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Register a new userElliptical slice sampling
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Added by
Iain Murray
Publication date
2009
fields
Mathematical Statistics
and research's language is
English
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Many probabilistic models introduce strong dependencies between variables using a latent multivariate Gaussian distribution or a Gaussian process. We present a new Markov chain Monte Carlo algorithm for performing inference in models with multivariate Gaussian priors. Its key properties are: 1) it has simple, generic code applicable to many models, 2) it has no free parameters, 3) it works well for a variety of Gaussian process based models. These properties make our method ideal for use while model building, removing the need to spend time deriving and tuning updates for more complex algorithms.
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Markov chain Monte Carlo (MCMC) methods asymptotically sample from complex probability distributions. The pseudo-marginal MCMC framework only requires an unbiased estimator of the unnormalized probability distribution function to construct a Markov chain. However, the resulting chains are harder to tune to a target distribution than conventional MCMC, and the types of updates available are limited. We describe a general way to clamp and update the random numbers used in a pseudo-marginal methods unbiased estimator. In this framework we can use slice sampling and other adaptive methods. We obtain more robust Markov chains, which often mix more quickly.
In this paper we introduce a new sampling algorithm which has the potential to be adopted as a universal replacement to the Metropolis--Hastings algorithm. It is related to the slice sampler, and motivated by an algorithm which is applicable to discrete probability distributions %which can be viewed as an alternative to the Metropolis--Hastings algorithm in this setting, which obviates the need for a proposal distribution, in that is has no accept/reject component. This paper looks at the continuous counterpart. A latent variable combined with a slice sampler and a shrinkage procedure applied to uniform density functions creates a highly efficient sampler which can generate random variables from very high dimensional distributions as a single block.
Monte Carlo (MC) methods are widely used for Bayesian inference and optimization in statistics, signal processing and machine learning. A well-known class of MC methods are Markov Chain Monte Carlo (MCMC) algorithms. In order to foster better exploration of the state space, specially in high-dimensional applications, several schemes employing multiple parallel MCMC chains have been recently introduced. In this work, we describe a novel parallel interacting MCMC scheme, called {it orthogonal MCMC} (O-MCMC), where a set of vertical parallel MCMC chains share information using some horizontal MCMC techniques working on the entire population of current states. More specifically, the vertical chains are led by random-walk proposals, whereas the horizontal MCMC techniques employ independent proposals, thus allowing an efficient combination of global exploration and local approximation. The interaction is contained in these horizontal iterations. Within the analysis of different implementations of O-MCMC, novel schemes in order to reduce the overall computational cost of parallel multiple try Metropolis (MTM) chains are also presented. Furthermore, a modified version of O-MCMC for optimization is provided by considering parallel simulated annealing (SA) algorithms. Numerical results show the advantages of the proposed sampling scheme in terms of efficiency in the estimation, as well as robustness in terms of independence with respect to initial values and the choice of the parameters.
Inference in continuous label Markov random fields is a challenging task. We use particle belief propagation (PBP) for solving the inference problem in continuous label space. Sampling particles from the belief distribution is typically done by using Metropolis-Hastings Markov chain Monte Carlo methods which involves sampling from a proposal distribution. This proposal distribution has to be carefully designed depending on the particular model and input data to achieve fast convergence. We propose to avoid dependence on a proposal distribution by introducing a slice sampling based PBP algorithm. The proposed approach shows superior convergence performance on an image denoising toy example. Our findings are validated on a challenging relational 2D feature tracking application.
Nested sampling (NS) computes parameter posterior distributions and makes Bayesian model comparison computationally feasible. Its strengths are the unsupervised navigation of complex, potentially multi-modal posteriors until a well-defined termination point. A systematic literature review of nested sampling algorithms and variants is presented. We focus on complete algorithms, including solutions to likelihood-restricted prior sampling, parallelisation, termination and diagnostics. The relation between number of live points, dimensionality and computational cost is studied for two complete algorithms. A new formulation of NS is presented, which casts the parameter space exploration as a search on a tree. Previously published ways of obtaining robust error estimates and dynamic variations of the number of live points are presented as special cases of this formulation. A new on-line diagnostic test is presented based on previous insertion rank order work. The survey of nested sampling methods concludes with outlooks for future research.
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