44 research outputs found
Suppressing Random Walks in Markov Chain Monte Carlo Using Ordered Overrelaxation
Markov chain Monte Carlo methods such as Gibbs sampling and simple forms of
the Metropolis algorithm typically move about the distribution being sampled
via a random walk. For the complex, high-dimensional distributions commonly
encountered in Bayesian inference and statistical physics, the distance moved
in each iteration of these algorithms will usually be small, because it is
difficult or impossible to transform the problem to eliminate dependencies
between variables. The inefficiency inherent in taking such small steps is
greatly exacerbated when the algorithm operates via a random walk, as in such a
case moving to a point n steps away will typically take around n^2 iterations.
Such random walks can sometimes be suppressed using ``overrelaxed'' variants of
Gibbs sampling (a.k.a. the heatbath algorithm), but such methods have hitherto
been largely restricted to problems where all the full conditional
distributions are Gaussian. I present an overrelaxed Markov chain Monte Carlo
algorithm based on order statistics that is more widely applicable. In
particular, the algorithm can be applied whenever the full conditional
distributions are such that their cumulative distribution functions and inverse
cumulative distribution functions can be efficiently computed. The method is
demonstrated on an inference problem for a simple hierarchical Bayesian model.Comment: uuencoded compressed postscript (with instructions on decoding
Fast Markov chain Monte Carlo sampling for sparse Bayesian inference in high-dimensional inverse problems using L1-type priors
Sparsity has become a key concept for solving of high-dimensional inverse
problems using variational regularization techniques. Recently, using similar
sparsity-constraints in the Bayesian framework for inverse problems by encoding
them in the prior distribution has attracted attention. Important questions
about the relation between regularization theory and Bayesian inference still
need to be addressed when using sparsity promoting inversion. A practical
obstacle for these examinations is the lack of fast posterior sampling
algorithms for sparse, high-dimensional Bayesian inversion: Accessing the full
range of Bayesian inference methods requires being able to draw samples from
the posterior probability distribution in a fast and efficient way. This is
usually done using Markov chain Monte Carlo (MCMC) sampling algorithms. In this
article, we develop and examine a new implementation of a single component
Gibbs MCMC sampler for sparse priors relying on L1-norms. We demonstrate that
the efficiency of our Gibbs sampler increases when the level of sparsity or the
dimension of the unknowns is increased. This property is contrary to the
properties of the most commonly applied Metropolis-Hastings (MH) sampling
schemes: We demonstrate that the efficiency of MH schemes for L1-type priors
dramatically decreases when the level of sparsity or the dimension of the
unknowns is increased. Practically, Bayesian inversion for L1-type priors using
MH samplers is not feasible at all. As this is commonly believed to be an
intrinsic feature of MCMC sampling, the performance of our Gibbs sampler also
challenges common beliefs about the applicability of sample based Bayesian
inference.Comment: 33 pages, 14 figure
Improved Algorithms for Simulating Crystalline Membranes
The physics of crystalline membranes, i.e. fixed-connectivity surfaces
embedded in three dimensions and with an extrinsic curvature term, is very rich
and of great theoretical interest. To understand their behavior, numerical
simulations are commonly used. Unfortunately, traditional Monte Carlo
algorithms suffer from very long auto-correlations and critical slowing down in
the more interesting phases of the model. In this paper we study the
performance of improved Monte Carlo algorithms for simulating crystalline
membrane, such as hybrid overrelaxation and unigrid methods, and compare their
performance to the more traditional Metropolis algorithm. We find that although
the overrelaxation algorithm does not reduce the critical slowing down, it
gives an overall gain of a factor 15 over the Metropolis algorithm. The unigrid
algorithm does, on the other hand, reduce the critical slowing down exponent to
z apprx. 1.7.Comment: 14 pages, 1 eps-figur
Monte Carlo Renormalization of 2d Simplicial Quantum Gravity Coupled to Gaussian Matter
We extend a recently proposed real-space renormalization group scheme for
dynamical triangulations to situations where the lattice is coupled to
continuous scalar fields. Using Monte Carlo simulations in combination with a
linear, stochastic blocking scheme for the scalar fields we are able to
determine the leading eigenvalues of the stability matrix with good accuracy
both for c = 1 and c = 10 theories.Comment: 17 pages, 7 figure
Zero Variance Markov Chain Monte Carlo for Bayesian Estimators
A general purpose variance reduction technique for Markov chain Monte Carlo (MCMC) estimators, based on the zero-variance principle introduced in the physics literature, is proposed to evaluate the expected value, of a function f with respect to a, possibly unnormalized, probability distribution . In this context, a control variate approach, generally used for Monte Carlo simulation, is exploited by replacing f with a dierent function, ~ f. The function ~ f is constructed so that its expectation, under , equals f , but its variance with respect to is much smaller. Theoretically, an optimal re-normalization f exists which may lead to zero variance; in practice, a suitable approximation for it must be investigated. In this paper, an ecient class of re-normalized ~ f is investigated, based on a polynomial parametrization. We nd that a low-degree polynomial (1st, 2nd or 3rd degree) can lead to dramatically huge variance reduction of the resulting zero-variance MCMC estimator. General formulas for the construction of the control variates in this context are given. These allow for an easy implementation of the method in very general settings regardless of the form of the target/posterior distribution (only dierentiability is required) and of the MCMC algorithm implemented (in particular, no reversibility is needed).Control variates, GARCH models, Logistic regression, Metropolis-Hastings algorithm, Variance reduction
Multivariate Stochastic Volatility Models: Bayesian Estimation and Model Comparison
In this paper we show that fully likelihood-based estimation and comparison of multivariate stochastic volatility (SV) models can be easily performed via a freely available Bayesian software called WinBUGS. Moreover, we introduce to the literature several new specifications which are natural extensions to certain existing models, one of which allows for time varying correlation coefficients. Ideas are illustrated by fitting, to a bivariate time series data of weekly exchange rates, nine multivariate SV models, including the specifications with Granger causality in volatility, time varying correlations, heavytailed error distributions, additive factor structure, and multiplicative factor structure. Empirical results suggest that the most adequate specifications are those that allow for time varying correlation coefficients.Multivariate stochastic volatility; Granger causality in volatility; Heavy-tailed distributions; Time varying correlations; Factors; MCMC; DIC.