Monte Carlo Methods in Bayesian Computation

Chen, Ming‐Hui; Shao, Qi-Man; Ibrahim, Joseph G.

doi:10.1007/978-1-4612-1276-8

Cited by 703 publications

(483 citation statements)

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“…Conversely, one can recycle the output of an Markov chain Monte Carlo algorithm towards estimating the evidence, with no or little additional programming effort; see for instance Gelfand & Dey (1994), Meng & Wong (1996), and Chen & Shao (1997). We describe below the solutions used in the subsequent comparison with nested sampling, but we do not pretend at an exhaustive coverage of those techniques, see Chen et al (2000) or Han & Carlin (2001) for a deeper coverage, nor at using the most efficient approach, see Meng & Schilling (2002).…”

Section: Nested Importance Samplingmentioning

confidence: 99%

Properties of nested sampling

Chopin¹,

Robert²

2010

Biometrika

100

129

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SUMMARYNested sampling is a simulation method for approximating marginal likelihoods proposed by Skilling (2006). We establish that nested sampling has an approximation error that vanishes at the standard Monte Carlo rate and that this error is asymptotically Gaussian. We show that the asymptotic variance of the nested sampling approximation typically grows linearly with the dimension of the parameter. We discuss the applicability and efficiency of nested sampling in realistic problems, and we compare it with two current methods for computing marginal likelihood. We propose an extension that avoids resorting to Markov chain Monte Carlo to obtain the simulated points.

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Section: Nested Importance Samplingmentioning

confidence: 99%

Properties of nested sampling

Chopin¹,

Robert²

2010

Biometrika

100

129

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“…We used posterior simulation techniques (17) to calculate structures and to simultaneously estimate the error of the lognormal model. Structures were parameterized in torsion angles; nonbonded interactions were represented with a purely repulsive potential (18).…”

Section: Resultsmentioning

confidence: 99%

Weighting of experimental evidence in macromolecular structure determination

Habeck

Rieping

Nilges

2006

Proc. Natl. Acad. Sci. U.S.A.

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The determination of macromolecular structures requires weighting of experimental evidence relative to prior physical information. Although it can critically affect the quality of the calculated structures, experimental data are routinely weighted on an empirical basis. At present, cross-validation is the most rigorous method to determine the best weight. We describe a general method to adaptively weight experimental data in the course of structure calculation. It is further shown that the necessity to define weights for the data can be completely alleviated. We demonstrate the method on a structure calculation from NMR data and find that the resulting structures are optimal in terms of accuracy and structural quality. Our method is devoid of the bias imposed by an empirical choice of the weight and has some advantages over estimating the weight by cross-validation.Bayesian probability theory ͉ Markov chain Monte Carlo E xperimental data are typically insufficient to determine a biomolecular structure in their own right but need to be complemented with prior physical information. Therefore, structure determination amounts to the search for conformations that have a low physical energy and that, at the same time, minimize a cost function E data quantifying the disagreement between a structural model X and the data. This approach is implemented as minimization of a hybrid energy (1, 2)where the force field E phys compensates a lack of data by imposing physical constraints on the structure. A target function of this form is widely used in macromolecular structure determination, notably from NMR data (3, 4) and from homologyderived restraints (5). The weight w data controls the contribution of the data relative to the force field. Its value can be critical: If it is too large, the contribution of the force field might be too small to avoid overfitting; if the weight is too small, the data contribute too little to define the structure. The choice of the weight also concerns the question of how to judge structural quality. Overfitted structures reach a low R value (6, 7) but exhibit a poor stereochemistry or an unlikely fold. Usually, experimental data are weighted empirically: w data is set ad hoc and held constant during structure calculation. However, already when introducing the hybrid energy concept, Jack and Levitt (1) remarked that correct weighting of the data ''is something of a problem.'' They proposed to adjust the weight to equalize E phys and w data E data ; this adjustment was later refined, for example, in ref. 8. At present, the most rigorous quantitative method to determine the optimal weight is complete crossvalidation (6, 7). However, cross-validation can become unstable and time-consuming in the case of sparse and heterogeneous data with several independent weights.In this work, we introduce an objective and unique way to weight experimental data. We show that a quantitative treatment does not necessitate heuristics like cross-validation: everything we need is contained in the rules of probability ...

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“…These methods are extensively documented in the statistical literature (see the books [47][48][49] and the references therein). The main objective of MCMC techniques is to simulate a stationary ergodic Markov chain whose samples asymptotically follow the posterior density p (S, C, θ|D).…”

Section: Estimation Via Markov Chain Monte Carlo Methodsmentioning

confidence: 99%

Bayesian analysis of spectral mixture data using Markov Chain Monte Carlo Methods

Moussaoui

Carteret

Brie

et al. 2006

Chemometrics and Intelligent Laboratory Systems

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This paper presents an original method for the analysis of multicomponent spectral data sets. The proposed algorithm is based on Bayesian estimation theory and Markov Chain Monte Carlo (MCMC) methods. Resolving spectral mixture analysis aims at recovering the unknown component spectra and at assessing the concentrations of the underlying species in the mixtures. In addition to non-negativity constraint, further assumptions are generally needed to get a unique resolution. The proposed statistical approach assumes mutually independent spectra and accounts for the non-negativity and the sparsity of both the pure component spectra and the concentration profiles. Gamma distribution priors are used to translate all these information in a probabilistic framework. The estimation is performed using MCMC methods which lead to an unsupervised algorithm, whose performances are assessed in a simulation study with a synthetic data set.

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Monte Carlo Methods in Bayesian Computation

Cited by 703 publications

References 0 publications

Properties of nested sampling

Properties of nested sampling

Weighting of experimental evidence in macromolecular structure determination

Bayesian analysis of spectral mixture data using Markov Chain Monte Carlo Methods

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