Hyper inverse wishart

Cholesky decomposition of a hyper inverse Wishart matrix. This is further used to generalise to the hyper inverse Wishart distribution some well-known properties of the inverse Wishart distribution. We propose an efficient solution to the problem of sampling from a hyper- inverse Wishart distribution in non-decomposable graphs. We will show that the type I and II Wisharts have properties similar to those of the hyper and hyper inverse Wishart. has an inverse Wishart distribution. 2 Hyper-Inverse Wishart Distributions The fully conjugate Bayesian analysis of decomposable Gaussian graphical models (Dawid and Lauritzen 1993) is based on the family of hyper-inverse Wishart (HIW) distributions for structured variance matrices. Normal-inverse-wishart Distribution. Aug 5, 2007 - SUMMARY. Structure estimation and Bayes factors 9. Original version: January, 2005 Final version: Biometrika 2007, 94:647-659 Hyper-inverse Wishart (HIW) distributions feature centrally in the analysis of Gaussian graphical models and related studies on structured variance matrices and covariance selection. The hyper-inverse Wishart distribution is a general class of hyper-Markov laws introduced by Dawid & Lauritzen (1993) for a covariance matrix 2M(G), where G= (V;E) is a decomposable graph. Introduction [3] proposed a general method to direct sample from the hyper-inverse Wishart for both decomposable and non-decomposable graphs. Simulation of hyper-inverse Wishart distributions in graphical models . It is the conjugate prior of a multivariate normal distribution with unknown mean and covariance matrix (the inverse of the precision matrix). The mean, inverse-mean, covariance and variance functions, as well as moments of higher order, are studied and their explicit formulas are given. The method relies on [3] proposed a general method to direct sample from the hyper-inverse Wishart for both decomposable and non-decomposable graphs. 3. Indeed, the inverse of the type II Wishart forms a conjugate family of priors for the covariance parameter of the graphical Gaussian model and is strong directed hyper Markov for every direction given to the graph by a perfect order of its cliques, while the type I Wishart is weak hyper Markov. 001,0. Carvalho, Carlos M. 21 (1993) 1272–1317]. Jump to navigation Jump to search. 214 Old Chemistry Bldg. Dawid (1981) provides a discussion of the relation of the matrix-valued normal distribution to other distributions, including the Wishart distribution, Inverse Wishart distribution and a "matrix-t distribution", but uses different notation from that employed here. 6 Hyper-inverse Wishart distributions 316 Decomposable graphical models 316 The hyper-inverse Wishart distribution 317 In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. Statist. (DOCX 100 KB)Hyper-exponential distribution; Inverse Gaussian distribution; Inverse multivariate gamma distribution; Inverse-chi-squared distribution; Inverse-gamma distribution; Inverse-Wishart distribution; Irwin–Hall distribution; K. Carlos Carvalho, Hélène Massam and Mike West . Distributions in Graphical Models. (2000). Sie ist die matrixvariate Entsprechung der χ 2 -Verteilung . and Rickershauser, J. In the context of graphical models, Roverato defined the hyper-inverse Wishart and Wang and West extended the inverse Wishart distribution for using hyper-Markov properties (see ), while Bryc proposed the compound Wishart and q-Wishart in graphical models. These laws occur naturally in connection with the analysis of decomposable log-linear and covariance selection models. 和. We show that the Type I and II Wisharts have properties similar to those of the hyper and hyper inverse Wishart. BibTeX @MISC{Carvalho07simulationof, author = {Carlos M. 1 Fixed variance (˙2), random mean ( ) Keeping ˙2 xed, the conjugate prior for is a Gaussian. (2010) Simulation of Hyper-Inverse Wishart Distributions for Non-decomposable Graphs. 且二者独立。 Inverse-Wishart分布: 如果一个正定矩阵B的逆矩阵 服从Wishart分布 ,那么称服从Inverse-Wishart分布 . July 2011. Joint high-dimensional Bayesian variable and covariance selection with an application to eQTL analysis 1 1. Hyper Markov Laws Identify θ ∈ Θ and P 10. Nonparametric Bayesian Matrix Completion Mingyuan Zhou, with a separate gamma hyper-prior inverse-Wishart prior, rather than via a factor model). This paper proposes a new algorithm for Bayesian model determination in Gaussian graphical models under \textit{G}-Wishart prior distributions. Finally the hyper-inverse Wishart distribution, defined by Dawid and Lauritzen (1993 Dawid, A. \A Monte Carlo method to compute the marginal likelihood in non decomposable Efficient Gaussian Graphical Model Determination under G-Wishart Prior Distributions (1) Hao Wang & (2) Sophia Zhengzi Li (1) University of South Carolina (2)Duke University . e. Indeed, the inverse of the type II Wishart forms a conjugate family of priors for the covariance parameter of the graphical Gaussian model and is strong directed hyper Markov for every direction given to the graph by a perfect order of its cliques, while the type I Wishart is weak hyper Markov. univariate) and , and the probability density function of the inverse-Wishart distribution becomes. Read in another language Watch this page Edit Bayesian Inference of a Multivariate Regression Model and inverse Wishart speci larger variance terms in the prior hyper- Ultimate bibles on probability distributions are: , Fisher F f, gamma gamma, geometric geom, hypergeometric hyper , logistic Normal Inverse Wishart "Hyper Inverse Wishart Distribution for Non-decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models," Scandinavian Journal of Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper Diagram showing queueing system equivalent of a hyper-Erlang distribution In probability theory , a hyper-Erlang distribution is a continuous probability distribution which takes a particular Erlang distribution E i with probability p i . Massam, Wishart distributions for decomposable graphs, The Annals of Statistics 35 (2007), 1278–1323. Received February 2001, in final form December 2001 Antonella Capitanio, Dipartimento di Scienze Statistiche, Universita` di Bologna, via delle densities have inverse–Wishart priors with scale In both cases, we assume that all hyper-parameters have fixed, known values, and that each training image has been labeled with the single object om it contains. Phân bố xác suất hậu nghiệm của một biến ngẫu nhiên khi cho trước giá trị của một biến khác Network meta-analysis of multiple outcome measures accounting for borrowing of information across outcomes. 2. 1111/1467-9469. Conjugate prior is normal-inverse-Wishart. A univariate specialization of the inverse-Wishart distribution is the inverse-gamma distribution. STAT (forthcoming) H. It can be obtained from a collection of clique specific marginal inverse Wisharts as follows: under the previous assumptions, the posterior distribution of the variance-covariance matrix \(\Sigma\) is a hyper Wishart distribution hyper inverse Wishart, the parameters ia', ft) of the / WpG are added to half of the . A second class of models considers and random quantities, giving rise to a hierarchical hyper inverse Wishart. hyper inverse wishartIn statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a Gaussian · Gompertz · half-logistic · half-normal · Hotelling's T-squared · hyper-Erlang · hyperexponential · hypoexponential · inverse chi-squared. Application to Bayesian Inference for Gaussian Graphical Models. 001) (Spiegelhalter et al. 5. They can be viewed as generalization of the hyper Wishart and the inverse of the hyper inverse Wishart as de ned by Dawid and Lauritzen(1993). "E cient Simulation of Hyper-Inverse Wishart Distributions in Graphical Mod-els", (2007), Biometrika, To appear, (with Carlos Carvalho and Mike West ). In probability theory and statistics, the normal-inverse-Wishart distribution (or Gaussian-inverse-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. This paper proposes a new algorithm for Bayesian model determination in Gaussian graphical models under \textit{G}-Wishart prior distributions. Fang Chen, SAS Institute Inc, Cary, NC ABSTRACT The MCMC procedure, such as the multivariate normal and inverse-Wishart distributions, and implements conjugate sampling methods when appropriate to improve sampling speed. Issue . Flat prior; Super-vague but proper prior: normal(0, 1e6); For a hierarchical covariance matrix, we suggest a Wishart (not inverse-Wishart) In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. hyper inverse wishart with hyper-parameters = n + 0, = n x T + 0 T n+ 0 Wishart and Inverse Wishart I The Wishart distribution with 0 degrees of freedom, over S+p the group of Bayesian Inference for Gaussian Semiparametric Multilevel Models Cauchy distribution is a Scaled Inverse- Wishart distribution for the of the hyper-priors for Simple Marginally Noninformative Prior Distributions for Covariance Matrices noninformative for particular hyper- to Inverse-Wishart priors for covariance "Hyper-Erlang Distribution Model and its Application in Wireless Mobile Networks". The method relies very naturally on the We use cookies to …For a distribution for [sum ], the strong hyper-Markov property is shown to be characterised by the mutual independence of the rows of Φ. We also note that the G-Wishart distribution introduced in [21] for undirected graphs, that is the inverse of the hyper-inverse Wishart of [7] which has a one-dimensional shape parameter δ, is In probability theory and statistics, the normal-inverse-Wishart distribution (or Gaussian-inverse-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. DOI: 10. The usual inverse Wishart is a speci c case, which is hyper Markov for the saturated model. The priors are formed by taking a Markov combination of non-sparse correlation matrix distributions, where these distributions come from marginalizing the diagonal elements out of an inverse Wishart or Wishart …Bayesian Structure Learning for Stationary Time Series: Supplementary Material Alex Tank University of Washington alextank@uw. If = 1 2 M(G), the hyper-inverse Wishart For a distribution for [sum ], the strong hyper-Markov property is shown to be characterised by the mutual independence of the rows of Φ. A. ALBERTO In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a Gaussian · Gompertz · half-logistic · half-normal · Hotelling's T-squared · hyper-Erlang · hyperexponential · hypoexponential · inverse chi-squared. More on structure estimation. Gibbs. The method relies very naturally on transformations under which the family of hyper inverse Wishart distributions is We finally assume the covariance matrix to have a hyper inverse Wishart We propose an efficient solution to the problem of direct sampling from a hyper-inverse Wishart distribution in non-decomposable graphs. Fang Chen, SAS Institute Inc. $\begingroup$ I didn't manage brute force with hyper-spherical coordinates and proofs about the Wishart distribution. Estimating its parameters using Bayesian inference and conjugate priors is also widely used. The approach is based. , 2003), proper distributions such as p(σ2 α) ∼ inverse-gamma(0. Hyper-inverse Wishart distribution 651 The key practical extension of the above structure to unrestricted graphs, including nondecomposable cases when some of …Simulation of Hyper-Inverse Wishart Distributions in Graphical Models By CARLOS M. The original technical report is also available. Für die Erläuterung wird zum besseren Verständnis zunächst von einer Zufallsvariablen ausgegangen: Man betrachtet eine standardnormalverteilte Zufallsvariable X, also mit dem Erwartungswert 0 und der Varianz 1. ALBERTO Simulation of Hyper-Inverse Wishart. incomplete, is the local hyper-inverse Wishart model in which the same basic form and density decomposition hold, but with modification to the compo- nent densities on incomplete components, as in Jones et al. 21 (1993) 1272--1317]. (2005). inverse Wishart distribution. K-distribution; Kumaraswamy distribution; L. This prior distribution depends on hyper-parameters. Yue, and H Zhao, Cojumps in China's Spot and Stock Index Futures Markets. Roverato, Cholesky decomposition of a hyper inverse Wishart matrix, Biometrika 87 (2000), 99–112. Hyper-exponential distribution; Inverse-gamma distribution; Inverse-Wishart distribution; Thanks for reading List of Continuous distributions. edu Λ0, µ, k0 are hyper-parameters of the model, trix Λ0 describes the scale of inverse-Wishart distribution. Given the prior distribution with associated posterior distributions a MCMC or variational sampler (i. A generalization is the inverse multivariate gamma distribution. ca Last updated October 3, 2007 1 Introduction The Gaussian or normal distribution is one of the most widely used in statistics. In the context of graphical models, Roverato (2002) de ned the hyper-inverse Wishart and Wang and West (2009) extended the inverse Wishart distribution for using hyper-Markov properties (see Dawid and Lauritzen (1993)), while Bryc (2008) proposed the compound Wishart and q-Wishart in graphical models. I am taking inverse chi square as my prior distribution for variance of normal sampling distribution. The Pareto distribution, is the inverse of the CDF. Directional Usually I have to use an inverse-Wishart distribution, which seems easy enough. MCMCpack, RMTstat, bayesm provides d, r functions , bayesm C19 : Lecture 4 : A Gibbs Sampler for Gaussian Mixture Models Frank Wood University of Oxford January, 2014 The parameters of the Inverse-Wishart prior, H= f 1 0; The hyper …In Bayesian statistics, the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random-vector. straint Wishart matrices. Yet inferring the condi-tional independence structure of a random vector presents a substantial problem inmal quasi-Wishart distribution. Note that if The Erlang distribution is a continuous probability distribution with wide The distribution is sometimes defined using the inverse of the inverse-Wishart We're upgrading the ACM DL, and would like your input. i. CPSC 540: Machine Learning choosing many hyper-parameters, handling non-IID data, etc. of Math. Carvalho and Hélène Massam and Mike West}, title = { Simulation of hyper-inverse Wishart distributions in graphical models}, year = {2007}} propose a hyper-inverse-Wishart-process prior for the covariance kernels of the in nite co-e cient sequences of the basis expansion, and establish its existence and uniqueness. Relationships to other distributions. of Wishart distributions, namely the type I and II Wisharts. A Non-parametric Learning Method for Confidently to validate and iteratively update the hyper-parameters the Normal-Inverse-Wishart prior distribution [11]: A new methodology for model determination in decomposable graphical Gaussian models is developed. I am confused as to how the inverse of a Novel Partitioning Algorithm for a Gaussian Inverse Wishart PHD Filter for Extended Target Tracking 0] = a constant) be a connected spacelike hypersurface in M and hence a codimension two-spacelike submanifold of [bar. In particular we show 5 Aug 2007 SUMMARY. 1 Introduction Gaussian graphical modeling offers a potent set of tools for shrinkage and regular-ization of covariance matrices in high-dimensional problems. the hyper complex inverse Wishart distribution, that serves as a conjugate prior for the spectral density matrices whose inverses have a zero pattern specified by a graph. Flexible Wishart distributions and their applications Hel´ ene Massam` York University with co-authors, G. (2010) Volatility in Prediction Markets: A Measure of Information Flow in Political Campaigns. 那么他们的分布是 . Stat. They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. We 0 for the hyper-parameters, They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. Murphy∗ murphyk@cs. 1 was partially inspired by A Bayesian model for repeated measures zero-inflated count data with application to outpatient psychiatric service use inverse-Wishart hyper -priors such as Fang Chen, SAS Institute Inc, Cary, NC ABSTRACT such as the multivariate normal and inverse-Wishart distributions, and implements conjugate sampling methods when First of all, independence of Y1 and Y2 introduced in section 2. We introduce and exemplify an efficient method for direct sampling from hyper-inverse Wishart distributions. We also prove the strong hyper Markov property and the conjugacy of this prior under a nite rank condition of the prior kernel parameter. With (i. Various noninformative prior distributions for σα have been suggested in Bayesian literature and software, including an improper uniform density on σα (Gelman et al. \A Monte Carlo method to compute the marginal likelihood in non decomposable graphical gaussian models" (with Aliye Atay-Kayis). [1] A new methodology for model determination in decomposable graphical Gaussian models (Dawid and Lauritzen in Ann. share | cite | improve this question. The normal-inverse Gaussian distribution (NIG) is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The Wishart distribution is related to the Inverse-Wishart distribution, denoted by , as follows: If and if we do the change of variables , then . ties similar to those of the hyper and hyper inverse Wishart. In statistics, the Wishart distribution is a generalization to multiple dimensions of the chi-square distribution, or, in the case of non-integer degrees of freedom, of the gamma distribution. 0001 I n. Novel Partitioning Algorithm for a Gaussian Inverse Wishart PHD Filter for Extended Target Tracking Some of the found results belonging to the obtained Pareto hypersurface are presented in Table VII, where I1 to I8 are the input parameters from Table I. 15) is fulfilled, meaning that there exists a j<isuch that S i = P i ∩H i−1 ⊂ P j, where H i−1 = i−1 j=1 P j. 81 10 [7] G. Roverato, A. Mike West Duke University Dynamic inverse Wishart processes (>25 years) Dynamic hyper-inverse Wishart models . Institute of Statistics and Decision Sciences, Duke May 3, 2010 Abstract. Second, we apply a class of priors that automatically handles the problem of Both the hyper inverse Wishart priors and the “Letac-Massam” priors have attractive properties which enable Bayesian inference, with the latter allowing multiple shape parameters and hence suitable in high-dimensional settings. Scandinavian Journal of Statistics. Hyper Markov Laws The hyper Markov property has a simple formulation in terms of junction trees: Arrange the prime components Q of G in a junction tree T with complete separators S and consider the extendedIn statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. Biometrika, 2005. The method relies on local computations based on the standard junction tree representation of graphs and distribution theoretical results of constraint Wishart matrices. Joint high-dimensional Bayesian variable and primarily using hyper-inverse Wishart type of priors. (2010) Volatility in Prediction Markets: A Measure of Information Flow in Political Campaigns. densities have inverse–Wishart priors with scale ∆p and νp degrees of freedom, while the means µz are given noninfor-mative priors. In probability theory and statistics, the normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. The Bayesian paradigm is used and, for each given graph, a hyper inverse Wishart prior distribution on the covariance matrix is considered. inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma The final test is to write a function that takes a multivariate Normal set of data and the 4 hyper paramters needed to specify a Normal Inverse Wishart prior for the multivariate Normal mean vector and covariance matrix. First, we develop a default version of the hyper-inverse Wishart prior for restricted covariance matrices, called the hyper-inverse Wishart g-prior, and show how it corresponds to the implied fractional prior for selecting a graph using fractional Bayes factors. Our aim is to nd conjugate prior distributions for these parameters. , the inverse-gamma distribution, where is the ordinary Gamma function. , Λ0, µ, k0 are hyper-parameters of the model, that is, parameters specifying our prior belief about param-eters Σ and µ before observing the data. 001; % from the inverse Wishart to account for the spread around that point picked on the line segment. MathSciNet CrossRef A very important area of financial risk management is systemic risk the hyper inverse Wishart distribution. Finally, the marginal prior on the space of graphs is a Bernoulli distribution with ψ = 2 ∕ n − 1 which provides a prior mode at n edges. Electronic Journal of Statistics 4 (2010):1467— 1470Template:ProbDistributions. However, they can be tricky to specify once the covariance matrix gets pretty big. First, we develop a default version of the hyper-inverse Wishart prior for restricted covariance matrices, called the hyper-inverse Wishart g-prior, and show how it corresponds to the implied fractional prior for selecting a graph using fractional Bayes factors. Simulation of hyper-inverse Wishart distributions for non-decomposable graphs. A univariate specialization of the inverse-Wishart distribution is the inverse-gamma distribution. If i have n=10 and i choose V0 as 9 it is a reasonable choice for this hyper parameter? bayesian. In this report, we summarize all of the most commonly used forms. Simulation of hyper-inverse Wishart distributions in graphical models, by Carlos Carvalho, Helene Massam and Mike West, Biometrika 2007 94:647-659. To ensure compatibility across models, such prior distributions are obtained by marginalisation from the prior conditional on the complete graph. the corresponding Wishart distribution has no Lebesgue density. The probability density function of the d-dimensional Inverse Wishart distribution is given by. Graphical models in applied multivariate statistics. 7. S. Roverato, Cholesky decomposition of a hyper inverse Wishart matrix, Biometrika 87 alization of the hyper Wishart and the inverse of the hyper inverse Wishart as de ned by Dawid and Lauritzen (1993). The priors are formed by taking a Markov combination of non-sparse correlation matrix distributions, where these distributions come from marginalizing the diagonal elements out of an inverse Wishart or Wishart …"E cient Simulation of Hyper-Inverse Wishart Distributions in Graphical Mod-els", (2007), Biometrika, To appear, (with Carlos Carvalho and Mike West ). This "Cited by" count includes citations to the following articles in Scholar. The method relies on 19 Dec 2002 Hyper Inverse Wishart Distribution for Non‐decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models. The contents covered in this lecture include; multivariate distributions, the basics of state space models, or DLMs, filtering and smoothing, the concept of discount factors, multivariate and matrix-variate extension, graphical models and modeling by hyper-(inverse) Wishart distribution. Characterization Probability density function Properties Marginal distributions. Hyper Inverse Wishart Distribution for Non-decomposable Graphs and its. Foti University of Washington The hyper complex inverse Wishart is then introduced in more detail and …Prior Choice Recommendations. Department of Statistical Science. Inverse-Wishart分布常作为Bayes中多元正态分布的协方差阵的 the hyper complex inverse Wishart distribution, that serves as a conjugate prior for the spectral density matrices whose inverses have a zero pattern specified by a graph. . The approach is based Wishart and hyper inverse Wishart, homogeneous cones, natural exponential families. We introduce and exemplify an efficient method for direct sampling from hyper- inverse Wishart distributions. Wiley, Chichester. def) تابع چگالی احتمال تابع توزیع تجمعی ‫تکیه‌گاه inverse(v) inverse of symmetric positive-definite matrix v logdet(v) log of determinant of v for symmetric positive-definite v mean(v) S i v i / n n = dim(v) eigen. In statistics, the inverse matrix gamma distribution is a generalization of the inverse gamma distribution to positive-definite matrices. متغیر می‌تواند برداری باشد = {,, …,}. 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. Initially we specified an Inverse-Wishart In probability theory, a hyper-exponential distribution is a continuous distribution such that the probability density function of the random variable is given by. Moreover, the family is quite simple and, for approximate natives to Inverse-Wishart priors for covariance matrices. Introduction The study of expression Normal-inverse-wishart Distribution. In probability theory and statistics, the normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. Add object-like properties hyper. g. By construction, the marginal distribution over is an inverse Wishart distribution, and the conditional distribution over given is a multivariate normal distribution. Downsides of inverse Wishart prior in hierarchical models. Biometrika, 87(1), 99–112. توجه کنید که در این حالت لزومی ندارد که بعد بردار متغیر با بعد بردار پارامتر یکسان باشد. "Wishart distributions for decomposable graphs, (2007), The Annals o f Statistics, 38, No 3, (with G. p is the df of wishart prior of inverse of the scale matrix of Dirichlet · Generalized Dirichlet · inverse-Wishart · Kent · matrix normal · multivariate normal · multivariate Student · von Mises-Fisher · Wigner quasi · Wishart Miscellaneous : bimodal · Cantor · conditional · equilibrium · exponential family · Infinite divisibility (probability) · location-scale family · marginal · maximum حالت پارامتر و متغیر برداری. Joint high-dimensional Bayesian variable and covariance selection with an Hyper-inverse Wishart distribution; Joint Variable and Joint high-dimensional Abstract. hyper-inverse Wishart model in which the same basic form and density decomposition hold, but with modification to the component densities on incomplete components, as in Jones et al. If C=1, the distribution is identical to the chi-square distribution with n degrees of freedom. (hyper) parameters for the Gaussian-inverse-Wishart prior: (scale matrix for the inverse Wishart). Bayesian Structure Learning for Stationary Time Series: Supplementary Material Alex Tank University of Washington The hyper complex inverse Wishart is Various noninformative prior distributions for σα have been suggested in Bayesian literature and software, including an improper uniform density on σα (Gelman et al. 3 The inverse Wishart distribution 314 Point estimates of variance matrices 315 10. Inverse Wishart distribution : Wishart distributions : Base R provides the r function for the Wishart distribution. uk/display/22874645Abstract. Please sign up to review new features, functionality and page designs. We study Natural Exponential Families (NEF) of Wishart distributions on convex cones [equation] Let G be the graph corresponding to the graphical model of nearest neighbor interaction in a Gaussian character. I'm estimating several inverse covariance matrices of a set of measurements across different subpopulations using an wishart prior in jags/rjags/R. Box 90251 Durham, NC 27708-0251 (919) 684-4210 The inverse Wishart distribution is based on the Wishart distribution. hyper-inverse Wishart distribution 1. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, For a distribution for [sum ], the strong hyper-Markov property is shown to be characterised by the mutual independence of the rows of Φ. We determine Riesz measures generating Wishart exponential families on \(Q_G\) and \(P_G\), and we give the quadratic construction of these Riesz measures and exponential families. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in …Wishart分布和样本协方差阵的关系: 设n个独立同分布的 ,有统计量 . The hyper-inverse Wishart distribution is a commonly used prior for Bayesian inference on covariance matrices in Gaussian Graphical models. Monte Carlo Feature Clustering0 are the degrees of freedom and the scale matrix for the inverse-Wishart distribution on , 0 is the prior mean, and is the number of prior measurements on the scale. , Xác suất hậu nghiệm (tiếng Anh: posterior probability) của một biến cố ngẫu nhiên hoặc một mệnh đề không chắc chắn là xác suất có điều kiện mà nó nhận được khi một bằng chứng có liên quan được xét đến. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution. Indeed, the inverse of the Type II Wishart forms a conjugate family of priors for the covariance HIW Sampler is an implementation of an efficient, direct simulation methods for hyper-inverse Wishart distributions arising in Gaussian graphical models, This is further used to generalise to the hyper inverse Wishart distribution some well-known properties of the inverse Wishart distribution. Letac and Massam (2007) as well as Atay-Kayis and Massam (2005) continue this development and call this distribution the G-Wishart. It is a more general version of the inverse Wishart distribution, and is used similarly, e. See also. Then has a normal-inverse-Wishart distribution, denoted as. [8] A. and Carvalho, Carlos M. Wang, H. Probability generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate normal • …Definitions of Hyper-exponential_distribution, synonyms, antonyms, derivatives of Hyper-exponential_distribution, analogical dictionary of Hyper-exponential_distribution (English) An example of a hyper-exponential random variable can be seen in the context of telephony, where, if someone has a modem and a phone, normal-inverse-Wishart The compound distribution resulting from compounding a matrix normal with an inverse matrix gamma prior over the covariance matrix is a generalized matrix t-distribution. 001,0. Publications "High dimensional Simulation of hyper inverse Wishart distributions, The moments and inverse moments of the complex Wishart distribution, 2003, the hyper inverse Wishart distribution to introduce hyper Markov priors for correlation matri-ces. The hyper Wishart distribution is a conjugate prior for a graphical Gaussian distribution with known mean. Simulation of Hyper-Inverse Wishart Distributions for Non-decomposable Graphs. The HIW prior (Dawid & Lauritzen,1993) enforces the hyper-Markov conditions speci ed by G, the hyper inverse Wishart distribution to introduce hyper Markov priors for correlation matri-ces. Directional The discrete phase-type distribution is a probability distribution that results from a system of one or more inter-related geometric distributions occurring in sequence, or phases. R defines the following functions: hcmm_hyperpar. BibTeX @INPROCEEDINGS{Carvalho05simulationof, author = {M. 001) (Spiegelhalter et al. 21(3), 1272-1317, 1993) is developed. The Bayesian paradigm is used and, for each given graph, a hyper-inverse Wishart prior distribution on the covariance matrix is considered. By CARLOS M. La famille des distributions Gamma inclut, entre autres, la loi du χ² et les distributions exponentielles. A. Some key words: Cholesky decomposition; Conjugate distribution; transformations under which the family of hyper inverse Wishart distributions is We finally assume the covariance matrix to have a hyper inverse Wishart In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on (0,∞). Examples include those inIn statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. Letac). The contents covered in this lecture include; multivariate distributions, the basics of state space models, or DLMs, filtering and smoothing, the concept of discount factors, multivariate and matrix-variate extension, graphical models and modeling by hyper-(inverse) Wishart distribution. 1 A non-hierarchical model Consider rst the case of xed hyperparameters. Carvalho and Héléne Massam and Mike West}, title = {Simulation of hyper-inverse Wishart distributions in graphical models}, booktitle = {ISDS Discussion Paper}, year = {2005}}Hyper Inverse Wishart Distribution for Non-decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical ModelsAbstract. Andrew Gelman edited this page Dec 31, 2018 · 44 revisions 5 levels of priors. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix (the inverse …To achieve this aim we consider a hyper inverse Wishart prior distribution on the concentration matrix for each given graph. 1. 1993. Simple Marginally Noninformative Prior Distributions for Covariance Matrices and correlation parameters being marginally noninformative for particular hyper-parameter choices. ( 2005 ). , the inverse-gamma distribution, where is the ordinary Gamma function. Die Exponentialverteilung (auch negative Exponentialverteilung) ist eine stetige Wahrscheinlichkeitsverteilung über der Menge der nicht-negativen reellen Zahlen, die durch eine Exponentialfunktion gegeben ist. hyper-Erlang; hyperexponential inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart. In this paper we derive the DY‐conjugate prior 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 . Hyper parameter for inverse chi square prior Informative Hyper Markov laws 7. hyper-Erlang; hyperexponential; hypoexponential; inverse chi-squared (scaled inverse chi-squared) inverse matrix gamma inverse-Wishart matrix normal matrix t Sparse Probabilistic Matrix Factorization by Laplace Distribution for Collaborative Filtering hyper parameters, BPMF can automatically control the model Shunsaku Horiuchi Present address Center for Neutrino Physics Telephone 540 231 0240 2015 { 2016 Laura Wishart (Virginia Tech) 2015 { 2016 Keegan Walkup (Virginia The Wishart distribution is the probability distribution of the maximum-likelihood estimator (MLE) of the covariance matrix of a multivariate normal distribution. vals(v) eigenvalues of matrix v ode(v1, v2, D(v3, s1), s2, s3) solution of system of ordinary differential equations at grid In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a generalization of the (ordinary) F-distribution. Pena, MBACT - Multiclass Bayesian Additive Classification Trees. kappa = 0. Structure estimation and Bayes factors 9. Rajaratnam, C. In decomposable models this decompositionWe also note that the G-Wishart distribution introduced in [21] for undirected graphs, that is the inverse of the hyper-inverse Wishart of [7] which has a one-dimensional shape parameter δ, is Inverse Wishart Distribution Definition. And the variables Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Hotelling's T-square • hyperbolic secant • hyper-exponential Wishart پارامترها > − درجه آزادی (آمار) (عدد حقیقی) > تجانس (هندسه) (× pos. Wang, H. Received May 2010. Biometrika 87, 99–112. Indeed, the inverse of the type II Wishart forms a conjugate family of priors for the covariance parameter ofFirst, we develop a default version of the hyper-inverse Wishart prior for restricted covariance matrices, called the hyper-inverse Wishart g-prior, and show how it corresponds to the implied fractional prior for covariance selection using fractional Bayes factors. , c k = 3 and C k = 0. ALBERTO ROVERATO; Article first published online: 19 DEC 2002. We introduce and exemplify an efficient method for direct sampling from hyper-inverse Wishart distributions. In particular, if G is an incomplete prime graph it constitutes a non‐trivial generalization of the inverse Wishart distribution. and Lauritzen, S. Letac and H. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix (the inverse of the covariance matrix). [1] choosing prior parameters for variational mixture of Gaussians. In the context of graphical models, Roverato defined the hyper-inverse Wishart and Wang and West extended the inverse Wishart distribution for using hyper-Markov properties (see ), while Bryc proposed the compound Wishart and q-Wishart in graphical models. For the Pareto distribution, Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate A rst class of models considers to be hyper inverse Wishart with respect to g, with xed hyperparameters and . The prior for the hyper-inverse Wishart distribution is also set to be fairly vague, i. A classic text on the subject is Mardia et al (1979), and one with more coverage of Bayesian approaches is Press (1982). reviewing and then extending the use of hyper-inverse Wishart distributions. Background. In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function En théorie des probabilités et en statistiques, une distribution Gamma ou loi Gamma est un type de loi de probabilité de variables aléatoires réelles positives. Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse Hierarchical Bayesian models. In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. [citation needed] This reduces to the inverse Wishart distribution with =, =. The scalar hyper-parameter ν0 describes the degrees of freedom and the ma-trix Λ0 describes the scale of inverse-Wishart distribution. The definition can be generalized to more R/hyperpars. Carvalho while the density of Y = X−1 is the inverse Wishart IWr The inverse of the hyper inverse Wishart has density WP GHyper Inverse Wishart Distribution for Non-decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models. Hyperprior distributions for the parameters (scale matrix and degrees of freedom) of a wishart prior to an inverse covariance matrix There's a good chance I think that it will puke if you Bereket P. Network meta-analysis of multiple outcome measures accounting for borrowing of information across outcomes. Implementing the Infinite GMM Michael Mandel May 6, 2005 are controls by a second level of hyper-parameters, become Wishart random matrices. [9] G. Die logistische Verteilung ist eine stetige Wahrscheinlichkeitsverteilung, die besonders für die analytische Beschreibung von Wachstumsprozessen mit einer Sättigungstendenz verwendet wird. Hyper-inverse Wishart distribution 649 (1996), p. para values of hyper-parameters and initial values of parameters for multivariate t to 10000. IfFor a distribution for [sum ], the strong hyper-Markov property is shown to be characterised by the mutual independence of the rows of Φ. P( j 0;˙2) / 1 ˙ 0 exp 1 2˙2 0 Inverse-Wishart distribution. 26) the choice of (a', ft) has the same kind of impact on the posterior mean as the choice of the shape parameters for the inverse or hyper inverse Wishart. Jump to bottom. In statistics , the inverse Wishart distribution , also called the inverted Wishart distribution , is a probability distribution defined on real-valued positive-definite matrices . As examples we construct a range of specific hyper Markov laws, including the hyper multinomial, hyper Dirichlet and the hyper Wishart and inverse Wishart laws. Jason Bentley, The University of Sydney, New South Wales, Australia The multivariate extension of the Half -Cauchy distribution is a Scaled Inverse- Wishart distribution for the with the exception of the hyper-priors for the scale parameters of the Inverse-Die Wishart-Verteilung ist eine Wahrscheinlichkeitsverteilung. In practice, when little is known about , it is common to set the location hyper-parameter to then the conjugate prior for ˙2 is an inverse Gamma distribution: The multivariate normal and Wishart distributions form the foundation of multivariate data analysis. “Hyper Markov Laws in the Statistical Analysis of Decomposable Graphical Models,”. 6. 4. we assign an inverse-Gamma prior to the parameter a parameter and a hyper-parameter . Efficient Gaussian Graphical Model Determination under G-Wishart Prior Distributions (1) Hao Wang & (2) Sophia Zhengzi Li (1) University of South Carolina (2)Duke University . Efficient Gaussian graphical model determination under G-Wishart prior distributions Hao Wang G-Wishart, hyper-inverse Wishart, Gibbs sampler, non-decomposable Simulation of Hyper-Inverse Wishart Distributions for Non-decomposable Graphs Hao Wang Department of Statistical Science, Duke University, Durham, North Carolina 27708-0251, U. A junction tree for G is a tree representation of the prime components. hyper Wishart distribution as the distribution of the maximum likelihood estimator of Eg, and the hyper inverse Wishart distribution as the Diaconis-Ylvisaker con jugate prior distribution for Eg- Subsequently Roverato [16] gave the distribution of^ = E_1=^-1(EG) when Eg follows the hyper inverse Wishart distribution. The Annals of Statistics, 21: 1272 – 1317. We achieve sparse estimation of the inverse covariance of the residuals in the graphical factor model by employing a hyper-inverse Wishart prior method for a decomposable graph and a Bayesian graphical lasso method for an unrestricted graph. En statistique, la règle 68-95-99,7 (ou règle des trois sigmas ou règle empirique) indique que pour une loi normale, presque toutes les valeurs se situent dans un intervalle centré autour de la moyenne et dont les bornes se situent à 3 écarts-types de part et d'autre. y = f Only random matrix generation is supported for the inverse Wishart, including both singular and nonsingular T. In Bayesian statistics, the Wishart distribution is the conjugate prior of the inverse covariance-matrix of a multivariate-normal random-vector. It describes the distribution of the quotient (X/n 1)/(Y/n 2), where the numerator X has a noncentral chi-squared distribution with n 1 degrees of freedom and the denominator Y has a central chi-squared distribution n 2 degrees The matrix generalized inverse Gaussian distribution (MGIG) is shown to arise as a conditional distribution of components of a Wishart distributio n. The priors are formed by taking a Markov combination of non-sparse correlation matrix distributions, where these distributions come from marginalizing the diagonal elements out of an inverse Wishart or Wishart prior. The method relies transformations under which the family of hyper inverse Wishart distributions is closed. Aug 1, 2007 We introduce and exemplify an efficient method for direct sampling from hyper-inverse Wishart distributions. Inverse-Wishart分布常作为Bayes中多元正态分布的协方差阵的 They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. YouTube Encyclopedic 1 / 5 hyper-Erlang; hyperexponential inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart. We introduce and exemplify an efficient method for direct sampling from hyperinverse Wishart distributions. We propose an efficient solution to the problem of direct sampling from a hyper-inverse Wishart distribution in non-decomposable graphs. The hyper Wishart distribution is a conjugate prior for a graphical Gaussian distribution with known mean. alization of the hyper Wishart and the inverse of the hyper inverse Wishart as de ned by Dawid and Lauritzen (1993). , 1994, 2003), and distributions that depend on the data-level variance (Box and A very important area of financial risk management is systemic risk modelling, propose a convenient prior for \(\Sigma\), the hyper inverse Wishart distribution Gand specifying a hyper-inverse Wishart (HIW) prior on l. (1990). Kindo, Hao Wang, Edsel A. edu Nicholas J. 36 (1935), 527–606. ubc. , 1994, 2003), and distributions that depend on the data-level variance (Box andBayesian Inference for Gaussian Semiparametric Multilevel Models . mal quasi-Wishart distribution. Electronic Journal of Statistics. Similarly, the reference position’s covariance prior is inverse–Wishart with hyperparameters ∆o and νo. L. 4 The normal, inverse Wishart distribution 315 10. The code is in Matlab, and is used as in the example below. Bayesian variable and covariance selection with an They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. Handbook of Applied Bayesian Analysis, . We want your feedback! Note that we can't provide technical support on individual packages. Roverato, A. , 2003), proper distributions such as p(σ2 α) ∼ inverse-gamma(0. In probability theory and statistics, the generalized inverse Gaussian distribution is a three-parameter family of continuous probability distributions with mvt. It is named in honor of John Wishart, who first formulated the distribution in 1928. 2 is ensured when at least one of a1 and a2 is the null vector and X12 is the null matrix, hence also the corresponding block, X12 , in the inverse matrix is 0. ac. the hyper complex inverse Wishart distribution, that serves as a conjugate prior for the spectral density matrices whose inverses have a zero pattern specified by a graph. (Redirected from Template:ProbDistributions). Normal-scaled inverse gamma distribution — Normal scaled inverse gamma parameters: location (real) (real) (real) (real) support … Wikipedia Inverse-Wishart distribution — In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability density function defined on matrices. Roverato (2002) generalizes the hyper inverse Wishart distribution of Dawid and Lauritzen (1993) to arbitrary graphs by deriving the Diaconis and Ylvisaker (1979) conjugate prior for K2P G. More on Hyper Markov Laws 8. We will investigate the hyper-parameter (prior parameter) update relations and the problem of predicting new data from old data: P(x new jx old). Simulation of Hyper-Inverse Wishart Distributions for Non-decomposable Graphs Hao Wang Department of Statistical Science, Duke University, Durham, North Carolina 27708-0251, U. If = 1 2 M(G), the hyper-inverse WishartConjugate Bayesian analysis of the Gaussian distribution use an inverse gamma or inverse chi-squared, etc), which can be very confusing for the student. Whittaker, J. To achieve this aim we consider a hyper inverse Wishart prior distribution on the concentration matrix for each given graph. The method relies Title: Hyper Inverse Wishart Distribution for Non-Decomposable Graphs and Its Application to Bayesian Inference for Gaussian Graphical Models Created Date Inverse-Wishart distribution. Indeed, the inverse of the type II Wishart forms a conjugate family of priors for the covariance parameter of Flexible Wishart distributions and their applications while the density of Y = X−1 is the inverse Wishart IWr The hyper inverse Wishart distribution Conjugate Bayesian analysis of the Gaussian distribution Kevin P. They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. 2 Hyper-Inverse Wishart Distributions The fully conjugate Bayesian analysis of decomposable Gaussian graphical models (Dawid and Lauritzen 1993) is based on the family of hyper-inverse Wishart (HIW) distributions for structured variance matrices. Related Models The graphical model of Fig. Abstract. Keywords and phrases: Hyper-inverse Wishart, junction trees, non-decomposable graphs, posterior simulation. extending the use of hyper-inverse Wishart distributions. This prior has the distinct advantage that it is a conjugate prior for this model but it suffers from lack of flexibility in high …En théorie des probabilités et en statistique, la loi de Wishart est la généralisation multidimensionnelle de la loi du La loi de Wishart est liée à la loi de Wishart inverse, notée W p −1, comme suit : Hyper Dirichlet distribution : provided in hyper2 package. e. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of …of Wishart distributions, namely the type I and II Wisharts. where is an exponentially distributed random variable with rate parameter , and is the probability that X will take on the form of the exponential distribution with rate . hood estimator of ΣG, and the hyper inverse Wishart distribution as the distributions, like the hyper inverse Wisharts, form a family of conjugate. as the conjugate prior of the covariance matrix of a multivariate normal distribution or matrix normal distribution. Box 90251 Durham, NC 27708-0251 (919) 684-4210Hyper Inverse Wishart Distribution for Non‐decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models. 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 ). Letac, B. On the structure of the inverse to Toeplitz-block Toeplitz matrices and of the corresponding polynomial reflection coefficients On Wishart and noncentral Wishart Schools: ranking schoolexamination (the inverse of the between-pupil variance) whilst the inverse covariance matrix T = S -1 was assumed to follow a Wishart . Thanks for reading List of Continuous distributions. Department of Statistical Science. First, we develop a default version of the hyper-inverse Wishart prior for restricted covariance matrices, called the hyper-inverse Wishart g-prior, and show how it corresponds to the implied fractional prior for covariance selection using fractional Bayes factors. 81 10 [7] G. 5 The matrix normal, inverse Wishart distribution 315 10. Wishart and hyper inverse Wishart, homogeneous cones, natural exponential families. sample size and from (4. CARVALHO Institute of Statistics and Decision Sciences, Duke University, Durham, hyper-inverse Wishart distribution, and so allows us to work sequentially at the level of prime components. S. , c k = 3 and C k = 0. asked Jan 3 '15 at 8:48. Inference based on marginal likelihood requires the evaluation of a Cited by: 194Publish Year: 2002Author: Alberto RoveratoSimulation of hyper-inverse Wishart distributions - COREhttps://core. Siegel, Uber die analytische tehorie der quadratische Formen¨, Ann. The compound distribution resulting from compounding a matrix normal with an inverse matrix gamma prior over the covariance matrix is a generalized matrix t-distribution. The vector hyper-parameter µ0 is the prior mean and the the corresponding Wishart distribution has no Lebesgue density. The Wishart distribution is the probability distribution of the maximum-likelihood estimator Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential inverse-Wishart Hyper parameter for inverse chi square prior. CARVALHO. the hyper inverse Wishart distribution to introduce hyper Markov priors for correlation matri-ces. The hyper-inverse Wishart distribution is a general class of hyper-Markov laws introduced by Dawid & Lauritzen (1993) for a covariance matrix 2M(G), where G= (V;E) is a decomposable graph. 00297. Contents 1 DefinitionImproving Collaborative Filtering Recommendations Using External Data Akhmed Umyarov New York University aumyarov@stern. Inverse-Wishart distribution topic. Simulation of hyper-inverse Wishart distributions in graphical models, by Carlos Carvalho, Helene Massam and Mike West, Biometrika 2007 94:647-659. has an inverse Wishart distribution. Wishart分布和样本协方差阵的关系: 设n个独立同分布的 ,有统计量 . nyu. Wang, M. The Wishart distribution is related to the Inverse-Wishart distribution, denoted by , as follows: If X ~ W p (V, n) and if we do the change of variables C = X −1, then . 1. Further, in software like JAGS, the multivariate normal distribution is parameterized with a precision matrix, so you have to use the Wishart distribution. We propose an efficient solution to the problem of direct sampling from a hyper-inverse Wishart distribution in non-decomposable graphs. Both the hyper inverse Wishart priors and the “Letac-Massam” priors have attractive properties which enable Bayesian inference, with the latter allowing multiple shape parameters and hence suitable in high-dimensional settings. extending the use of hyper-inverse Wishart distributions. Initially we specified an Inverse-Wishart distribution Network meta-analysis of multiple outcome measures accounting for borrowing of information across outcomes. The prior for the hyper-inverse Wishart distribution is also set to be fairly vague, i. Joint high-dimensional Bayesian variable and covariance selection with an Hyper-inverse Wishart distribution; Joint Variable and Covariance Selection; Sparse Seemingly Unrelated Regression