Home Mathematicsematical Statistics • Download e-book for iPad: Advanced Markov chain Monte Carlo methods by Faming Liang, Chuanhai Liu, Raymond Carroll

Download e-book for iPad: Advanced Markov chain Monte Carlo methods by Faming Liang, Chuanhai Liu, Raymond Carroll

By Faming Liang, Chuanhai Liu, Raymond Carroll

ISBN-10: 0470748265

ISBN-13: 9780470748268

Markov Chain Monte Carlo (MCMC) equipment at the moment are an necessary software in clinical computing. This publication discusses fresh advancements of MCMC tools with an emphasis on these utilising earlier pattern details in the course of simulations. the appliance examples are drawn from diversified fields comparable to bioinformatics, computing device studying, social technology, combinatorial optimization, and computational physics.

Key good points:

  • Expanded insurance of the stochastic approximation Monte Carlo and dynamic weighting algorithms which are primarily resistant to neighborhood catch difficulties.
  • A designated dialogue of the Monte Carlo Metropolis-Hastings set of rules that may be used for sampling from distributions with intractable normalizing constants.
  • Up-to-date money owed of modern advancements of the Gibbs sampler.
  • Comprehensive overviews of the population-based MCMC algorithms and the MCMC algorithms with adaptive proposals.
  • Accompanied through a assisting site that includes datasets utilized in the ebook, in addition to codes used for a few simulation examples.

This publication can be utilized as a textbook or a reference booklet for a one-semester graduate path in records, computational biology, engineering, and computing device sciences. utilized or theoretical researchers also will locate this ebook precious.

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Extra info for Advanced Markov chain Monte Carlo methods

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To speed up DA can be achieved by boosting the conditional variability (or variance if THE GIBBS SAMPLER 36 exists) of θ given Xmis and Xobs . A way of doing this is to consider different data augmentation schemes. Much work has been done in the EM world; see Meng and van Dyk (1997) and Liu et al . (1998). Meng and van Dyk (1997) seek within a given class of complete-data models to get the optimal one in terms of the fraction of missing information (see Little and Rubin, 1987). This idea of ‘imputing less’ is powerful.

A Markov chain, named after Andrey Markov, is a sequence of random variables {Xi : i = 0, 1, 2, . } with the Markov property that given the present state, the future and past states are independent, that is, for all measurable sets A in X, Pr (Xt+1 ∈ A|X0 = x0 , . . , Xt = xt ) = Pr (Xt+1 ∈ A|Xt = xt ) holds for time t = 0, 1, . . 26) MARKOV CHAIN MONTE CARLO 19 A convenient way of handling both discrete and continuous variables is to use the notation π(dy) to denote the probability measure π on (X, X).

Most important for this method is to choose g(x) for both simplicity in generating Monte Carlo samples and accuracy in estimating Ef [h(X)] by controlling the associated Monte Carlo errors. For Monte Carlo accuracy, a natural way is to choose g(x) to minimize the ˜ variance of h(X) with X ∼ g(x). Theoretical results on optimal g(x) are also available. The following result is due to Rubinstein (1981); see also Robert and Casella (2004). 18) is g∗ (x) = |h(x)|f(x) . 1 is left as an exercise. As always, theoretical results provide helpful guidance.

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Advanced Markov chain Monte Carlo methods by Faming Liang, Chuanhai Liu, Raymond Carroll

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