Monte 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... 

WK Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97-109, 1970. 

Lowry, G. G., Ed. Markov Chains and Monte Carlo Calculations in Polymer Science Marcel Dekker New York, 1970. 

Lowry GG (ed) (1989) Markov chains and Monte Carlo calculations in polymer science. Marcel Dekker Inc, New York... 

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... 

Presently Monte Carlo calculations are based on the technique proposed by Metropolis [22] in 1953 which involves selecting the successive configurations in such a way that they build up a Markov chain [23], The one-step transition probabilities pij are defined as the probability that beginning from the i configuration with qj(N), the configuration j with qj,N> is reached in one step. These probabilities are the elements of the one-step probability matrix associated to the Markov chain and they must fulfill the following conditions ... 

Monte Carlo method within this ensemble [29,30,32,33] differs from the canonical ensemble (N,V,T) in that the configuration variables are c]successive steps of the Markov chain not only q,N> must be perturbed but also the volume. For computational reasons it is convenient to introduce scaled coordinates ... 

The difficulty arises from the fact that the one-step transition probabilities of the Markov chain involve only ratios of probability densities, in which Z(N,V,T) cancels out. This way, the Metropolis Markov chain procedure intentionally avoids the calculation of the configurational integral, the Monte Carlo method not being able to directly apply equation (31). 

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. 

Figure 9.21. Cloud of points from a Monte Carlo Markov chain sampling of the likelihood of models fit to the WMAP plus other CMB datasets. The size of the points indicates how consistent the model is with the HST Key Project on the Distance Scale value for the Hubble constant. The contours show the likelihood computed for 230 Type la supernovae (Tonry et al., 2003).

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. 

---