Markov chain Monte Carlo methods

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... 

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

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). 

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. 

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. 

Mixed effects models under a Bayesian framework have been widely studied and used with the use of Markov chain Monte Carlo methods (10). These methods have gained particular popularity as complex problems became easily formulated using the WinBUGS software (11). See Congdon (12) for an extensive coverage of topics and examples and implementation in WinBUGS. 

BAYESIAN HIERARCHICAL MODELING WITH MARKOV CHAIN MONTE CARLO METHODS... 

B. P. Carlin and S. Chib, Bayesian model choice via Markov chain Monte Carlo methods. / Roy Stat Soc Br 57 473 84 (1995). 

An HMM is essentially a Markov chain (—> Monte Carlo methods). Each state inside the Markov chain can produce a letter, and it does so by a stochastic Poisson process that chooses one of a finite number of letters in an alphabet. Each letter has a characteristic probability of being chosen. That probability depends on the state and on the letter. After the state produces a letter, the Markov chain moves to another state. This happens according to a transition probability that depends on the prior and succeeding state. 

Smith AFM, Roberts GO (1993) Bayesian computation via the Gibbs sampler and related Markov-Chain Monte-Carlo methods. Journal of the Royal Statistical Society Series B - Methodological 55 3-23. 

One example of the NMR reconstraction problem employs the reversible-jump Markov chain Monte-Carlo method [16]. It assumes that the model spectram S Fi,F2) is made up of a limited number m of two-dimensional Gaussian resonance lines. Then m, the linewidths, intensities, and frequency co-ordinates are varied until the Markov chain reaches convergence. The allowed transitions between the current map M and the new map M comprise movement, merging or splitting of resonance lines, and birth or death of component responses. Compatibility with the experimental traces is checked by projecting M at the appropriate angles. The procedure has been found to be stable and reproducible [16]. 

Computational Issues Transitional Markov Chain Monte Carlo Method... 

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