Inverse Wishart Distribution Research Articles

Simulation of hyper-inverse Wishart distributions in graphical models

We introduce and exemplify an efficient method for direct sampling from hyper-inverse Wishart distributions. The method relies very naturally on the use of standard junction-tree representation of graphs, and couples these with matrix results for inverse Wishart distributions. We describe the theory and resulting computational algorithms for both decomposable and nondecomposable graphical models. An example drawn from financial time series demonstrates application in a context where inferences on a structured covariance model are required. We discuss and investigate questions of scalability of the simulation methods to higher-dimensional distributions. The paper concludes with general comments about the approach, including its use in connection with existing Markov chain Monte Carlo methods that deal with uncertainty about the graphical model structure.

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.

Inverse Wishart distributions based on singular elliptically contoured distribution

Assuming that the random matrix X has a singular or non-singular matrix variate elliptically contoured distribution, the density function of the Moore–Penrose inverse Z = ( X ′ X ) + is given with respect to the Hausdorff measure. The result is applied to Bayesian inference for a general multivariate linear regression model with matrix variate elliptically distributed errors. Some results concerning the posterior joint and marginal distributions of the parameters are obtained.

Multivariate Bayesian analysis of Gaussian, right censored Gaussian, ordered categorical and binary traits using Gibbs sampling.

A fully Bayesian analysis using Gibbs sampling and data augmentation in a multivariate model of Gaussian, right censored, and grouped Gaussian traits is described. The grouped Gaussian traits are either ordered categorical traits (with more than two categories) or binary traits, where the grouping is determined via thresholds on the underlying Gaussian scale, the liability scale. Allowances are made for unequal models, unknown covariance matrices and missing data. Having outlined the theory, strategies for implementation are reviewed. These include joint sampling of location parameters; efficient sampling from the fully conditional posterior distribution of augmented data, a multivariate truncated normal distribution; and sampling from the conditional inverse Wishart distribution, the fully conditional posterior distribution of the residual covariance matrix. Finally, a simulated dataset was analysed to illustrate the methodology. This paper concentrates on a model where residuals associated with liabilities of the binary traits are assumed to be independent. A Bayesian analysis using Gibbs sampling is outlined for the model where this assumption is relaxed.

Multivariate Bayesian analysis of Gaussian, right censored Gaussian, ordered categorical and�binary traits using Gibbs sampling

A fully Bayesian analysis using Gibbs sampling and data augmentation in a multivariate model of Gaussian, right censored, and grouped Gaussian traits is described. The grouped Gaussian traits are either ordered categorical traits (with more than two categories) or binary traits, where the grouping is determined via thresholds on the underlying Gaussian scale, the liability scale. Allowances are made for unequal models, unknown covariance matrices and missing data. Having outlined the theory, strategies for implementation are reviewed. These include joint sampling of location parameters; efficient sampling from the fully conditional posterior distribution of augmented data, a multivariate truncated normal distribution; and sampling from the conditional inverse Wishart distribution, the fully conditional posterior distribution of the residual covariance matrix. Finally, a simulated dataset was analysed to illustrate the methodology. This paper concentrates on a model where residuals associated with liabilities of the binary traits are assumed to be independent. A Bayesian analysis using Gibbs sampling is outlined for the model where this assumption is relaxed.

Hyper Inverse Wishart Distribution for Non‐decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models

While conjugate Bayesian inference in decomposable Gaussian graphical models is largely solved, the non‐decomposable case still poses difficulties concerned with the specification of suitable priors and the evaluation of normalizing constants. In this paper we derive the DY‐conjugate prior (Diaconis & Ylvisaker, 1979) for non‐decomposable models and show that it can be regarded as a generalization to an arbitrary graph G of the hyper inverse Wishart distribution (Dawid & Lauritzen, 1993). In particular, if G is an incomplete prime graph it constitutes a non‐trivial generalization of the inverse Wishart distribution. Inference based on marginal likelihood requires the evaluation of a normalizing constant and we propose an importance sampling algorithm for its computation. Examples of structural learning involving non‐decomposable models are given. In order to deal efficiently with the set of all positive definite matrices with non‐decomposable zero‐pattern we introduce the operation of triangular completion of an incomplete triangular matrix. Such a device turns out to be extremely useful both in the proof of theoretical results and in the implementation of the Monte Carlo procedure.

Non-conjugate Prior Distribution Assessment for Multivariate Normal Sampling

Summary Elicitation methods are proposed for quantifying expert opinion about a multivariate normal sampling model. The natural conjugate prior family imposes a relationship between the mean vector and the covariance matrix that can portray an expert's opinion poorly. Instead we assume that opinions about the mean and the covariance are independent and suggest innovative forms of question which enable the expert to quantify separately his or her opinion about each of these parameters. Prior opinion about the mean vector is modelled by a multivariate normal distribution and about the covariance matrix by both an inverse Wishart distribution and a generalized inverse-Wishart (GIW) distribution. To construct the latter, results are developed that give insight into the GIW parameters and their interrelationships. Certain of the elicitation methods exploit unconditional assessments as fully as possible, since these can reflect an expert's beliefs more accurately than conditional assessments. Methods are illustrated through an example.

Anomaly detection in communication networks using wavelets

An algorithm is proposed for network anomaly detection based on the undecimated discrete wavelet transform and Bayesian analysis. The proposed algorithm checks the wavelet coefficients across resolution levels, and locates smooth and abrupt changes in variance and frequency in the given time series, by using the wavelet coefficients at these levels. The unknown variance of the wavelet coefficients is considered as a stochastic nuisance parameter. Marginalisation is then used to remove this nuisance parameter by using three different priors: flat, Jeffreys' and the inverse Wishart distribution (scalar case). The different versions of the proposed algorithm are evaluated using synthetic data, and compared with autoregressive models and thresholding techniques. The proposed algorithm is applied to monitor events in a Dial Internet Protocol service. The results show that the proposed algorithm is able to identify the presence of abnormal network behaviours in advance of reported network anomalies.

Cholesky decomposition of a hyper inverse Wishart matrix

The canonical parameter of a covariance selection model is the inverse covariance matrix Σ -1 whose zero pattern gives the conditional independence structure characterising the model. In this paper we consider the upper triangular matrix Φ obtained by the Cholesky decomposition Σ - 1 = Φ T Φ. This provides an interesting alternative parameterisation of decomposable models since its upper triangle has the same zero structure as Σ -1 and its elements have an interpretation as parameters of certain conditional distributions. For a distribution for Σ, the strong hyper-Markov property is shown to be characterised by the mutual independence of the rows of Φ. This is further used to generalise to the hyper inverse Wishart distribution some well-known properties of the inverse Wishart distribution. In particular we show that a hyper inverse Wishart matrix can be decomposed into independent normal and chi-squared random variables, and we describe a family of transformations under which the family of hyper inverse Wishart distributions is closed.

A useful reparameterisation to obtain samples from conditional inverse Wishart distributions

A useful reparameterisation to obtain samples from conditional inverse Wishart distributions

A useful reparameterisation to obtain samples from conditional inverse Wishart distributions

A Bayesian joint analysis of normally distributed traits and binary traits, using the Gibbs sampler, requires the drawing of samples from a conditional inverse Wishart distribution. This is the fully conditional posterior distribution of the residual covariance matrix of the normally distributed traits and liabilities of the binary traits. Obtaining samples from the conditional inverse Wishart distribution is not straightforward. However, combining well-known matrix results and properties of the Wishart distribution, it is shown that this can be easily carried out by successively drawing from Wishart and normally distributed random variables. © Inra/Elsevier,

An elicitation method for multivariate normal distributions

This paper presents an elicitation method for quantifying an expert’s subjective opinion about a multivariate normal distribution. It is assumed that the expert’s opinion can be adequately represented by a natural conjugate distribution (a normal inverse-Wishart distribution), and the expert performs various assessment tasks that enable the hyperparameters of the distribution to be estimated. There is some choice in the way hyperparameters are determined from the elicited assessments and empirical work underlies the choices made. The method is implemented in an interactive computer program that questions the expert and forms the subjective distribution. An example illustrating use of the method is given.

Random continued fractions and inverse Gaussian distribution on a symmetric cone

In this paper we introduce the inverse Gaussian and Wishart distributions on the cone of real (n, n) symmetric positive definite matricesHn+(ℝ) and more generally on an irreducible symmetric coneC. Then we study the convergence of random continued fractions onHn+(ℝ) andC by means of real Lagrangians forHn+(ℝ) and by new algebraic identities on symmetric cones forC. Finally we get a characterization of the inverse Gaussian distribution onHn+(ℝ) andC.

Some Matrix-Variate Distribution Theory: Notational Considerations and a Bayesian Application

We introduce and justify a convenient notation for certain matrix-variate distributions which, by its emphasis on the important underlying parameters, and the theory on which it is based, eases greatly the task of manipulating such distributions. Important examples include the matrix-variate normal, t, F and beta, and the Wishart and inverse Wishart distributions. The theory is applied to compound matrix distributions and to Bayesian prediction in the multivariate linear model.

Some matrix-variate distribution theory: Notational considerations and a Bayesian application

SUMMARY We introduce and justify a convenient notation for certain matrix-variate distributions which, by its emphasis on the important underlying parameters, and the theory on which it is based, eases greatly the task of manipulating such distributions. Important examples include the matrix-variate normal, t, F and beta, and the Wishart and inverse Wishart distributions. The theory is applied to compound matrix distributions and to Bayesian prediction in the multivariate linear model.