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Kevin Smith | Reversible Jump Markov Chain Monte Carlo (RJMCMC) Tutorial A point estimate is a single multi-object configuration caculated from the Markov Chain which serves as the output of the tracker. Initialize the MH sampler by choosing a sample from the t-1 Markov Chain and move all targets according to their state evoution. To generate a new sample in the Markov Chain, the first step is to choose a move type from the set of move types. www.idiap.ch /~smith/RJMCMC_flash7.php   (4097 words)

Markov chain Summary Markov chains are often described by a directed graph, where the edges are labeled by the probabilities of going from one state to the other states. Markov chains also have many applications in biological modelling, particularly population processes, which are useful in modelling processes that are (at least) analogous to biological populations. Markov chains are related to Brownian motion and the ergodic hypothesis, two topics in physics which were important in the early years of the twentieth century, but Markov appears to have pursued this out of a mathematical motivation, namely the extension of the law of large numbers to dependent events. www.bookrags.com /Markov_chain   (2434 words)

CiteULike: Bayesian phylogenetic model selection using reversible jump Markov chain Monte Carlo.   (Site not responding. Last check: 2007-11-07) Moreover, almost all of the models that are chosen as best do not constrain a transition rate to be the same as a transversion rate, suggesting that it is the transition/transversion rate bias that plays the largest role in determining which models are selected. Importantly, the reversible jump Markov chain Monte Carlo algorithm described here allows estimation of phylogeny (and other phylogenetic model parameters) to be performed while accounting for uncertainty in the model of DNA substitution. Importantly, the reversible jump Markov chain Monte Carlo algorithm described here allows estimation of phylogeny (and other phylogenetic model parameters) to be performed while accounting for uncertainty in the model of DNA substitution.}, address = {Section of Ecology, Behavior and Evolution, Division of Biological Sciences, University of California, San Diego, USA. www.citeulike.org /user/aprasad/article/349460   (635 words)

Reversible Markov Chains DC MetaData pour: Markov chains with positive transitions are not determined by... Stability of the Tail Markov Chain and the Evaluation of... Markov Chain Monte Carlo: innovations and applications in statistics, physics, a... www.scienceoxygen.com /math/595.html   (231 words)

SIMAX Volume 14 Issue 4 A simple upper bound for the second-largest eigenvalue of a finite reversible time-homogeneous Markov chain is presented as a function of the transition probabilities, the equilibrium distribution, and the underlying structure of the chain. Furthermore, a lower bound for the smallest eigenvalue of a reversible chain is also presented, thereby providing a bound on the spectral gap of such chains. These eigenvalue bounds are fairly easy to compute for a variety of reversible chains by using known results on eigenvalues of certain matrices associated with graphs or random walks on graphs. locus.siam.org /SIMAX/volume-14/art_0614063.html   (217 words)

Computing, image analysis, and spatial statistics, Master's projects, Math Stat, MC, LU/LTH Typical examples are mixture distributions -- the latent variables are then labels of the components from which the observations were drawn -- and hidden Markov models -- the latent variables are then the unobserved Markov chain. This project could involve constructing spatio-temporal Bayesian hierarchical models for investigating the effect of rainfall on the population dynamics of the red kangaroo in South Australia, or of migratory birds spending their winter quarters in the Sahel area south of the Sahara desert. With the aid of Markov chain Monte Carlo methods, non-parametric Bayesian statistics has emerged from a field of mostly theoretical interest, to a collection of powerful tools for analysing data without making too restrictive distributional assumptions. www.maths.lth.se /matstat/education/exjobb/computation.html   (1389 words)

BioMed Central | Full text | Inference of demographic history from genealogical trees using reversible jump Markov ... Markov chain Monte Carlo (MCMC) is one particularly useful sampling algorithm as it doesn't require calculation of the sum (or integral) in the nominator of Eq. Briefly, sampling via MCMC is done by constructing a Markov chain with the possible combinations of parameter values as "states", and the desired posterior as its stationary distribution. First, the Markov chain is started with an initial state that corresponds to a completely flat demographic function, i.e. www.biomedcentral.com /1471-2148/5/6   (5668 words)

Markov Chain Monte Carlo   (Site not responding. Last check: 2007-11-07) 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). from the Markov chain beginning at the current state x. www.mathcs.duq.edu /larget/math496/mcmc.html   (351 words)

Citations: Exploiting random walks for learning - Bartlett, Fischer, Hoffgen (ResearchIndex) Extension of the PAC Framework to Finite and Countable Markov.. The PAC learning for a Markov chain with countably infinite state space is more complicated. The proof is based on establishing an analogue of Lemmas 3.1, 3.2 [4] However, unlike [4] we do not assume that the Markov chain is in steady state. citeseer.ist.psu.edu /context/728676/142331   (850 words)

Reversible jump Markov chain Monte Carlo computation and Bayesian model determination - Green (ResearchIndex) Abstract: This article proposes a new framework for the construction of reversible Markov chain samplers that jump between parameter subspaces of differing dimensionality, which is flexible and entirely constructive. The methodology is illustrated with applications to multiple change-point analysis in one and two dimensions, and to a Bayesian comparison of binomial experiments. Green, P. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. citeseer.ist.psu.edu /green95reversible.html   (619 words)

Fastest Mixing Markov Chain on a Graph The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e., the mixing rate of the Markov chain, is determined by the second largest eigenvalue modulus (SLEM) of the transition probability matrix. For many of the examples considered, the fastest mixing Markov chain is substantially faster than those obtained using these heuristic methods. We derive the Lagrange dual of the fastest mixing Markov chain problem, which gives a sophisticated method for obtaining (arbitrarily good) bounds on the optimal mixing rate, as well as the optimality conditions. epubs.siam.org /sam-bin/dbq/article/42326   (349 words)

95-41 A semidefinite bound for mixing rates of Markov chains   (Site not responding. Last check: 2007-11-07) We study the method of bounding the spectral gap of a reversible Markov chain by establishing canonical paths between the states. Further generalization using multicommodity flow yields a bound which is an invariant of the Markov chain, and which can be computed at an arbitrary precision in polynomial time via semidefinite programming. We show that, for any reversible Markov chain on n states, this bound is off by a factor of at most O(log^2), and that this is tight. dimacs.rutgers.edu /TechnicalReports/abstracts/1995/95-41.html   (125 words)

Minimum-Entropy Data Partitioning Using Reversible Jump Markov Chain Monte Carlo Problems in data analysis often require the unsupervised partitioning of a data set into classes. [17] L. Tierney, “Markov Chains for Exploring Posterior Distributions,” Annals of Statistics, vol. Citation: Stephen J. Roberts, Chris Holmes, Dave Denison, "Minimum-Entropy Data Partitioning Using Reversible Jump Markov Chain Monte Carlo," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/trans/tp/&toc=comp/trans/tp/2001/08/i8toc.xml&DOI=10.1109/34.946994   (508 words)

Reversible jump Markov chain Monte Carlo computation and Bayesian model determination   (Site not responding. Last check: 2007-11-07) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination, by Peter J. Green. Markov chain Monte Carlo methods for Bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure. This article proposes a new framework for the construction of reversible Markov chain samplers that jump between parameter subspaces of differing dimensionality, which is flexible and entirely constructive. www.stats.bris.ac.uk /~peter/abstracts/revjump.html   (154 words)

Gibbs sampling - MLpedia Gibbs sampling is a special case of the Metropolis-Hastings algorithm, and thus an example of a Markov chain Monte Carlo algorithm. It can be shown (see, for example, Gelman et al.) that the sequence of samples comprises a Markov chain, and the stationary distribution of that Markov chain is just the sought-after joint distribution. Then it can be shown that this is a reversible markov chain with invariant distribution g as follows. www.mlpedia.org /index.php?title=Gibbs_sampling   (413 words)

Abstract for ``Analysis of a Non-Reversible Markov Chain Sampler'' Abstract for ``Analysis of a Non-Reversible Markov Chain Sampler'' We analyse the convergence to stationarity of a simple non-reversible Markov chain that serves as a model for several non-reversible Markov chain sampling methods that are used in practice. Diaconis, P., Holmes, S., and Neal, R. (2000) ``Analysis of a nonreversible Markov chain sampler'', Annals of Applied Probability, vol. www.cs.toronto.edu /~radford/nonrev.abstract.html   (152 words)

Electronic Communications in Probability - Vol. 2 (1997) Various notions of geometric ergodicity for Markov chains on general state spaces exist. We then apply these results to a collection of chains commonly used in Markov chain Monte Carlo simulation algorithms, the so-called hybrid chains. Meyn and R. Tweedie (1993), Markov chains and stochastic stability, Springer-Verlag, London. www.emis.de /journals/EJP-ECP/_ejpecp/ECP/viewarticle28f1.html?id=1564&layout=abstract   (415 words)

Bayesian Phylogenetic Model Selection Using Reversible Jump Markov Chain Monte Carlo -- Huelsenbeck et al. 21 (6): 1123 ... Markov chain is a matrix of rates, specifying the rate of change Markov chain Monte Carlo algorithms for the Bayesian analysis of phylogenetic trees. Tierney, L. Markov chains for exploring posterior distributions. mbe.oxfordjournals.org /cgi/content/full/21/6/1123   (5445 words)

in-cites - An Interview With Professor Peter Green The paper introduces Reversible jump Markov chain Monte Carlo, a new simulation-based methodology for fitting statistical models that have variable-dimension parameters. Reversible jump Markov chain Monte Carlo is a class of methods for doing that, which is feasible even in very complex models, and which is not too cumbersome to set up and use. More fundamentally, there are a host of challenging theoretical questions about how to summarize inference in complex variable-dimension models: Reversible jump helps you compute what you want, but can’t tell you what to compute. in-cites.com /papers/PeterGreen.html   (1127 words)

in-cites - An Interview With Professor Peter Green The paper introduces Reversible jump Markov chain Monte Carlo, a new simulation-based methodology for fitting statistical models that have variable-dimension parameters. It caught on simply because people found this useful in their statistical modelling work, and indeed even now it must be the most widely used method for treating these problems. Reversible jump Markov chain Monte Carlo is a class of methods for doing that, which is feasible even in very complex models, and which is not too cumbersome to set up and use. www.in-cites.com /papers/PeterGreen.html   (1139 words)

Xrate Format < Main < Biowiki indicating a general irreversible chain, the rates and initial probabilities of which are not free variables, but rather algebraic functions of the separately-declared grammar parameters. (chain ; declare a Markov chain (update-policy irrev) ; EM update policy (terminal (RX)) ; abstract state label (initial (state (a)) (prob 1)) ; initial distribution (mutate (from (a)) (to (c)) (rate 1)) (mutate (from (c)) (to (g)) (rate 1)) (mutate (from (g)) (to (t)) (rate 1)) (mutate (from (t)) (to (a)) (rate 1))) The "#=GS" tag in the Stockholm format alignment, which is used to specify by-sequence annotation, selects the component chain that will be used on a particular branch. biowiki.org /twiki/bin/view/Main/XgramFormat   (3871 words)

Citebase - Improving Asymptotic Variance of MCMC Estimators: Non-reversible Chains are Better The non-reversible chain achieves this improvement by avoiding (to the extent possible) transitions that backtrack to the state from which the chain just came. The proof that this modification cannot increase the asymptotic variance of an MCMC estimator uses a new technique that can also be used to prove Peskun's (1973) theorem that modifying a reversible chain to reduce the probability of staying in the same state cannot increase asymptotic variance. A non-reversible chain that avoids backtracking will often take little or no more computation time per transition than the original reversible chain, and can sometime produce a large reduction in asymptotic variance, though for other chains the improvement is slight. www.citebase.org /abstract?id=oai:arXiv.org:math/0407281   (223 words)

Worth noting statistically: Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination Abstract: Markov chain Monte Carlo methods for Bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure. They have therefore not been available for application to Bayesian model determination, where the dimensionality of the parameter vector is typically not fixed. This paper proposes a new framework for the construction of reversible Markov chain samplers that jump between parameter subspaces of differing dimensionality, which is flexible and entirely constructive. www.stat.columbia.edu /~tzheng/tianblog/2005/08/reversible-jump-markov-chain-monte.html   (174 words)

Fastest mixing Markov chain on a graph   (Site not responding. Last check: 2007-11-07) The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e., the mixing rate of the Markov chain, is determined by the second largest (in magnitude) eigenvalue of the transition matrix. We compare the fastest mixing Markov chain to those obtained using two commonly used heuristics: the maximum-degree method, and the Metropolis-Hastings algorithm. We derive the Lagrange dual of the fastest mixing Markov chain problem, which gives a sophisticated method for obtaining (arbitrarily good) bounds on the optimal mixing rate, as well the optimality conditions. www.stanford.edu /~boyd/fmmc.html   (328 words)

6.856 -- Randomized Algorithms   (Site not responding. Last check: 2007-11-07) Suppose that a Markov chain is doubly stochastic. (it follows the probability of a given state sequence occurring is equal to the probability of the reversal of that state sequence, so running time backwards doesn't affect the distribution at all). Prove that the stationary distribution of any time-reversible Markov chain is uniform. theory.lcs.mit.edu /classes/6.856/00/Handouts/hw11.html   (267 words)

Electronic Communications in Probability - Vol. 10 (2005) In a recent work, Boyd, Diaconis and Xiao introduced a semidefinite programming approach for computing the fastest mixing Markov chain on a graph of allowed transitions, given a target stationary distribution. Diaconis and D. Stroock, Geometric bounds for eigenvalues of Markov chains, Ann. Sun, J., Boyd, S., Xiao, L., and Diaconis, P., The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem. www.emis.de /journals/EJP-ECP/_ejpecp/ECP/viewarticle09ea.html?id=1761&layout=abstract   (311 words)

Colloquium - 27 September 2001 - Department of Mathematics - University of Montana   (Site not responding. Last check: 2007-11-07) random walk on a graph represents a reversible Markov chain, whose transition probabilities depend on the degrees of the vertics. A graph is said to be recurrent on transient, according to wheather the corresponding Markov chain is recurrent or transient. We shall discuss how random walks on graphs can be used to classify Rievann surfaces, as to their hyperbolicity or parabolicity. www.umt.edu /math/Colloq/fall01/092701.html   (104 words)

Univariate Polynomial Inference by Monte Carlo Message Length Approximation The orthonormal polynomial parameters are sampled using reversible jump Markov chain Monte Carlo methods. A reversible jump Markov chain Monte Carlo algorithm for sampling polynomials is described in Section 4. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. www.csse.monash.edu.au /~lloyd/tildeMML/Structured/2002-ICML/paper.html   (3435 words)

Michael Alfaro   (Site not responding. Last check: 2007-11-07) A simulation study comparing the performance of Bayesian Markov chain Monte Carlo sampling and bootstrapping in assessing phylogenetic confidence. Evolutionary consequences of a redundant map of morphology to mechanics: an example using the jaws of labrid fishes. Shown is the majority rules consensus of 90,000 post-burnin states visited by a million generation Bayesian Markov Monte Carlo reanalysis of previously published data (Alfaro and Arnold, 2001) performed using MrBayes (Huelsenbeck, 2000). brahms.ucsd.edu /alfaro.html   (850 words)

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