Approximate reference priors for Gaussian random fields

Reference priors are theoretically attractive for the analysis of geostatistical data since they enable automatic Bayesian analysis and have desirable Bayesian and frequentist properties. But their use is hindered by computational hurdles that make their application in practice challenging. In this work, we derive a new class of default priors that approximate reference priors for the parameters of some Gaussian random fields. It is based on an approximation to the integrated likelihood of the covariance parameters derived from the spectral approximation of stationary random fields. This prior depends on the structure of the mean function and the spectral density of the model evaluated at a set of spectral points associated with an auxiliary regular grid. In addition to preserving the desirable Bayesian and frequentist properties, these approximate reference priors are more stable, and their computations are much less onerous than those of exact reference priors. Unlike exact reference priors, the marginal approximate reference prior of correlation parameter is always proper, regardless of the mean function or the smoothness of the correlation function. This property has important consequences for covariance model selection. An illustration comparing default Bayesian analyses is provided with a dataset of lead pollution in Galicia, Spain.     

Bayesian analysis of covariance under inverse Gaussian model

This paper considers the problem of analysis of covariance (ANCOVA) under the assumption of inverse Gaussian distribution for response variable from the Bayesian point of view. We develop a fully Bayesian model for ANCOVA based on the conjugate prior distributions for parameters contained in the model. The Bayes estimator of parameters, ANCOVA model and adjusted effects for both treatments and covariates along with predictive distribution of future observations are developed. We also provide the essentials for comparing adjusted treatments effects and adjusted factor effects. A simulation study and a real world application are also performed to illustrate and evaluate the proposed Bayesian model.     

A permutation-based Bayesian approach for inverse covariance estimation

Abstract

Covariance estimation and selection for multivariate datasets in a high-dimensional regime is a fundamental problem in modern statistics. Gaussian graphical models are a popular class of models used for this purpose. Current Bayesian methods for inverse covariance matrix estimation under Gaussian graphical models require the underlying graph and hence the ordering of variables to be known. However, in practice, such information on the true underlying model is often unavailable. We therefore propose a novel permutation-based Bayesian approach to tackle the unknown variable ordering issue. In particular, we utilize multiple maximum a posteriori estimates under the DAG-Wishart prior for each permutation, and subsequently construct the final estimate of the inverse covariance matrix. The proposed estimator has smaller variability and yields order-invariant property. We establish posterior convergence rates under mild assumptions and illustrate that our method outperforms existing approaches in estimating the inverse covariance matrices via simulation studies.     

Equivalence of Gaussian measures for some nonstationary random fields

Gaussian random fields whose covariance structures are described by a power law model provide a simple and flexible class of models for isotropic random fields. This class includes fractional Brownian fields as a special case. Because these random fields are nonstationary, the extensive results available on equivalence of Gaussian measures for stationary models do not apply to them. This work shows that results on equivalence for two stationary Gaussian random field models extend in a natural way to the equivalence of a stationary model and a power law model. This result is used to show that if we use a power law model for predicting a random field at unobserved locations when in fact the random field is stationary, we can obtain asymptotically optimal predictions as long as the high frequency behavior of the true spectral density is sufficiently close to the high frequency behavior of the spectral density of the power law model.     

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.     

Bayesian Analysis of Skew Gaussian Spatial Models Based on Censored Data

The customary approach to spatial data modeling in the presence of censored data, is to assume the underlying random field is Gaussian. However, in practice, we often faced data that the exploratory data analysis shows the skewness and consequently, it violates the normality assumption. In such setting, the skew Gaussian (SG) spatial model is used to overcome this issue. In this article, the SG model is fitted based on censored observations. For this purpose, we adopt the Bayesian approach and utilize the Markov chain Monte Carlo algorithms and data augmentations to carry out calculations. A numerical example illustrates the methodology.     

Histogram estimation of radon-nikodym derivatives for strong mixing data

Nonparametric inference for point processes is discussed by way of histograms, which provide a nice tool for the analysis of on-line data. The construction of histograms depends on a sequence of partitions, which we take tc be nonenibedded to allow partitions with sets of equal measure. This presents some theoretical problems, which are addressed with an assumption on the decomposition of second order moments. In another direction, we drop the usual independence assumption on the sample, replacing it by a strong mixing assumption. Under this setting, we study the convergence of the histogram in probability, which depends on approximation conditions between the distributions of random pairs and the product of their marginal distributions, and^almost completely, which is based on the decomposition of the second order moments. This last convergence is stated on two versions according to the assumption of Laplace transforms or the Cramer moment conditions. These are somewhat stronger, but enable us to recover the usual condition on the decrease rate of sets on each partition. In the final section we prove that the finite dimensional distributions converge in distribution to a Gaussian centered vector with a specified covariance.     

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.     

Learning a multivariate Gaussian mixture model with the reversible jump MCMC algorithm

This paper is a contribution to the methodology of fully Bayesian inference in a multivariate Gaussian mixture model using the reversible jump Markov chain Monte Carlo algorithm. To follow the constraints of preserving the first two moments before and after the split or combine moves, we concentrate on a simplified multivariate Gaussian mixture model, in which the covariance matrices of all components share a common eigenvector matrix. We then propose an approach to the construction of the reversible jump Markov chain Monte Carlo algorithm for this model. Experimental results on several data sets demonstrate the efficacy of our algorithm.     

Bayesian nonparametric estimation of a copula

A copula can fully characterize the dependence of multiple variables. The purpose of this paper is to provide a Bayesian nonparametric approach to the estimation of a copula, and we do this by mixing over a class of parametric copulas. In particular, we show that any bivariate copula density can be arbitrarily accurately approximated by an infinite mixture of Gaussian copula density functions. The model can be estimated by Markov Chain Monte Carlo methods and the model is demonstrated on both simulated and real data sets.     

Bayesian modelling of spatial compositional data

Compositional data are vectors of proportions, specifying fractions of a whole. Aitchison (1986) defines logistic normal distributions for compositional data by applying a logistic transformation and assuming the transformed data to be multi- normal distributed. In this paper we generalize this idea to spatially varying logistic data and thereby define logistic Gaussian fields. We consider the model in a Bayesian framework and discuss appropriate prior distributions. We consider both complete observations and observations of subcompositions or individual proportions, and discuss the resulting posterior distributions. In general, the posterior cannot be analytically handled, but the Gaussian base of the model allows us to define efficient Markov chain Monte Carlo algorithms. We use the model to analyse a data set of sediments in an Arctic lake. These data have previously been considered, but then without taking the spatial aspect into account.     

Robust Bayesian synthetic likelihood via a semi-parametric approach

Bayesian synthetic likelihood (BSL) is now a well-established method for performing approximate Bayesian parameter estimation for simulation-based models that do not possess a tractable likelihood function. BSL approximates an intractable likelihood function of a carefully chosen summary statistic at a parameter value with a multivariate normal distribution. The mean and covariance matrix of this normal distribution are estimated from independent simulations of the model. Due to the parametric assumption implicit in BSL, it can be preferred to its nonparametric competitor, approximate Bayesian computation, in certain applications where a high-dimensional summary statistic is of interest. However, despite several successful applications of BSL, its widespread use in scientific fields may be hindered by the strong normality assumption. In this paper, we develop a semi-parametric approach to relax this assumption to an extent and maintain the computational advantages of BSL without any additional tuning. We test our new method, semiBSL, on several challenging examples involving simulated and real data and demonstrate that semiBSL can be significantly more robust than BSL and another approach in the literature.     

Using conditional independence for parsimonious model-based Gaussian clustering

In the framework of model-based cluster analysis, finite mixtures of Gaussian components represent an important class of statistical models widely employed for dealing with quantitative variables. Within this class, we propose novel models in which constraints on the component-specific variance matrices allow us to define Gaussian parsimonious clustering models. Specifically, the proposed models are obtained by assuming that the variables can be partitioned into groups resulting to be conditionally independent within components, thus producing component-specific variance matrices with a block diagonal structure. This approach allows us to extend the methods for model-based cluster analysis and to make them more flexible and versatile. In this paper, Gaussian mixture models are studied under the above mentioned assumption. Identifiability conditions are proved and the model parameters are estimated through the maximum likelihood method by using the Expectation-Maximization algorithm. The Bayesian information criterion is proposed for selecting the partition of the variables into conditionally independent groups. The consistency of the use of this criterion is proved under regularity conditions. In order to examine and compare models with different partitions of the set of variables a hierarchical algorithm is suggested. A wide class of parsimonious Gaussian models is also presented by parameterizing the component-variance matrices according to their spectral decomposition. The effectiveness and usefulness of the proposed methodology are illustrated with two examples based on real datasets.     

Estimating the Spot Covariation of Asset Prices—Statistical Theory and Empirical Evidence

ABSTRACT

We propose a new estimator for the spot covariance matrix of a multi-dimensional continuous semimartingale log asset price process, which is subject to noise and nonsynchronous observations. The estimator is constructed based on a local average of block-wise parametric spectral covariance estimates. The latter originate from a local method of moments (LMM), which recently has been introduced by Bibinger et al.. We prove consistency and a point-wise stable central limit theorem for the proposed spot covariance estimator in a very general setup with stochastic volatility, leverage effects, and general noise distributions. Moreover, we extend the LMM estimator to be robust against autocorrelated noise and propose a method to adaptively infer the autocorrelations from the data. Based on simulations we provide empirical guidance on the effective implementation of the estimator and apply it to high-frequency data of a cross-section of Nasdaq blue chip stocks. Employing the estimator to estimate spot covariances, correlations, and volatilities in normal but also unusual periods yields novel insights into intraday covariance and correlation dynamics. We show that intraday (co-)variations (i) follow underlying periodicity patterns, (ii) reveal substantial intraday variability associated with (co-)variation risk, and (iii) can increase strongly and nearly instantaneously if new information arrives. Supplementary materials for this article are available online.     

Gaussian mixture analysis of covariance

ABSTRACT

In many real-world applications, the traditional theory of analysis of covariance (ANCOVA) leads to inadequate and unreliable results because of violation of the response variable observations from the essential Gaussian assumption that may be due to the heterogeneity of population, the presence of outlier or both of them. In this paper, we develop a Gaussian mixture ANCOVA model for modelling heterogeneous populations with a finite number of subpopulation. We provide the maximum likelihood estimates of the model parameters via an EM algorithm. We also drive the adjusted effects estimators for treatments and covariates. The Fisher information matrix of the model and asymptotic confidence intervals for the parameter are also discussed. We performed a simulation study to assess the performance of the proposed model. A real-world example is also worked out to explained the methodology.     

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.     

Posterior consistency for the spectral density of non-Gaussian stationary time series

Various nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation involving a nonparametric correction of a parametric likelihood has been proposed in the literature with a proof of posterior consistency for spectral density estimation in combination with the Bernstein–Dirichlet process prior for Gaussian time series. In this article, we will extend the posterior consistency result to non-Gaussian time series by employing a general consistency theorem for dependent data and misspecified models. As a special case, posterior consistency for the spectral density under the Whittle likelihood is also extended to non-Gaussian time series. Small sample properties of this approach are illustrated with several examples of non-Gaussian time series.     

Extensions of a Conflict Measure of Inconsistencies in Bayesian Hierarchical Models

Abstract.  In a recent paper we extended and refined some tools introduced by O'Hagan for criticism of Bayesian hierarchical models. Especially, avoiding double use of data by a data-splitting approach was a main concern. Such tools can be applied at each node of the model, with a view to diagnosing problems of model fit at any point in the model structure. As O'Hagan, we investigated a Gaussian model of one-way analysis of variance. Through extensive Markov chain Monte Carlo simulations it was shown that our method detects model misspecification about as well as the one of O'Hagan, when this is properly calibrated, while retaining the desired false warning probability for data generated from the assumed model. In the present paper, we suggest some new measures of conflict based on tail probabilities of the so-called integrated posterior distributions introduced in our recent paper. These new measures are equivalent to the measure applied in the latter paper in simple Gaussian models, but seem more appropriately adjusted to deviations from normality and to conflicts not concerning location parameters. A general linear normal model with known covariance matrices is considered in detail.     

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.     

A comparison of centring parameterisations of Gaussian process-based models for Bayesian computation using MCMC

Markov chain Monte Carlo (MCMC) algorithms for Bayesian computation for Gaussian process-based models under default parameterisations are slow to converge due to the presence of spatial- and other-induced dependence structures. The main focus of this paper is to study the effect of the assumed spatial correlation structure on the convergence properties of the Gibbs sampler under the default non-centred parameterisation and a rival centred parameterisation (CP), for the mean structure of a general multi-process Gaussian spatial model. Our investigation finds answers to many pertinent, but as yet unanswered, questions on the choice between the two. Assuming the covariance parameters to be known, we compare the exact rates of convergence of the two by varying the strength of the spatial correlation, the level of covariance tapering, the scale of the spatially varying covariates, the number of data points, the number and the structure of block updating of the spatial effects and the amount of smoothness assumed in a Matérn covariance function. We also study the effects of introducing differing levels of geometric anisotropy in the spatial model. The case of unknown variance parameters is investigated using well-known MCMC convergence diagnostics. A simulation study and a real-data example on modelling air pollution levels in London are used for illustrations. A generic pattern emerges that the CP is preferable in the presence of more spatial correlation or more information obtained through, for example, additional data points or by increased covariate variability.