Bayesian Inference on Multivariate Normal Covariance and Precision Matrices in a Star-Shaped Model with Missing Data

In this article, we study Bayesian estimation for the covariance matrix Σ and the precision matrix Ω (the inverse of the covariance matrix) in the star-shaped model with missing data. Based on a Cholesky-type decomposition of the precision matrix Ω = Ψ′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements, we develop the Jeffreys prior and a reference prior for Ψ. We then introduce a class of priors for Ψ, which includes the invariant Haar measures, Jeffreys prior, and reference prior. The posterior properties are discussed and the closed-form expressions for Bayesian estimators for the covariance matrix Σ and the precision matrix Ω are derived under the Stein loss, entropy loss, and symmetric loss. Some simulation results are given for illustration.     

Bayesian model-averaged regularization for Gaussian graphical models

The graphical lasso has now become a useful tool to estimate high-dimensional Gaussian graphical models, but its practical applications suffer from the problem of choosing regularization parameters in a data-dependent way. In this article, we propose a model-averaged method for estimating sparse inverse covariance matrices for Gaussian graphical models. We consider the graphical lasso regularization path as the model space for Bayesian model averaging and use Markov chain Monte Carlo techniques for the regularization path point selection. Numerical performance of our method is investigated using both simulated and real datasets, in comparison with some state-of-art model selection procedures.     

Structural learning with time-varying components: tracking the cross-section of financial time series

Summary.  When modelling multivariate financial data, the problem of structural learning is compounded by the fact that the covariance structure changes with time. Previous work has focused on modelling those changes by using multivariate stochastic volatility models. We present an alternative to these models that focuses instead on the latent graphical structure that is related to the precision matrix. We develop a graphical model for sequences of Gaussian random vectors when changes in the underlying graph occur at random times, and a new block of data is created with the addition or deletion of an edge. We show how a Bayesian hierarchical model incorporates both the uncertainty about that graph and the time variation thereof.     

Singular Gaussian graphical models: Structure learning

ABSTRACT

The goal of this article is to introduce singular Gaussian graphical models and their conditional independence properties. In fact, we extend the concept of Gaussian Markov Random Field to the case of a multivariate normally distributed vector with a singular covariance matrix. We construct, then, the associated graph’s structure from the covariance matrix’s pseudo-inverse on the basis of a characterization of the pairwise conditional independence. The proposed approach can also be used when the covariance matrix is ill-conditioned, through projecting data on a smaller subspace. In this case, our method ensures numerical stability and consistency of the constructed graph and significantly reduces the inference problem’s complexity. These aspects are illustrated using numerical experiments.     

Shared frailty models with baseline generalized Pareto distribution

Abstract

In this article, we have considered three different shared frailty models under the assumption of generalized Pareto Distribution as baseline distribution. Frailty models have been used in the survival analysis to account for the unobserved heterogeneity in an individual risks to disease and death. These three frailty models are with gamma frailty, inverse Gaussian frailty and positive stable frailty. Then we introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters. We applied these three models to a kidney infection data and find the best fitted model for kidney infection data. We present a simulation study to compare true value of the parameters with the estimated values. Model comparison is made using Bayesian model selection criterion and a well-fitted model is suggested for the kidney infection data.     

Spatial sampling design for parameter estimation of the covariance function

We study the spatial optimal sampling design for covariance parameter estimation. The spatial process is modeled as a Gaussian random field and maximum likelihood (ML) is used to estimate the covariance parameters. We use the log determinant of the inverse Fisher information matrix as the design criterion and run simulations to investigate the relationship between the inverse Fisher information matrix and the covariance matrix of the ML estimates. A simulated annealing algorithm is developed to search for an optimal design among all possible designs on a fine grid. Since the design criterion depends on the unknown parameters, we define relative efficiency of a design and consider minimax and Bayesian criteria to find designs that are robust for a range of parameter values. Simulation results are presented for the Matérn class of covariance functions.     

Large Bayesian VARs: A Flexible Kronecker Error Covariance Structure

ABSTRACT

We introduce a class of large Bayesian vector autoregressions (BVARs) that allows for non-Gaussian, heteroscedastic, and serially dependent innovations. To make estimation computationally tractable, we exploit a certain Kronecker structure of the likelihood implied by this class of models. We propose a unified approach for estimating these models using Markov chain Monte Carlo (MCMC) methods. In an application that involves 20 macroeconomic variables, we find that these BVARs with more flexible covariance structures outperform the standard variant with independent, homoscedastic Gaussian innovations in both in-sample model-fit and out-of-sample forecast performance.     

Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

Abstract. We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor (BF), requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper‐parameter, which can be set to its minimal value. We show that our approach produces genuine BFs. The implied prior on the concentration matrix of any complete graph is a data‐dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models and show that in this case they coincide with those recently obtained using limiting versions of hyper‐inverse Wishart distributions as priors on the graph‐constrained covariance matrices.     

Fisher information and maximum-likelihood estimation of covariance parameters in Gaussian stochastic processes

The inverse of the Fisher information matrix is commonly used as an approximation for the covariance matrix of maximum-likelihood estimators. We show via three examples that for the covariance parameters of Gaussian stochastic processes under infill asymptotics, the covariance matrix of the limiting distribution of their maximum-likelihood estimators equals the limit of the inverse information matrix. This is either proven analytically or justified by simulation. Furthermore, the limiting behaviour of the trace of the inverse information matrix indicates equivalence or orthogonality of the underlying Gaussian measures. Even in the case of singularity, the estimator of the process variance is seen to be unbiased, and also its variability is approximated accurately from the information matrix.     

Strategies for Fitting Large,Geostatistical Data in MCMC Simulation

Models for geostatistical data introduce spatial dependence in the covariance matrix of location-specific random effects. This is usually defined to be a parametric function of the distances between locations. Bayesian formulations of such models overcome asymptotic inference and estimation problems involved in maximum likelihood-based approaches and can be fitted using Markov chain Monte Carlo (MCMC) simulation. The MCMC implementation, however, requires repeated inversions of the covariance matrix which makes the problem computationally intensive, especially for large number of locations. In the present work, we propose to convert the spatial covariance matrix to a sparse matrix and compare a number of numerical algorithms especially suited within the MCMC framework in order to accelerate large matrix inversion. The algorithms are assessed empirically on simulated datasets of different size and sparsity. We conclude that the band solver applied after ordering the distance matrix reduces the computational time in inverting covariance matrices substantially.     

Edge selection based on the geometry of dually flat spaces for Gaussian graphical models

We propose a method for selecting edges in undirected Gaussian graphical models. Our algorithm takes after our previous work, an extension of Least Angle Regression (LARS), and it is based on the information geometry of dually flat spaces. Non-diagonal elements of the inverse of the covariance matrix, the concentration matrix, play an important role in edge selection. Our iterative method estimates these elements and selects covariance models simultaneously. A sequence of pairs of estimates of the concentration matrix and an independence graph is generated, whose length is the same as the number of non-diagonal elements of the matrix. In our algorithm, the next estimate of the graph is the nearest graph to the latest estimate of the concentration matrix. The next estimate of the concentration matrix is not just the projection of the latest estimate, and it is shrunk to the origin. We describe the algorithm and show results for some datasets. Furthermore, we give some remarks on model identification and prediction.     

Investigating the sensitivity of Gaussian processes to the choice of their correlation function and prior specifications

A Gaussian process (GP) can be thought of as an infinite collection of random variables with the property that any subset, say of dimension n, of these variables have a multivariate normal distribution of dimension n, mean vector β and covariance matrix Σ [O'Hagan, A., 1994, Kendall's Advanced Theory of Statistics, Vol. 2B, Bayesian Inference (John Wiley & Sons, Inc.)]. The elements of the covariance matrix are routinely specified through the multiplication of a common variance by a correlation function. It is important to use a correlation function that provides a valid covariance matrix (positive definite). Further, it is well known that the smoothness of a GP is directly related to the specification of its correlation function. Also, from a Bayesian point of view, a prior distribution must be assigned to the unknowns of the model. Therefore, when using a GP to model a phenomenon, the researcher faces two challenges: the need of specifying a correlation function and a prior distribution for its parameters. In the literature there are many classes of correlation functions which provide a valid covariance structure. Also, there are many suggestions of prior distributions to be used for the parameters involved in these functions. We aim to investigate how sensitive the GPs are to the (sometimes arbitrary) choices of their correlation functions. For this, we have simulated 25 sets of data each of size 64 over the square [0, 5]×[0, 5] with a specific correlation function and fixed values of the GP's parameters. We then fit different correlation structures to these data, with different prior specifications and check the performance of the adjusted models using different model comparison criteria.     

Shrinkage tuning parameter selection in precision matrices estimation

Recent literature provides many computational and modeling approaches for covariance matrices estimation in a penalized Gaussian graphical models but relatively little study has been carried out on the choice of the tuning parameter. This paper tries to fill this gap by focusing on the problem of shrinkage parameter selection when estimating sparse precision matrices using the penalized likelihood approach. Previous approaches typically used K-fold cross-validation in this regard. In this paper, we first derived the generalized approximate cross-validation for tuning parameter selection which is not only a more computationally efficient alternative, but also achieves smaller error rate for model fitting compared to leave-one-out cross-validation. For consistency in the selection of nonzero entries in the precision matrix, we employ a Bayesian information criterion which provably can identify the nonzero conditional correlations in the Gaussian model. Our simulations demonstrate the general superiority of the two proposed selectors in comparison with leave-one-out cross-validation, 10-fold cross-validation and Akaike information criterion.     

Mixtures of modified t-factor analyzers for model-based clustering,classification, and discriminant analysis

A novel family of mixture models is introduced based on modified t-factor analyzers. Modified factor analyzers were recently introduced within the Gaussian context and our work presents a more flexible and robust alternative. We introduce a family of mixtures of modified t-factor analyzers that uses this generalized version of the factor analysis covariance structure. We apply this family within three paradigms: model-based clustering; model-based classification; and model-based discriminant analysis. In addition, we apply the recently published Gaussian analogue to this family under the model-based classification and discriminant analysis paradigms for the first time. Parameter estimation is carried out within the alternating expectation-conditional maximization framework and the Bayesian information criterion is used for model selection. Two real data sets are used to compare our approach to other popular model-based approaches; in these comparisons, the chosen mixtures of modified t-factor analyzers model performs favourably. We conclude with a summary and suggestions for future work.     

New HEAVY Models for Fat-Tailed Realized Covariances and Returns

ABSTRACT

We develop a new score-driven model for the joint dynamics of fat-tailed realized covariance matrix observations and daily returns. The score dynamics for the unobserved true covariance matrix are robust to outliers and incidental large observations in both types of data by assuming a matrix-F distribution for the realized covariance measures and a multivariate Student's t distribution for the daily returns. The filter for the unknown covariance matrix has a computationally efficient matrix formulation, which proves beneficial for estimation and simulation purposes. We formulate parameter restrictions for stationarity and positive definiteness. Our simulation study shows that the new model is able to deal with high-dimensional settings (50 or more) and captures unobserved volatility dynamics even if the model is misspecified. We provide an empirical application to daily equity returns and realized covariance matrices up to 30 dimensions. The model statistically and economically outperforms competing multivariate volatility models out-of-sample. Supplementary materials for this article are available online.     

Standard errors for the parameters of graphical Gaussian models

The comparison of an estimated parameter to its standard error, the Wald test, is a well known procedure of classical statistics. Here we discuss its application to graphical Gaussian model selection. First we derive the Fisher information matrix and its inverse about the parameters of any graphical Gaussian model. Both the covariance matrix and its inverse are considered and a comparative analysis of the asymptotic behaviour of their maximum likelihood estimators (m.l.e.s) is carried out. Then we give an example of model selection based on the standard errors. The method is shown to produce almost identical inference to likelihood ratio methods in the example considered.     

Robust Bayesian analysis for parallel system with masked data under inverse Weibull lifetime distribution

Abstract

This paper considers the statistical analysis of masked data in a parallel system with inverse Weibull distributed components under type II censoring. Based on Gamma conjugate prior, the Bayesian estimation as well as the hierarchical Bayesian estimation for the parameters and the reliability function of system are obtained by using the Bayesian theory and the hierarchical Bayesian method. Finally, Monte Carlo simulations are provided to compare the performances of the estimates under different masking probabilities and effective sample sizes.     

Posterior consistency of logistic Gaussian process priors in density estimation

We establish weak and strong posterior consistency of Gaussian process priors studied by Lenk [1988. The logistic normal distribution for Bayesian, nonparametric, predictive densities. J. Amer. Statist. Assoc. 83 (402), 509–516] for density estimation. Weak consistency is related to the support of a Gaussian process in the sup-norm topology which is explicitly identified for many covariance kernels. In fact we show that this support is the space of all continuous functions when the usual covariance kernels are chosen and an appropriate prior is used on the smoothing parameters of the covariance kernel. We then show that a large class of Gaussian process priors achieve weak as well as strong posterior consistency (under some regularity conditions) at true densities that are either continuous or piecewise continuous.     

Poisson-Driven Stationary Markov Models

ABSTRACT

We propose a simple yet powerful method to construct strictly stationary Markovian models with given but arbitrary invariant distributions. The idea is based on a Poisson-type transform modulating the dependence structure in the model. An appealing feature of our approach is the possibility to control the underlying transition probabilities and, therefore, incorporate them within standard estimation methods. Given the resulting representation of the transition density, a Gibbs sampler algorithm based on the slice method is proposed and implemented. In the discrete-time case, special attention is placed to the class of generalized inverse Gaussian distributions. In the continuous case, we first provide a brief treatment of the class of gamma distributions, and then extend it to cover other invariant distributions, such as the generalized extreme value class. The proposed approach and estimation algorithm are illustrated with real financial datasets. Supplementary materials for this article are available online.     

Covariance structure approximation via gLasso in high-dimensional supervised classification

Recent work has shown that the Lasso-based regularization is very useful for estimating the high-dimensional inverse covariance matrix. A particularly useful scheme is based on penalizing the ?₁ norm of the off-diagonal elements to encourage sparsity. We embed this type of regularization into high-dimensional classification. A two-stage estimation procedure is proposed which first recovers structural zeros of the inverse covariance matrix and then enforces block sparsity by moving non-zeros closer to the main diagonal. We show that the block-diagonal approximation of the inverse covariance matrix leads to an additive classifier, and demonstrate that accounting for the structure can yield better performance accuracy. Effect of the block size on classification is explored, and a class of asymptotically equivalent structure approximations in a high-dimensional setting is specified. We suggest a variable selection at the block level and investigate properties of this procedure in growing dimension asymptotics. We present a consistency result on the feature selection procedure, establish asymptotic lower an upper bounds for the fraction of separative blocks and specify constraints under which the reliable classification with block-wise feature selection can be performed. The relevance and benefits of the proposed approach are illustrated on both simulated and real data.