Monte-Carlo Markov Chain

In the next subsection, I describe how the basic elements of Bayesian analysis are formulated mathematically. I also describe the methods for deriving posterior distributions from the model, either in terms of conjugate prior likelihood forms or in terms of simulation using Markov chain Monte Carlo (MCMC) methods. The utility of Bayesian methods has expanded greatly in recent years because of the development of MCMC methods and fast computers. I also describe the basics of hierarchical and mixture models. 

F. Simulation via Markov Chain Monte Carlo Methods... 

In some cases, we may not be able to draw directly from the posterior distribution. The difficulty lies in calculating the denominator of Eq. (18), the marginal data distribution p(y). But usually we can evaluate the ratio of the probabilities of two values for the parameters, p(Q, y)/p(Qu y), because the denominator in Eq. (18) cancels out in the ratio. The Markov chain Monte Carlo method [40] proceeds by generating draws from some distribution of the parameters, referred to as the proposal distribution, such that the new draw depends only on the value of the old draw, i.e., some function We accept... 

Harmon R, Challenor P (1997) A Markov chain Monte Carlo method for estimation and assimilation into models. Ecol Model 101 41 19... 

Geyer, C. J. Thompson, E. A., Annealing markov chain Monte Carlo with applications to ancestral inference, J. Am. Stat. Soc. 1995, 90, 909-920... 

Mark-Houwink-Sakurada relationship, 1 309, 310t 20 439-440 Markov chain, 26 1006, 1018, 1024, 1025 HSTA algorithm and, 26 1030-1031 Markov chain Monte Carlo (MCMC) sampling method, 26 1017-1018 Markovnikov addition, in silicone network preparation, 22 563... 

I3G. Winkler, Image Analysis, Random Fields and Markov Chain Monte Carlo Methods, Springer-Verlag, New York, 2003. 

The preceding discussion applied implicitly to what we classify as dynamical simulations — namely, those simulations in which all correlations in the final trajectory arise because each configuration is somehow generated from the previous one. This time-correlated picture applies to a broad class of algorithms MD, Langevin and Brownian dynamics, as well as traditional Monte Carlo (MC, also known as Markov-chain Monte Carlo). Even though MC may not lead to true physical dynamics, all the correlations are sequential. 

Berg, B.A. Markov Chain Monte Carlo Simulations and Their Statistical Analysis, World Scientific,... 

Brown, S., Head-Cordon, T. Cool walking a new Markov chain Monte Carlo sampling method. J. Comput. Chem. 2003, 24, 68-76. 

The use of estimates of treatment effect based on indirect comparisons when there is a common comparator has recently been shown on many occasions to agree with the results of head-to-head clinical trials (Song et al. 2003). Clearly a more challenging situation exists where there is not a common parameter, for example, in a recent study of the relative cost effectiveness of newer drugs for treatment of epilepsy (Wilby et al. 2003). In this study, Bayesian Markov chain Monte Carlo models for multiparameter synthesis were used (Ades 2003). Here, complex models were used to analyze a set of clinical studies involving a series of clinical alternatives, including the two alternatives of interest. 

Gamer C, Mclnnes LA, Service SK, et al. Linkage analysis of a complex pedigree with severe bipolar disorder, using a Markov chain Monte Carlo method. Am J Hum Genet 2001 68(4) 1061-1064. 

The assessment of the error of the approximation depends on the posterior variance of akj for which we do not have a closed form expression. Empirical comparisons that we conducted on gene expression data sets suggest that the results based on our numerical approximation are virtually indistinguishable from those obtained by Markov chain Monte Carlo methods when ti, 2 > 10. Details are described by Sebastiani et al. (2005). 

Covington TR, Gentry PR, Van Landingham CB, Andersen ME, Kester JE, Clewell HJ. 2007. The use of Markov chain Monte Carlo uncertainty analysis to support a public health goal for perchloroethylene. Regul Toxicol Pharmacol 47 1-18. 

Lyons M, Yang RSH, Mayeno AN, Reisfeld B. 2008. Computational toxicology of chloroform reverse dosimetry using Bayesian inference, Markov chain Monte Carlo simulation, and human biomonitoring data. Environ Health Perspect 116 1040-1046. 

Suppose that configurations visited by a thermal trajectory, for example from Markov Chain Monte Carlo, sample points uniformly in the allowed area. Ixt M be the total number of points involved and m(A) be the number... 

Spiegelhalter D.J., Best N. G., Gilks W. R., Inskip H. (1995c). Hepatitis a case study in MCMC methods. In Markov chain Monte Carlo Methods in practice. (ed. W. R. Gilks, S. Richardson, and D. J. Spiegelhalter), pp. 21-43. Chapman and Hall, New York. 

B. A. Berg (2003) Multicanonical simulations step by step. Comp. Phys. Comm. 153, pp. 397-406 ibid. (2004) Markov Chain Monte Carlo Simulations and their Statistical Analysis. World Scientific P. 380... 

Population models describe the relationship between individuals and a population. Individual parameter sets are considered to arise from a joint population distribution described by a set of means and variances. The conditional dependencies among individual data sets, individual variables, and population variables can be represented by a graphical model, which can then be translated into the probability distributions in Bayes theorem. For most cases of practical interest, the posterior distribution is obtained via numerical simulation. It is also the case that the complexity of the posterior distribution for most PBPK models is such that standard MC sampling is inadequate leading instead to the use of Markov Chain Monte Carlo (MCMC) methods... 

More recently Brochot et al. [89] reported an extension of the isobolographic approach to interaction studies for convulsant interaction among pelloxacin, norfloxacin, and theophylline in rats. Their contribution is unique in that they started out by explaining pharmacodynamic interactions for two drugs, but then extended the approach to derive an isobol for three drug interaction. In addition they included Bayesian analysis and developed a population model with Markov chain Monte Carlo methods. 

Gelman A, Rubin DB. Markov chain Monte Carlo methods in biostatistics. Stat Meth Med Res 1996 5 339-55. 

Geyer, C. J., Markov chain Monte Carlo maximum likelihood. In Computing Science and Statistics Proceedings of the 23rd Symposium on the Interface. American Statistical Association, New York, 1991, pp. 156-163. 

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