On the stability and ergodicity of adaptive scaling Metropolis algorithms

The stability and ergodicity properties of two adaptive random walk Metropolis algorithms are considered. Both algorithms adjust the scaling of the proposal distribution continuously based on the observed acceptance probability. Unlike the previously proposed forms of the algorithms, the adapted scaling parameter is not constrained within a predefined compact interval. The first algorithm is based on scale adaptation only, while the second one also incorporates covariance adaptation. A strong law of large numbers is shown to hold assuming that the target density is smooth enough and has either compact support or super-exponentially decaying tails.     

Optimal scaling of random-walk metropolis algorithms on general target distributions

One main limitation of the existing optimal scaling results for Metropolis–Hastings algorithms is that the assumptions on the target distribution are unrealistic. In this paper, we consider optimal scaling of random-walk Metropolis algorithms on general target distributions in high dimensions arising from practical MCMC models from Bayesian statistics. For optimal scaling by maximizing expected squared jumping distance (ESJD), we show the asymptotically optimal acceptance rate 0.234 can be obtained under general realistic sufficient conditions on the target distribution. The new sufficient conditions are easy to be verified and may hold for some general classes of MCMC models arising from Bayesian statistics applications, which substantially generalize the product i.i.d. condition required in most existing literature of optimal scaling. Furthermore, we show one-dimensional diffusion limits can be obtained under slightly stronger conditions, which still allow dependent coordinates of the target distribution. We also connect the new diffusion limit results to complexity bounds of Metropolis algorithms in high dimensions.     

Adaptive Markov Chain Monte Carlo for Bayesian Variable Selection

We describe adaptive Markov chain Monte Carlo (MCMC) methods for sampling posterior distributions arising from Bayesian variable selection problems. Point-mass mixture priors are commonly used in Bayesian variable selection problems in regression. However, for generalized linear and nonlinear models where the conditional densities cannot be obtained directly, the resulting mixture posterior may be difficult to sample using standard MCMC methods due to multimodality. We introduce an adaptive MCMC scheme that automatically tunes the parameters of a family of mixture proposal distributions during simulation. The resulting chain adapts to sample efficiently from multimodal target distributions. For variable selection problems point-mass components are included in the mixture, and the associated weights adapt to approximate marginal posterior variable inclusion probabilities, while the remaining components approximate the posterior over nonzero values. The resulting sampler transitions efficiently between models, performing parameter estimation and variable selection simultaneously. Ergodicity and convergence are guaranteed by limiting the adaptation based on recent theoretical results. The algorithm is demonstrated on a logistic regression model, a sparse kernel regression, and a random field model from statistical biophysics; in each case the adaptive algorithm dramatically outperforms traditional MH algorithms. Supplementary materials for this article are available online.     

Examples of Adaptive MCMC

We investigate the use of adaptive MCMC algorithms to automatically tune the Markov chain parameters during a run. Examples include the Adaptive Metropolis (AM) multivariate algorithm of Haario, Saksman, and Tamminen (2001), Metropolis-within-Gibbs algorithms for nonconjugate hierarchical models, regionally adjusted Metropolis algorithms, and logarithmic scalings. Computer simulations indicate that the algorithms perform very well compared to nonadaptive algorithms, even in high dimension.     

Dynamically Updated Spatially Varying Parameterizations of Hierarchical Bayesian Models for Spatial Data

Fitting hierarchical Bayesian models to spatially correlated datasets using Markov chain Monte Carlo (MCMC) techniques is computationally expensive. Complicated covariance structures of the underlying spatial processes, together with high-dimensional parameter space, mean that the number of calculations required grows cubically with the number of spatial locations at each MCMC iteration. This necessitates the need for efficient model parameterizations that hasten the convergence and improve the mixing of the associated algorithms. We consider partially centred parameterizations (PCPs) which lie on a continuum between what are known as the centered (CP) and noncentered parameterizations (NCP). By introducing a weight matrix we remove the conditional posterior correlation between the fixed and the random effects, and hence construct a PCP which achieves immediate convergence for a three-stage model, based on multiple Gaussian processes with known covariance parameters. When the covariance parameters are unknown we dynamically update the parameterization within the sampler. The PCP outperforms both the CP and the NCP and leads to a fully automated algorithm which has been demonstrated in two simulation examples. The effectiveness of the spatially varying PCP is illustrated with a practical dataset of nitrogen dioxide concentration levels. Supplemental materials consisting of appendices, datasets, and computer code to reproduce the results are available online.     

MCMC Estimation of Restricted Covariance Matrices

This article is motivated by the difficulty of applying standard simulation techniques when identification constraints or theoretical considerations induce covariance restrictions in multivariate models. To deal with this difficulty, we build upon a decomposition of positive definite matrices and show that it leads to straightforward Markov chain Monte Carlo samplers for restricted covariance matrices. We introduce the approach by reviewing results for multivariate Gaussian models without restrictions, where standard conjugate priors on the elements of the decomposition induce the usual Wishart distribution on the precision matrix and vice versa. The unrestricted case provides guidance for constructing efficient Metropolis–Hastings and accept-reject Metropolis–Hastings samplers in more complex settings, and we describe in detail how simulation can be performed under several important constraints. The proposed approach is illustrated in a simulation study and two applications in economics. Supplemental materials for this article (appendixes, data, and computer code) are available online.     

Proposal adaptation in simulated annealing for continuous optimization problems

In recent years, adaptive Markov Chain Monte Carlo (MCMC) methods have become a standard tool for Bayesian parameter estimation. In adaptive MCMC, the past iterations are used to tune the proposal distribution of the algorithm. The same adaptation mechanisms can be used in Simulated Annealing (SA), a popular optimization method based on MCMC. The difficulty in using adaptation directly in SA is that the target function changes along the iterations in the annealing process, and the adaptation should keep up with the annealing. In this paper, a mechanism for automatically tuning the proposal distribution in SA is proposed. The approach is based on the Adaptive Metropolis algorithm of Haario et al. (Bernoulli 7(2):223–242, 2001), combined with a weighting mechanism to account for the cooling target. The proposed adaptation mechanism does not add any computational complexity to the problem in terms of objective function evaluations. The effect of adaptation is demonstrated using two benchmark problems, showing that the proposed adaptation mechanism can significantly improve optimization results compared to non-adaptive SA. The approach is presented for continuous optimization problems and generalization to integer and mixed-integer problems is a topic of future research.     

A New Algorithm for Simulating a Correlation Matrix Based on Parameter Expansion and Reparameterization

The correlation matrix (denoted by R) plays an important role in many statistical models. Unfortunately, sampling the correlation matrix in Markov chain Monte Carlo (MCMC) algorithms can be problematic. In addition to the positive definite constraint of covariance matrices, correlation matrices have diagonal elements fixed at one. In this article, we propose an efficient two-stage parameter expanded reparameterization and Metropolis-Hastings (PX-RPMH) algorithm for simulating R. Using this algorithm, we draw all elements of R simultaneously by first drawing a covariance matrix from an inverse Wishart distribution, and then translating it back to a correlation matrix through a reduction function and accepting it based on a Metropolis-Hastings acceptance probability. This algorithm is illustrated using multivariate probit (MVP) models and multivariate regression (MVR) models with a common correlation matrix across groups. Via both a simulation study and a real data example, the performance of the PX-RPMH algorithm is compared with those of other common algorithms. The results show that the PX-RPMH algorithm is more efficient than other methods for sampling a correlation matrix.     

Scaling Analysis of Delayed Rejection MCMC Methods

In this paper, we study the asymptotic efficiency of the delayed rejection strategy. In particular, the efficiency of the delayed rejection Metropolis–Hastings algorithm is compared to that of the regular Metropolis algorithm. To allow for a fair comparison, the study is carried under optimal mixing conditions for each of these algorithms. After introducing optimal scaling results for the delayed rejection (DR) algorithm, we outline the fact that the second proposal after the first rejection is discarded, with a probability tending to 1 as the dimension of the target density increases. To overcome this drawback, a modification of the delayed rejection algorithm is proposed, in which the direction of the different proposals is fixed once for all, and the Metropolis–Hastings accept-reject mechanism is used to select a proper scaling along the search direction. It is shown that this strategy significantly outperforms the original DR and Metropolis algorithms, especially when the dimension becomes large. We include numerical studies to validate these conclusions.     

Automatically tuned general-purpose MCMC via new adaptive diagnostics

Adaptive Markov Chain Monte Carlo (MCMC) algorithms attempt to ‘learn’ from the results of past iterations so the Markov chain can converge quicker. Unfortunately, adaptive MCMC algorithms are no longer Markovian, so their convergence is difficult to guarantee. In this paper, we develop new diagnostics to determine whether the adaption is still improving the convergence. We present an algorithm which automatically stops adapting once it determines further adaption will not increase the convergence speed. Our algorithm allows the computer to tune a ‘good’ Markov chain through multiple phases of adaption, and then run conventional non-adaptive MCMC. In this way, the efficiency gains of adaptive MCMC can be obtained while still ensuring convergence to the target distribution.     

An Adaptive Parallel Tempering Algorithm

Parallel tempering is a generic Markov chain Monte Carlo sampling method which allows good mixing with multimodal target distributions, where conventional Metropolis-Hastings algorithms often fail. The mixing properties of the sampler depend strongly on the choice of tuning parameters, such as the temperature schedule and the proposal distribution used for local exploration. We propose an adaptive algorithm with fixed number of temperatures which tunes both the temperature schedule and the parameters of the random-walk Metropolis kernel automatically. We prove the convergence of the adaptation and a strong law of large numbers for the algorithm under general conditions. We also prove as a side result the geometric ergodicity of the parallel tempering algorithm. We illustrate the performance of our method with examples. Our empirical findings indicate that the algorithm can cope well with different kinds of scenarios without prior tuning. Supplementary materials including the proofs and the Matlab implementation are available online.     

Parameter estimation in stochastic differential equations with Markov chain Monte Carlo and non-linear Kalman filtering

This paper is concerned with parameter estimation in linear and non-linear Itô type stochastic differential equations using Markov chain Monte Carlo (MCMC) methods. The MCMC methods studied in this paper are the Metropolis–Hastings and Hamiltonian Monte Carlo (HMC) algorithms. In these kind of models, the computation of the energy function gradient needed by HMC and gradient based optimization methods is non-trivial, and here we show how the gradient can be computed with a linear or non-linear Kalman filter-like recursion. We shall also show how in the linear case the differential equations in the gradient recursion equations can be solved using the matrix fraction decomposition. Numerical results for simulated examples are presented and discussed in detail.     

Componentwise adaptation for high dimensional MCMC

Summary  We introduce a new adaptive MCMC algorithm, based on the traditional single component Metropolis-Hastings algorithm and on our earlier adaptive Metropolis algorithm (AM). In the new algorithm the adaption is performed component by component. The chain is no more Markovian, but it remains ergodic. The algorithm is demonstrated to work well in varying test cases up to 1000 dimensions.     

Efficient MCMC estimation of inflated beta regression models

This paper introduces a new and computationally efficient Markov chain Monte Carlo (MCMC) estimation algorithm for the Bayesian analysis of zero, one, and zero and one inflated beta regression models. The algorithm is computationally efficient in the sense that it has low MCMC autocorrelations and computational time. A simulation study shows that the proposed algorithm outperforms the slice sampling and random walk Metropolis–Hastings algorithms in both small and large sample settings. An empirical illustration on a loss given default banking model demonstrates the usefulness of the proposed algorithm.     

Improving the Graphical Lasso Estimation for the Precision Matrix Through Roots of the Sample Covariance Matrix

In this article, we focus on the estimation of a high-dimensional inverse covariance (i.e., precision) matrix. We propose a simple improvement of the graphical Lasso (glasso) framework that is able to attain better statistical performance without increasing significantly the computational cost. The proposed improvement is based on computing a root of the sample covariance matrix to reduce the spread of the associated eigenvalues. Through extensive numerical results, using both simulated and real datasets, we show that the proposed modification improves the glasso procedure. Our results reveal that the square-root improvement can be a reasonable choice in practice. Supplementary material for this article is available online.     

A Sparse Singular Value Decomposition Method for High-Dimensional Data

We present a new computational approach to approximating a large, noisy data table by a low-rank matrix with sparse singular vectors. The approximation is obtained from thresholded subspace iterations that produce the singular vectors simultaneously, rather than successively as in competing proposals. We introduce novel ways to estimate thresholding parameters, which obviate the need for computationally expensive cross-validation. We also introduce a way to sparsely initialize the algorithm for computational savings that allow our algorithm to outperform the vanilla singular value decomposition (SVD) on the full data table when the signal is sparse. A comparison with two existing sparse SVD methods suggests that our algorithm is computationally always faster and statistically always at least comparable to the better of the two competing algorithms. Supplementary materials for the article are available in an online appendix. An R package ssvd implementing the algorithms introduced in this article is available on CRAN.     

Efficient Covariance Approximations for Large Sparse Precision Matrices

The use of sparse precision (inverse covariance) matrices has become popular because they allow for efficient algorithms for joint inference in high-dimensional models. Many applications require the computation of certain elements of the covariance matrix, such as the marginal variances, which may be nontrivial to obtain when the dimension is large. This article introduces a fast Rao–Blackwellized Monte Carlo sampling-based method for efficiently approximating selected elements of the covariance matrix. The variance and confidence bounds of the approximations can be precisely estimated without additional computational costs. Furthermore, a method that iterates over subdomains is introduced, and is shown to additionally reduce the approximation errors to practically negligible levels in an application on functional magnetic resonance imaging data. Both methods have low memory requirements, which is typically the bottleneck for competing direct methods.     

Automated Factor Slice Sampling

Markov chain Monte Carlo (MCMC) algorithms offer a very general approach for sampling from arbitrary distributions. However, designing and tuning MCMC algorithms for each new distribution can be challenging and time consuming. It is particularly difficult to create an efficient sampler when there is strong dependence among the variables in a multivariate distribution. We describe a two-pronged approach for constructing efficient, automated MCMC algorithms: (1) we propose the “factor slice sampler,” a generalization of the univariate slice sampler where we treat the selection of a coordinate basis (factors) as an additional tuning parameter, and (2) we develop an approach for automatically selecting tuning parameters to construct an efficient factor slice sampler. In addition to automating the factor slice sampler, our tuning approach also applies to the standard univariate slice samplers. We demonstrate the efficiency and general applicability of our automated MCMC algorithm with a number of illustrative examples. This article has online supplementary materials.     

A GREEDY ALGORITHM FOR SPARSE PRECISION MATRIX APPROXIMATION

Precision matrix estimation is an important problem in statistical data analysis.This paper proposes a sparse precision matrix estimation approach,based on CLIME estimator and an efficient algorithm GISSρ that was originally proposed for l1 sparse signal recov-ery in compressed sensing.The asymptotic convergence rate for sparse precision matrix estimation is analyzed with respect to the new stopping criteria of the proposed GISSρ algorithm.Finally,numerical comparison of GISSρ with other sparse recovery algorithms,such as ADMM and HTP in three settings of precision matrix estimation is provided and the numerical results show the advantages of the proposed algorithm.     

On the Relationship Between Markov chain Monte Carlo Methods for Model Uncertainty

This article considers Markov chain computational methods for incorporating uncertainty about the dimension of a parameter when performing inference within a Bayesian setting. A general class of methods is proposed for performing such computations, based upon a product space representation of the problem which is similar to that of Carlin and Chib. It is shown that all of the existing algorithms for incorporation of model uncertainty into Markov chain Monte Carlo (MCMC) can be derived as special cases of this general class of methods. In particular, we show that the popular reversible jump method is obtained when a special form of Metropolis–Hastings (M–H) algorithm is applied to the product space. Furthermore, the Gibbs sampling method and the variable selection method are shown to derive straightforwardly from the general framework. We believe that these new relationships between methods, which were until now seen as diverse procedures, are an important aid to the understanding of MCMC model selection procedures and may assist in the future development of improved procedures. Our discussion also sheds some light upon the important issues of “pseudo-prior” selection in the case of the Carlin and Chib sampler and choice of proposal distribution in the case of reversible jump. Finally, we propose efficient reversible jump proposal schemes that take advantage of any analytic structure that may be present in the model. These proposal schemes are compared with a standard reversible jump scheme for the problem of model order uncertainty in autoregressive time series, demonstrating the improvements which can be achieved through careful choice of proposals.