this is about: Seminar in Data Analysis: Discrete Markov Chain Monte Carlo; Stat 344; Mt. Holyoke College; Spring 2001; Prof. What you'll find here: Markov chain monte carlo is neat stuff. If you grab the whole pdf, you can see them. wonka.hampshire.edu /~jason/math/markov/index.html   (86 words)

 Markov Chain Monte Carlo   (Site not responding. Last check: 2007-11-04) Bayesian Computation Using Markov Chain Monte Carlo Methods Iterative simulations such as Markov chain Monte Carlo Chuanhai Liu and Donald B. Rubin are continuing their work on "Markov Analysis of Iterative Simulations Before Their Convergence" (Liu and Rubin, 1996), which can be used to create an overdispersed starting distribution for running multiple chains. cm.bell-labs.com /cm/ms/departments/sia/project/mcmc   (199 words)

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