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In Bayesian analysis, the Markov Chain Monte Carlo (MCMC) algorithm is an efficient and simple method to compute posteriors. However, the chain may appear to converge while the posterior is improper, which will leads to incorrect statistical inferences. In this paper, we focus on the necessary and sufficient conditions for which improper hierarchical priors can yield proper posteriors in a multivariate linear model. In addition, we carry out a simulation study to illustrate the theoretical results, in which the Gibbs sampling and Metropolis-Hasting sampling are employed to generate the posteriors.     

Tensor-on-Tensor Regression

I propose a framework for the linear prediction of a multiway array (i.e., a tensor) from another multiway array of arbitrary dimension, using the contracted tensor product. This framework generalizes several existing approaches, including methods to predict a scalar outcome from a tensor, a matrix from a matrix, or a tensor from a scalar. I describe an approach that exploits the multiway structure of both the predictors and the outcomes by restricting the coefficients to have reduced PARAFAC/CANDECOMP rank. I propose a general and efficient algorithm for penalized least-squares estimation, which allows for a ridge (L₂) penalty on the coefficients. The objective is shown to give the mode of a Bayesian posterior, which motivates a Gibbs sampling algorithm for inference. I illustrate the approach with an application to facial image data. An R package is available at https://github.com/lockEF/MultiwayRegression.     

Efficient algorithms for generating truncated multivariate normal distributions

Sampling from a truncated multivariate normal distribution (TMVND) constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient methods, an iterative data augmentation (DA) algorithm and a non-iterative inverse Bayes formulae (IBF) sampler, to simulate TMVND and generalize them to multivariate normal distributions with linear inequality constraints. By creating a Bayesian incomplete-data structure, the posterior step of the DA algorithm directly generates random vector draws as opposed to single element draws, resulting obvious computational advantage and easy coding with common statistical software packages such as S-PLUS, MATLAB and GAUSS. Furthermore, the DA provides a ready structure for implementing a fast EM algorithm to identify the mode of TMVND, which has many potential applications in statistical inference of constrained parameter problems. In addition, utilizing this mode as an intermediate result, the IBF sampling provides a novel alternative to Gibbs sampling and eliminates problems with convergence and possible slow convergence due to the high correlation between components of a TMVND. The DA algorithm is applied to a linear regression model with constrained parameters and is illustrated with a published data set. Numerical comparisons show that the proposed DA algorithm and IBF sampler are more efficient than the Gibbs sampler and the accept-reject algorithm.     

Bayesian analysis of Birnbaum–Saunders distribution via the generalized ratio-of-uniforms method

This paper deals with the Bayesian inference for the parameters of the Birnbaum–Saunders distribution. We adopt the inverse-gamma priors for the shape and scale parameters because the continuous conjugate joint prior distribution does not exist and the reference prior (or independent Jeffreys’ prior) results in an improper posterior distribution. We propose an efficient sampling algorithm via the generalized ratio-of-uniforms method to compute the Bayesian estimates and the credible intervals. One appealing advantage of the proposed procedure over other sampling techniques is that it efficiently generates independent samples from the required posterior distribution. Simulation studies are conducted to investigate the behavior of the proposed method, and two real-data applications are analyzed for illustrative purposes.     

Discrete Gabor transforms: The Gabor-Gram matrix approach

The fundamental problem ofdiscrete Gabor transforms is to compute a set ofGabor coefficients in efficient ways. Recent study on the subject is an indirect approach: in order to compute the Gabor coefficients, one needs to find an auxiliary bi-orthogonal window function γ. We are seeking a direct approach in this paper. We introduce concepts ofGabor-Gram matrices and investigate their structural properties. We propose iterative methods to compute theGabor coefficients. Simple solutions for critical sampling, certain oversampling, and undersampling cases are developed. Acknowledgements and Notes. The author was with University of Connecticut, Storrs, CT 06269-3009.     

Evolutionary Monte Carlo Methods for Clustering

The problem of clustering a group of observations according to some objective function (e.g., K-means clustering, variable selection) or a density (e.g., posterior from a Dirichlet process mixture model prior) can be cast in the framework of Monte Carlo sampling for cluster indicators. We propose a new method called the evolutionary Monte Carlo clustering (EMCC) algorithm, in which three new “crossover moves,” based on swapping and reshuffling sub cluster intersections, are proposed. We apply the EMCC algorithm to several clustering problems including Bernoulli clustering, biological sequence motif clustering, BIC based variable selection, and mixture of normals clustering. We compare EMCC's performance both as a sampler and as a stochastic optimizer with Gibbs sampling, “split-merge” Metropolis–Hastings algorithms, K-means clustering, and the MCLUST algorithm.     

An Efficient Algorithm for Bayesian Nearest Neighbours

K-Nearest Neighbours (k-NN) is a popular classification and regression algorithm, yet one of its main limitations is the difficulty in choosing the number of neighbours. We present a Bayesian algorithm to compute the posterior probability distribution for k given a target point within a data-set, efficiently and without the use of Markov Chain Monte Carlo (MCMC) methods or simulation—alongside an exact solution for distributions within the exponential family. The central idea is that data points around our target are generated by the same probability distribution, extending outwards over the appropriate, though unknown, number of neighbours. Once the data is projected onto a distance metric of choice, we can transform the choice of k into a change-point detection problem, for which there is an efficient solution: we recursively compute the probability of the last change-point as we move towards our target, and thus de facto compute the posterior probability distribution over k. Applying this approach to both a classification and a regression UCI data-sets, we compare favourably and, most importantly, by removing the need for simulation, we are able to compute the posterior probability of k exactly and rapidly. As an example, the computational time for the Ripley data-set is a few milliseconds compared to a few hours when using a MCMC approach.

     

Functional Compatibility,Markov Chains,and Gibbs Sampling with Improper Posteriors

Abstract

The members of a set of conditional probability density functions are called compatible if there exists a joint probability density function that generates them. We generalize this concept by calling the conditionals functionally compatible if there exists a non-negative function that behaves like a joint density as far as generating the conditionals according to the probability calculus, but whose integral over the whole space is not necessarily finite. A necessary and sufficient condition for functional compatibility is given that provides a method of calculating this function, if it exists. A Markov transition function is then constructed using a set of functionally compatible conditional densities and it is shown, using the compatibility results, that the associated Markov chain is positive recurrent if and only if the conditionals are compatible. A Gibbs Markov chain, constructed via “Gibbs conditionals” from a hierarchical model with an improper posterior, is a special case. Therefore, the results of this article can be used to evaluate the consequences of applying the Gibbs sampler when the posterior's impropriety is unknown to the user. Our results cannot, however, be used to detect improper posteriors. Monte Carlo approximations based on Gibbs chains are shown to have undesirable limiting behavior when the posterior is improper. The results are applied to a Bayesian hierarchical one-way random effects model with an improper posterior distribution. The model is simple, but also quite similar to some models with improper posteriors that have been used in conjunction with the Gibbs sampler in the literature.     

Influence measures and robust estimators of dependence in multivariate extremes

We develop a simple influence measure to assess whether Bayesian estimators in multivariate extreme value problems are sensitive to outliers. The proposed measure is easy to compute by importance sampling and successfully captures two effects on the functional: the “data effect” and the “parameter uncertainty effect”. We also propose a new Bayesian estimator which is easy to implement and is robust. The methods are tested and illustrated using simulated data and then applied to stock market data.     

Efficient Bayesian Inference for Multivariate Probit Models With Sparse Inverse Correlation Matrices

We propose a Bayesian approach for inference in the multivariate probit model, taking into account the association structure between binary observations. We model the association through the correlation matrix of the latent Gaussian variables. Conditional independence is imposed by setting some off-diagonal elements of the inverse correlation matrix to zero and this sparsity structure is modeled using a decomposable graphical model. We propose an efficient Markov chain Monte Carlo algorithm relying on a parameter expansion scheme to sample from the resulting posterior distribution. This algorithm updates the correlation matrix within a simple Gibbs sampling framework and allows us to infer the correlation structure from the data, generalizing methods used for inference in decomposable Gaussian graphical models to multivariate binary observations. We demonstrate the performance of this model and of the Markov chain Monte Carlo algorithm on simulated and real datasets. This article has online supplementary materials.     

A note on some algorithms for the Gibbs posterior

Jiang and Tanner (2008) consider a method of classification using the Gibbs posterior which is directly constructed from the empirical classification errors. They propose an algorithm to sample from the Gibbs posterior which utilizes a smoothed approximation of the empirical classification error, via a Gibbs sampler with augmented latent variables. In this paper, we note some drawbacks of this algorithm and propose an alternative method for sampling from the Gibbs posterior, based on the Metropolis algorithm. The numerical performance of the algorithms is examined and compared via simulated data. We find that the Metropolis algorithm produces good classification results at an improved speed of computation.     

Bayesian cylindrical data modeling using Abe–Ley mixtures

This paper proposes a Metropolis–Hastings algorithm based on Markov chain Monte Carlo sampling, to estimate the parameters of the Abe–Ley distribution, which is a recently proposed Weibull-Sine-Skewed-von Mises mixture model, for bivariate circular-linear data. Current literature estimates the parameters of these mixture models using the expectation-maximization method, but we will show that this exhibits a few shortcomings for the considered mixture model. First, standard expectation-maximization does not guarantee convergence to a global optimum, because the likelihood is multi-modal, which results from the high dimensionality of the mixture’s likelihood. Second, given that expectation-maximization provides point estimates of the parameters only, the uncertainties of the estimates (e.g., confidence intervals) are not directly available in these methods. Hence, extra calculations are needed to quantify such uncertainty. We propose a Metropolis–Hastings based algorithm that avoids both shortcomings of expectation-maximization. Indeed, Metropolis–Hastings provides an approximation to the complete (posterior) distribution, given that it samples from the joint posterior of the mixture parameters. This facilitates direct inference (e.g., about uncertainty, multi-modality) from the estimation. In developing the algorithm, we tackle various challenges including convergence speed, label switching and selecting the optimum number of mixture components. We then (i) verify the effectiveness of the proposed algorithm on sample datasets with known true parameters, and further (ii) validate our methodology on an environmental dataset (a traditional application domain of Abe–Ley mixtures where measurements are function of direction). Finally, we (iii) demonstrate the usefulness of our approach in an application domain where the circular measurement is periodic in time.     

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.     

Laplace Expansions in Markov chain Monte Carlo Algorithms

Complex hierarchical models lead to a complicated likelihood and then, in a Bayesian analysis, to complicated posterior distributions. To obtain Bayes estimates such as the posterior mean or Bayesian confidence regions, it is therefore necessary to simulate the posterior distribution using a method such as an MCMC algorithm. These algorithms often get slower as the number of observations increases, especially when the latent variables are considered. To improve the convergence of the algorithm, we propose to decrease the number of parameters to simulate at each iteration by using a Laplace approximation on the nuisance parameters. We provide a theoretical study of the impact that such an approximation has on the target posterior distribution. We prove that the distance between the true target distribution and the approximation becomes of order O(N^?a) with a ∈ (0, 1), a close to 1, as the number of observations N increases. A simulation study illustrates the theoretical results. The approximated MCMC algorithm behaves extremely well on an example which is driven by a study on HIV patients.     

Computing Normalizing Constants for Finite Mixture Models via Incremental Mixture Importance Sampling (IMIS)

This article proposes a method for approximating integrated likelihoods in finite mixture models. We formulate the model in terms of the unobserved group memberships, z, and make them the variables of integration. The integral is then evaluated using importance sampling over the z. We propose an adaptive importance sampling function which is itself a mixture, with two types of component distributions, one concentrated and one diffuse. The more concentrated type of component serves the usual purpose of an importance sampling function, sampling mostly group assignments of high posterior probability. The less concentrated type of component allows for the importance sampling function to explore the space in a controlled way to find other, unvisited assignments with high posterior probability. Components are added adaptively, one at a time, to cover areas of high posterior probability not well covered by the current importance sampling function. The method is called incremental mixture importance sampling (IMIS).

IMIS is easy to implement and to monitor for convergence. It scales easily for higher dimensional mixture distributions when a conjugate prior is specified for the mixture parameters. The simulated values on which the estimate is based are independent, which allows for straightforward estimation of standard errors. The self-monitoring aspects of the method make it easier to adjust tuning parameters in the course of estimation than standard Markov chain Monte Carlo algorithms. With only small modifications to the code, one can use the method for a wide variety of mixture distributions of different dimensions. The method performed well in simulations and in mixture problems in astronomy and medical research.     

Properties of Prior and Posterior Distributions for Multivariate Categorical Response Data Models

In this article, we model multivariate categorical (binary and ordinal) response data using a very rich class of scale mixture of multivariate normal (SMMVN) link functions to accommodate heavy tailed distributions. We consider both noninformative as well as informative prior distributions for SMMVN-link models. The notation of informative prior elicitation is based on available similar historical studies. The main objectives of this article are (i) to derive theoretical properties of noninformative and informative priors as well as the resulting posteriors and (ii) to develop an efficient Markov chain Monte Carlo algorithm to sample from the resulting posterior distribution. A real data example from prostate cancer studies is used to illustrate the proposed methodologies.     

An approach for the Steiner problem in directed graphs

We present a scheme to solve the Steiner problem in directed graphs using a heuristic method to obtain upper bounds and thek shortest arborescences algorithm to compute lower bounds. We propose to combine these ideas in an enumerative algorithm. Computational results are presented for both thek shortest arborescences algorithm and the heuristic method, including reduction tests for the problem.This work was partially supported by CNP_q, FINEP, CAPES and IBM do Brasil.     

Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

This paper studies the matrix completion problem under arbitrary sampling schemes. We propose a new estimator incorporating both max-norm and nuclear-norm regularization, based on which we can conduct efficient low-rank matrix recovery using a random subset of entries observed with additive noise under general non-uniform and unknown sampling distributions. This method significantly relaxes the uniform sampling assumption imposed for the widely used nuclear-norm penalized approach, and makes low-rank matrix recovery feasible in more practical settings. Theoretically, we prove that the proposed estimator achieves fast rates of convergence under different settings. Computationally, we propose an alternating direction method of multipliers algorithm to efficiently compute the estimator, which bridges a gap between theory and practice of machine learning methods with max-norm regularization. Further, we provide thorough numerical studies to evaluate the proposed method using both simulated and real datasets.     

Post-Processing Accept-Reject Samples: Recycling and Rescaling

Abstract

This article proposes alternative methods for constructing estimators from accept-reject samples by incorporating the variables rejected by the algorithm. The resulting estimators are quick to compute, and turn out to be variations of importance sampling estimators, although their derivations are quite different. We show that these estimators are superior asymptotically to the classical accept-reject estimator, which ignores the rejected variables. In addition, we consider the issue of rescaling of estimators, a topic that has implications beyond accept-reject and importance sampling. We show how rescaling can improve an estimator and illustrate the domination of the standard importance sampling techniques in different setups.     

Uncertainty Quantification for Modern High-Dimensional Regression via Scalable Bayesian Methods

Tremendous progress has been made in the last two decades in the area of high-dimensional regression, especially in the “large p, small n” setting. Such sample starved settings inevitably lead to models which are potentially very unstable and hence quite unreliable. To this end, Bayesian shrinkage methods have generated a lot of recent interest in the modern high-dimensional regression and model selection context. Such methods span the wide spectrum of modern regression approaches and include among others, spike-and-slab priors, the Bayesian lasso, ridge regression, and global-local shrinkage priors such as the Horseshoe prior and the Dirichlet–Laplace prior. These methods naturally facilitate tractable uncertainty quantification and have thus been used extensively across diverse applications. A common unifying feature of these models is that the corresponding priors on the regression coefficients can be expressed as a scale mixture of normals. This property has been leveraged extensively to develop various three-step Gibbs samplers to explore the corresponding intractable posteriors. The convergence of such samplers however is very slow in high dimensions settings, making them disconnected to the very setting that they are intended to work in. To address this challenge, we propose a comprehensive and unifying framework to draw from the same family of posteriors via a class of tractable and scalable two-step blocked Gibbs samplers. We demonstrate that our proposed class of two-step blocked samplers exhibits vastly superior convergence behavior compared to the original three-step sampler in high-dimensional regimes on simulated data as well as data from a variety of applications including gene expression data, infrared spectroscopy data, and socio-economic/law enforcement data. We also provide a detailed theoretical underpinning to the new method by deriving explicit upper bounds for the (geometric) rate of convergence, and by proving that the proposed two-step sampler has superior spectral properties. Supplementary material for this article is available online.