151 research outputs found

    Metropolis Sampling

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    Monte Carlo (MC) sampling methods are widely applied in Bayesian inference, system simulation and optimization problems. The Markov Chain Monte Carlo (MCMC) algorithms are a well-known class of MC methods which generate a Markov chain with the desired invariant distribution. In this document, we focus on the Metropolis-Hastings (MH) sampler, which can be considered as the atom of the MCMC techniques, introducing the basic notions and different properties. We describe in details all the elements involved in the MH algorithm and the most relevant variants. Several improvements and recent extensions proposed in the literature are also briefly discussed, providing a quick but exhaustive overview of the current Metropolis-based sampling's world.Comment: Wiley StatsRef-Statistics Reference Online, 201

    HYDRA: a Java library for Markov Chain Monte Carlo

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    Hydra is an open-source, platform-neutral library for performing Markov Chain Monte Carlo. It implements the logic of standard MCMC samplers within a framework designed to be easy to use, extend, and integrate with other software tools. In this paper, we describe the problem that motivated our work, outline our goals for the Hydra pro ject, and describe the current features of the Hydra library. We then provide a step-by-step example of using Hydra to simulate from a mixture model drawn from cancer genetics, first using a variable-at-a-time Metropolis sampler and then a Normal Kernel Coupler. We conclude with a discussion of future directions for Hydra.

    Approximating Probability Densities by Iterated Laplace Approximations

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    The Laplace approximation is an old, but frequently used method to approximate integrals for Bayesian calculations. In this paper we develop an extension of the Laplace approximation, by applying it iteratively to the residual, i.e., the difference between the current approximation and the true function. The final approximation is thus a linear combination of multivariate normal densities, where the coefficients are chosen to achieve a good fit to the target distribution. We illustrate on real and artificial examples that the proposed procedure is a computationally efficient alternative to current approaches for approximation of multivariate probability densities. The R-package iterLap implementing the methods described in this article is available from the CRAN servers.Comment: to appear in Journal of Computational and Graphical Statistics, http://pubs.amstat.org/loi/jcg

    Adaptive Optimal Scaling of Metropolis-Hastings Algorithms Using the Robbins-Monro Process

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    We present an adaptive method for the automatic scaling of Random-Walk Metropolis-Hastings algorithms, which quickly and robustly identifies the scaling factor that yields a specified overall sampler acceptance probability. Our method relies on the use of the Robbins-Monro search process, whose performance is determined by an unknown steplength constant. We give a very simple estimator of this constant for proposal distributions that are univariate or multivariate normal, together with a sampling algorithm for automating the method. The effectiveness of the algorithm is demonstrated with both simulated and real data examples. This approach could be implemented as a useful component in more complex adaptive Markov chain Monte Carlo algorithms, or as part of automated software packages

    Ensemble Transport Adaptive Importance Sampling

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    Markov chain Monte Carlo methods are a powerful and commonly used family of numerical methods for sampling from complex probability distributions. As applications of these methods increase in size and complexity, the need for efficient methods increases. In this paper, we present a particle ensemble algorithm. At each iteration, an importance sampling proposal distribution is formed using an ensemble of particles. A stratified sample is taken from this distribution and weighted under the posterior, a state-of-the-art ensemble transport resampling method is then used to create an evenly weighted sample ready for the next iteration. We demonstrate that this ensemble transport adaptive importance sampling (ETAIS) method outperforms MCMC methods with equivalent proposal distributions for low dimensional problems, and in fact shows better than linear improvements in convergence rates with respect to the number of ensemble members. We also introduce a new resampling strategy, multinomial transformation (MT), which while not as accurate as the ensemble transport resampler, is substantially less costly for large ensemble sizes, and can then be used in conjunction with ETAIS for complex problems. We also focus on how algorithmic parameters regarding the mixture proposal can be quickly tuned to optimise performance. In particular, we demonstrate this methodology's superior sampling for multimodal problems, such as those arising from inference for mixture models, and for problems with expensive likelihoods requiring the solution of a differential equation, for which speed-ups of orders of magnitude are demonstrated. Likelihood evaluations of the ensemble could be computed in a distributed manner, suggesting that this methodology is a good candidate for parallel Bayesian computations

    Change-point of multiple biomarkers in women with ovarian cancer

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    To date several algorithms for longitudinal analysis of ovarian cancer biomarkers have been proposed in the literature. An issue of specific interest is to determine whether the baseline level of a biomarker changes significantly at some time instant (change-point) prior to the clinical diagnosis of cancer. Such change-points in the serum biomarker Cancer Antigen 125 (CA125) time series data have been used in ovarian cancer screening, resulting in earlier detection with a sensitivity of 85% in the most recent trial, the UK Collaborative Trial of Ovarian Cancer Screening (UKCTOCS, number ISRCTN22488978; NCT00058032). Here we propose to apply a hierarchical Bayesian change-point model to jointly study the features of time series from multiple biomarkers. For this model we have analytically derived the conditional probability distribution of every unknown parameter, thus enabling the design of efficient Markov Chain Monte Carlo methods for their estimation. We have applied these methods to the estimation of change-points in time series data of multiple biomarkers, including CA125 and others, using data from a nested case-control study of women diagnosed with ovarian cancer in UKCTOCS. In this way we assess whether any of these additional biomarkers can play a role in change-point detection and, therefore, aid in the diagnosis of the disease in patients for whom the CA125 time series does not display a change-point. We have also investigated whether the change-points for different biomarkers occur at similar times for the same patient. The main conclusion of our study is that the combined analysis of a group of specific biomarkers may possibly improve the detection of change-points in time series data (compared to the analysis of CA125 alone) which, in turn, are relevant for the early diagnosis of ovarian cancer
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