19,308 research outputs found
Bayesian computational methods
In this chapter, we will first present the most standard computational
challenges met in Bayesian Statistics, focussing primarily on mixture
estimation and on model choice issues, and then relate these problems with
computational solutions. Of course, this chapter is only a terse introduction
to the problems and solutions related to Bayesian computations. For more
complete references, see Robert and Casella (2004, 2009), or Marin and Robert
(2007), among others. We also restrain from providing an introduction to
Bayesian Statistics per se and for comprehensive coverage, address the reader
to Robert (2007), (again) among others.Comment: This is a revised version of a chapter written for the Handbook of
Computational Statistics, edited by J. Gentle, W. Hardle and Y. Mori in 2003,
in preparation for the second editio
A Shuffled Complex Evolution Metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters
Markov Chain Monte Carlo (MCMC) methods have become increasingly popular for estimating the posterior probability distribution of parameters in hydrologic models. However, MCMC methods require the a priori definition of a proposal or sampling distribution, which determines the explorative capabilities and efficiency of the sampler and therefore the statistical properties of the Markov Chain and its rate of convergence. In this paper we present an MCMC sampler entitled the Shuffled Complex Evolution Metropolis algorithm (SCEM-UA), which is well suited to infer the posterior distribution of hydrologic model parameters. The SCEM-UA algorithm is a modified version of the original SCE-UA global optimization algorithm developed by Duan et al. [1992]. The SCEM-UA algorithm operates by merging the strengths of the Metropolis algorithm, controlled random search, competitive evolution, and complex shuffling in order to continuously update the proposal distribution and evolve the sampler to the posterior target distribution. Three case studies demonstrate that the adaptive capability of the SCEM-UA algorithm significantly reduces the number of model simulations needed to infer the posterior distribution of the parameters when compared with the traditional Metropolis-Hastings samplers
On computational tools for Bayesian data analysis
While Robert and Rousseau (2010) addressed the foundational aspects of
Bayesian analysis, the current chapter details its practical aspects through a
review of the computational methods available for approximating Bayesian
procedures. Recent innovations like Monte Carlo Markov chain, sequential Monte
Carlo methods and more recently Approximate Bayesian Computation techniques
have considerably increased the potential for Bayesian applications and they
have also opened new avenues for Bayesian inference, first and foremost
Bayesian model choice.Comment: This is a chapter for the book "Bayesian Methods and Expert
Elicitation" edited by Klaus Bocker, 23 pages, 9 figure
Joint segmentation of multivariate astronomical time series : bayesian sampling with a hierarchical model
Astronomy and other sciences often face the problem of detecting and characterizing structure in two or more related time series. This paper approaches such problems using Bayesian priors to represent relationships between signals with various degrees of certainty, and not just rigid constraints. The segmentation is conducted by using a hierarchical Bayesian approach to a piecewise constant Poisson rate model. A Gibbs sampling strategy allows joint estimation of the unknown parameters and hyperparameters. Results obtained with synthetic and real photon counting data illustrate the performance of the proposed algorithm
Consistent estimation of the spectrum of trace class data augmentation algorithms
Markov chain Monte Carlo is widely used in a variety of scientific
applications to generate approximate samples from intractable distributions. A
thorough understanding of the convergence and mixing properties of these Markov
chains can be obtained by studying the spectrum of the associated Markov
operator. While several methods to bound/estimate the second largest eigenvalue
are available in the literature, very few general techniques for consistent
estimation of the entire spectrum have been proposed. Existing methods for this
purpose require the Markov transition density to be available in closed form,
which is often not true in practice, especially in modern statistical
applications. In this paper, we propose a novel method to consistently estimate
the entire spectrum of a general class of Markov chains arising from a popular
and widely used statistical approach known as Data Augmentation. The transition
densities of these Markov chains can often only be expressed as intractable
integrals. We illustrate the applicability of our method using real and
simulated data.Comment: 43 pages (including Appendix), 3 figures; final versio
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