sampling decomposable graphs using a markov chain on junction trees this paper makes two contributions to the computational geometry of decomposable graphs aimed primarily at facilitating statistical inference about such graphs where they arise as assumed conditional independence structures in stochastic models the rst of these provides sucient conditions under which it is possible to completely connect two disconnected cliques of vertices or perform the reverse procedure yet maintain decomposability of the graph the second is a new markov chain monte carlo sampler for arbitrary positive distributions on decomposable graphs taking a junction tree representing the graph as its state variable the resulting methodology is illustrated with two numerical experiments eld model determination

a monte carlo method for computing the marginal likelihood in nondecomposable gaussian graphical models summary a centred gaussian model that is markov with respect to an undirected graph g is characterised by the parameter set of its precision matrices which is the cone m g of positive definite matrices with entries corresponding to the missing edges of g constrained to be equal to zero in a bayesian framework the conjugate family for the precision parameter is the distribution with wishart density with respect to the lebesgue measure restricted to m g we call this distribution the g wishart when g is nondecomposable the normalising constant of the g wishart cannot be computed in closed form in this paper we give a simple monte carlo method for computing this normalising constant the main feature of our method is that the sampling distribution is exact and consists of a product of independent univariate standard normal and chi squared distributions that can be read off the graph g computing this normalising constant is necessary for obtaining the posterior distribution of g or the marginal likelihood of the corresponding graphical gaussian model our method also gives a way of sampling from the posterior distribution of the precision matrix

decomposable graphical gaussian model determination we propose a methodology for bayesian model determination in decomposable graphical gaussian models to achieve this aim we consider a hyper inverse wishart prior distribution on the concentration matrix for each given graph to ensure compatibility across models such prior distributions are obtained by marginalisation from the prior conditional on the complete graph we explore alternative structures for the hyperparameters of the latter and their consequences for the model model determination is carried out by implementing a reversible jump markov chain monte carlo sampler in particular the dimension changing move we propose involves adding or dropping an edge from the graph we characterise the set of moves which preserve the decomposability of the graph giving a fast algorithm for maintaining the junction tree representation of the graph at each sweep as state variable we use the incomplete variance covariance matrix containing only the elements for which the corresponding element of the inverse is nonzero this allows all computations to be performed locally at the clique level which is a clear advantage for the analysis of large and complex datasets finally the statistical and computational performance of the procedure is illustrated by mean of both artificial and real datasets

joint mean covariance models with applications to longitudinal data unconstrained parameterisation summary we provide unconstrained parameterisation for and model a covariance using covariates the cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix the entries of the lower triangular and the log of the diagonal matrix are unconstrained and have meaning as regression coefficients and prediction variances when regressing a measurement on its predecessors an extended generalised linear model is introduced for joint modelling of the vectors of predictors for the mean and covariance subsuming the joint modelling strategy for mean and variance heterogeneity gabriel s antedependence models dempster s covariance selection models and the class of graphical models the likelihood function and maximum likelihood estimators of the covariance and the mean parameters are studied when the observations are normally distributed applications to modelling nonstationary dependence structures and multivariate data are discussed and illustrated using real data a graphical method similar to that based on the correlogram in time series is developed and used to identify parametric models for nonstationary covariances

geometric representations of random hypergraphs abstractwe introduce a novel parameterization of distributions on hypergraphs based on the geometry of points in rd the idea is to induce distributions on hypergraphs by placing priors on point configurations via spatial processes this specification is then used to infer conditional independence models or markov structure for multivariate distributions this approach results in a broader class of conditional independence models beyond standard graphical models factorizations that cannot be retrieved via a graph are possible inference of nondecomposable graphical models is possible without requiring decomposability or the need of gaussian assumptions this approach leads to new metropolis hastings markov chain monte carlo algorithms with both local and global moves in graph space generally offers greater control on the distribution of graph features than currently possible and naturally extends to hypergraphs we provide a comparative performance evaluation against state of the art approaches and

isoperimetric and weingarten surfaces in the schwarzschild manifold we show that any star shaped convex hypersurface with constant weingarten curvature in the desitter schwarzschild manifold is a sphere of symmetry moreover we study an isoperimetric problem for bounded domains in the doubled schwarzschild manifold we prove the existence of an isoperimetric surface for any value of the enclosed volume and we completely describe the isoperimetric surfaces for very large enclosed volume this complements work in h bray s thesis where isoperimetric surfaces homologous to the horizon are studied

on the radius pinching estimate and uniqueness of the cmc foliation in asymptotically flat 3 manifolds abstract in this paper we consider the uniqueness problem of the constant mean curvature spheres in asymptotically flat 3 manifolds we require the metric to have the form g i j i j h i j with h i j o 4 r 1 and r o r 3 0 we do not require the metric to be close to schwarzschild metric in any sense or to satisfy rt conditions we prove that when the mass is not 0 stable cmc spheres that separate a certain compact part from the infinity satisfy the radius pinching estimate r 1 c r 0 which in many cases is critical to prove the uniqueness of the cmc spheres as applications of this estimate we remove the radius conditions of the uniqueness result in 8 and 15 in some special cases

shotgun stochastic search for large p regression model search in regression with very large numbers of candidate predictors raises challenges for both model specification and computation for which standard approaches such as markov chain monte carlo mcmc methods are often infeasible or ineffective we describe a novel shotgun stochastic search sss approach that explores interesting regions of the resulting high dimensional model spaces and quickly identifies regions of high posterior probability over models we describe algorithmic and modeling aspects priors over the model space that induce sparsity and parsimony over and above the traditional dimension penalization implicit in bayesian and likelihood analyses and parallel computation using cluster computers we discuss an example from gene expression cancer genomics comparisons with mcmc and other methods and theoretical and simulation based aspects of performance characteristics in large scale regression model searches we also provide software implementing the methods

estimation of a covariance matrix using the reference prior estimation of a covariance matrix sum is a notoriously difficult problem the standard unbiased estimator can be substantially suboptimal we approach the problem from a noninformative prior bayesian perspective developing the reference noninformative prior for a covariance matrix and obtaining expressions for the resulting bayes estimators these expressions involve the computation of high dimensional posterior expectations which is done using a recent markov chain simulation tool the hit and run sampler frequentist risk comparisons with previously suggested estimators are also given and determination of the accuracy of the estimators is addressed

hyper markov laws in the statistical analysis of decomposable graphical models this paper introduces and investigates the notion of a hyper markov law which is a probability distribution over the set of probability measures on a multivariate space that i is concentrated on the set of markov probabilities over some decomposable graph and ii satisfies certain conditional independence restrictions related to that graph a stronger version of this hyper markov property is also studied our analysis starts by reconsidering the properties of markov probabilities using an abstract approach which thereafter proves equally applicable to the hyper markov case next it is shown constructively that hyper markov laws exist that they appear as sampling distributions of maximum likelihood estimators in decomposable graphical models and also that they form natural conjugate prior distributions for a bayesian analysis of these models 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 these laws occur naturally in connection with the analysis of decomposable log linear and covariance selection models

a nonparametric prior for simultaneous covariance estimation in the modeling of longitudinal data from several groups appropriate handling of the dependence structure is of central importance standard methods include specifying a single covariance matrix for all groups or independently estimating the covariance matrix for each group without regard to the others but when these model assumptions are incorrect these techniques can lead to biased mean effects or loss of efficiency respectively thus it is desirable to develop methods to simultaneously estimate the covariance matrix for each group that will borrow strength across groups in a way that is ultimately informed by the data in addition for several groups with covariance matrices of even medium dimension it is difficult to manually select a single best parametric model among the huge number of possibilities given by incorporating structural zeros and or commonality of individual parameters across groups in this paper we develop a family of nonparametric priors using the matrix stick breaking process of dunson et al 2008 that seeks to accomplish this task by parameterizing the covariance matrices in terms of the parameters of their modified cholesky decomposition pourahmadi 1999 we establish some theoretic properties of these priors examine their effectiveness via a simulation study and illustrate the priors using data from a longitudinal clinical trial

covariance matrix selection and estimation via penalised normal likelihood summary we propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix we reparameterise a covariance matrix through the modified cholesky decomposition of its inverse or the one step ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models the cholesky factor containing these regression coefficients is likely to have many off diagonal elements that are zero or close to zero penalised normal likelihoods in this situation with l1 and l2 penalities are shown to be closely related to tibshirani s 1996 lasso approach and to ridge regression adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the cholesky factor while because of its singularity the l1 penalty will set some elements to zero and produce interpretable models an algorithm is developed for computing the estimator and selecting the tuning parameter the proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 x 102 covariance matrix

bayesian inference for a covariance matrix a flexible class of prior distributions is proposed for the covariance matrix of a multivariate normal distribution yielding much more general hierarchical and empirical bayes smoothing and inference when compared with a conjugate analysis involving an inverted wishart distribution a likelihood approximation is obtained for the matrix logarithm of the covariance matrix via bellman s iterative solution to a volterra integral equation exact and approximate bayesian empirical and hierarchical bayesian estimation and finite sample inference techniques are developed some risk and asymptotic frequency properties are investigated a subset of the project talent american high school data is analyzed applications and extensions to multivariate analysis including a generalized linear model for covariance matrices are indicated

nonconjugate bayesian estimation of covariance matrices and its use in hierarchical models abstract the problem of estimating a covariance matrix in small samples has been considered by several authors following early work by stein this problem can be especially important in hierarchical models where the standard errors of fixed and random effects depend on estimation of the covariance matrix of the distribution of the random effects we propose a set of hierarchical priors hps for the covariance matrix that produce posterior shrinkage toward a specified structure here we examine shrinkage toward diagonality we then address the computational difficulties raised by incorporating these priors and nonconjugate priors in general into hierarchical models we apply a combination of approximation gibbs sampling possibly with a metropolis step and importance reweighting to fit the models and compare this hybrid approach to alternative markov chain monte carlo methods our investigation involves three alternative hps the first works with the spectral decomposition of the covariance matrix an

equivalent relaxations of optimal power flow several convex relaxations of the optimal power flow opf problem have recently been developed using both bus injection models and branch flow models in this paper we prove relations among three convex relaxations a semidefinite relaxation that computes a full matrix a chordal relaxation based on a chordal extension of the network graph and a second order cone relaxation that computes the smallest partial matrix we prove a bijection between the feasible sets of the opf in the bus injection model and the branch flow model establishing the equivalence of these two models and their second order cone relaxations our results imply that for radial networks all these relaxations are equivalent and one should always solve the second order cone relaxation for mesh networks the semidefinite relaxation and the chordal relaxation are equally tight and both are strictly tighter than the second order cone relaxation therefore for mesh networks one should either solve the chordal relaxation or the socp relaxation trading off tightness and the required computational effort simulations are used to illustrate these results

characterization of large isoperimetric regions in asymptotically hyperbolic initial data let m g be a complete riemannian 3 manifold asymptotic to schwarzschild anti desitter and with scalar curvature r geq 6 building on work of a neves and g tian and of the first named author we show that the leaves of the canonical foliation of m g are the unique solutions of the isoperimetric problem for their area the assumption r geq 6 is necessary this is the first characterization result for large isoperimetric regions in the asymptotically hyperbolic setting that does not assume exact rotational symmetry at infinity

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

parsimonious covariance matrix estimation for longitudinal data this article proposes a data driven method to identify parsimony in the covariance matrix of longitudinal data and to exploit any such parsimony to produce a statistically efficient estimator of the covariance matrix the approach parameterizes the covariance matrix through the cholesky decomposition of its inverse for longitudinal data this is a one step ahead predictive representation and the cholesky factor is likely to have off diagonal elements that are zero or close to zero a hierarchical bayesian model is used to identify any such zeros in the cholesky factor similar to approaches that have been successful in bayesian variable selection the model is estimated using a markov chain monte carlo sampling scheme that is computationally efficient and can be applied to covariance matrices of high dimension it is demonstrated through simulations that the proposed method compares favorably in terms of statistical efficiency with a highly regarded competing approach the estimator is applied to three

state space theory of linear time invariant systems with delays in state control and observation variables ii abstract part i of this paper studies a coupled mathematical system which provides a good model for important families of linear time invariant hereditary systems delay differential equations integro differential equations volterra stieltjes integral equations functional differential equations of retarded and neutral types etc appropriate states are constructed and associated semigroups and abstract differential equations are obtained in part ii we emphasize the structural operator approach as in delfour and manitius control operators are added to the coupled mathematical system allowing delays in the control variables again structural operators are introduced to define the state and obtain abstract differential equations without delays in the control variable as in the work of vinter and kwong finally observation operators are added which allow for delays in the observation variable and delayed control variables in the observation equation again a new state and a state equations are constructed in such a way that no delay appears in the new observation operator thus generalizing the construction of a pritchard and d salamon

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

bayesian covariance selection in generalized linear mixed models repeated measures and longitudinal data are commonly collected for analysis in epidemiology clinical trials biology sociology and economic sciences in such studies a response is measured repeatedly over time for each subject under study and the number and timing of the observations often varies among subjects in contrast to cross sectional studies that collect a single measurement per subject longitudinal studies have the extra complication of within subject dependence in the repeated measures such dependence can be thought to arise due to the impact of unmeasured predictors main effects of unmeasured predictors lead to variation in the average level of response among subjects while interactions with measured predictors lead to heterogeneity in the regression coefficients this justification has motivated random effects models which allow the intercept and slopes in a regression model to be subject specific random effects models are broadly useful for modeling of dependence not only for longitudinal data but also in multicenter studies meta analysis and functional data analysis

large isoperimetric surfaces in initial data sets we study the isoperimetric structure of asymptotically flat riemannian 3 manifolds m g that are mathcal c 0 asymptotic to schwarzschild of mass m gt 0 refining an argument due to h bray we obtain an effective volume comparison theorem in schwarzschild we use it to show that isoperimetric regions exist in m g for all sufficiently large volumes and that they are close to centered coordinate spheres this implies that the volume preserving stable constant mean curvature spheres constructed by g huisken and s t yau as well as r ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume this confirms a conjecture of h bray our results are consistent with the uniqueness results for volume preserving stable constant mean curvature surfaces in initial data sets obtained by g huisken and s t yau and strengthened by j qing and g tian the additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work

extremal bipartite matrices abstract let g be an undirected graph on vertices 1 n let m g be the convex cone of all positive semidefinite hermitian matrices a satisfying aij 0 if i j is not an edge of g in the case that g is a complete bipartite graph we characterize all extreme rays of m g in doing so we shall also find the maximum rank that an extreme matrix in m g can have

the real positive definite completion problem for a simple cycle abstract we consider the question of whether a real partial positive definite matrix in which the specified off diagonal entries consist of a full n cycle has a positive definite completion this lies in contrast to the previously studied chordal case we give two solutions in one we describe about n 2 independent conditions on angles associated with a normalization of the data that are necessary and sufficient the second is more computational and allows presentation of all positive definite completions as well as answering the existence question

inferring network structure in non normal and mixed discrete continuous genomic data summary inferring dependence structure through undirected graphs is crucial for uncovering the major modes of multivariate interaction among high dimensional genomic markers that are potentially associated with cancer traditionally conditional independence has been studied using sparse gaussian graphical models for continuous data and sparse ising models for discrete data however there are two clear situations when these approaches are inadequate the first occurs when the data are continuous but display non normal marginal behavior such as heavy tails or skewness rendering an assumption of normality inappropriate the second occurs when a part of the data is ordinal or discrete e g presence or absence of a mutation and the other part is continuous e g expression levels of genes or proteins in this case the existing bayesian approaches typically employ a latent variable framework for the discrete part that precludes inferring conditional independence among the data that are actually observed the current article overcomes these two challenges in a unified framework using gaussian scale mixtures our framework is able to handle continuous data that are not normal and data that are of mixed continuous and discrete nature while still being able to infer a sparse conditional sign independence structure among the observed data extensive performance comparison in simulations with alternative techniques and an analysis of a real cancer genomics data set demonstrate the effectiveness of the proposed approach

an improved estimation to make markowitz s portfolio optimization theory users friendly and estimation accurate with application on the us stock market investment abstract using the markowitz mean variance portfolio optimization theory researchers have shown that the traditional estimated return greatly overestimates the theoretical optimal return especially when the dimension to sample size ratio p n is large bai et al 2009 propose a bootstrap corrected estimator to correct the overestimation but there is no closed form for their estimator to circumvent this limitation this paper derives explicit formulas for the estimator of the optimal portfolio return we also prove that our proposed closed form return estimator is consistent when n and p n y 0 1 our simulation results show that our proposed estimators dramatically outperform traditional estimators for both the optimal return and its corresponding allocation under different values of p n ratios and different inter asset correlations especially when p n is close to 1 we also find that our proposed estimators perform better than the bootstrap corrected estimators for both the optimal return and its corresponding allocation another advantage of our improved estimation of returns is that we can also obtain an explicit formula for the standard deviation of the improved return estimate and it is smaller than that of the traditional estimate especially when p n is large in addition we illustrate the applicability of our proposed estimate on the us stock market investment

bayesian hierarchical spatially correlated functional data analysis with application to colon carcinogenesis in this article we present new methods to analyze data from an experiment using rodent models to investigate the role of p27 an important cell cycle mediator in early colon carcinogenesis the responses modeled here are essentially functions nested within a two stage hierarchy standard functional data analysis literature focuses on a single stage of hierarchy and conditionally independent functions with near white noise however in our experiment there is substantial biological motivation for the existence of spatial correlation among the functions which arise from the locations of biological structures called colonic crypts this possible functional correlation is a phenomenon we term crypt signaling thus as a point of general methodology we require an analysis that allows for functions to be correlated at the deepest level of the hierarchy our approach is fully bayesian and uses markov chain monte carlo methods for inference and estimation analysis of this data set gives new insights into the structure of p27 expression in early colon carcinogenesis and suggests the existence of significant crypt signaling our methodology uses regression splines and because of the hierarchical nature of the data dimension reduction of the covariance matrix of the spline coefficients is important we suggest simple methods for overcoming this problem

large scale semidefinite programming via a saddle point mirror prox algorithm in this paper we first demonstrate that positive semidefiniteness of a large well structured sparse symmetric matrix can be represented via positive semidefiniteness of a bunch of smaller matrices linked in a linear fashion to the matrix we derive also the dual counterpart of the outlined representation which expresses the possibility of positive semidefinite completion of a well structured partially defined symmetric matrix in terms of positive semidefiniteness of a specific bunch of fully defined submatrices of the matrix using the representations we then reformulate well structured large scale semidefinite problems into smooth convex concave saddle point problems which can be solved by a prox method developed in 6 with efficiency mathcal o epsilon 1 implementations and some numerical results for large scale lovasz capacity and maxcut problems are finally presented

sampled data h8 state feedback control of systems with state delays in the present paper we adopt the second approach to the nite horizon sampled data state delay h 1 control problem note that in the case of state delays the lifting technique leads to complicated discrete system with in nite dimensional state variable we obtain the piecewise constant state feedback controllers by solving coupled rpdes between sampling and updating the terminal conditions at the sampling instants we derive an asymptotic approximation to the solution of these rpdes by expanding it in the powers of the largest delay the resulting approximation is obtained by solving uncoupled low order partial dia erential equations the performance of the system with the controller that has been obtained using the zero approximation the one that corresponds to zero delay is analysed when the open loop system possesses a non zero delay bounded real lemmas for systems with state delays and jumps are obtained 2 problem formulation

quadratic control of evolution equations with delays in control the quadratic cost problem of evolution equations with delays in control is considered a semigroup model which involves no explicit delays in control but contains an unbounded control operator is introduced with the aid of a family of approximating systems it is shown that the optimal feedback control and the minimum cost are characterized by the solution of a riccati equation three examples are given to illustrate the theory the filtering problem of evolution equations with observation delays is also solved through the duality relation

unique isoperimetric foliations of asymptotically flat manifolds in all dimensions the question of isoperimetry what is the largest amount of volume that can be enclosed by a given amount of area can be traced back to antiquity 1 the first mathematically rigorous results are as recent as the nineteenth century the question of isoperimetry and its close relative the analysis of minimal surfaces are two of the model problems of the geometric calculus of variations the list of geometries where an explicit answer to the question of isoperimetry is available is short we provide an overview of available results in appendix h in this paper and 22 we extend this list by a class of riemannian manifolds m g for which we describe all large isoperimetric regions completely we refer the reader to sect 2 for the precise definitions of all terms in the statement of our first main theorem

blow up phenomena for the yamabe equation it is well known that the pde 1 has at least one positive solution for any choice of m g if n 6 and m g is not locally conformally flat this follows from results of t aubin 3 the remaining cases were solved by r schoen using the positive mass theorem 16 solutions to 1 are not usually unique as an example consider the product metric on 51 l x 5n 1 l if l is sufficiently small then the yamabe pde has a unique solution on the other hand there are many non minimizing solutions if l is large d pollack 14 has used gluing techniques to construct high energy solutions on more general background manifolds given any conformai class with positive yamabe constant and any positive integer iv there exists a new conformai class which is close to the original one in the c norm and contains at least n metrics of constant scalar curvature see 14 theorem 0 1 it is an interesting question whether the set of all solutions to the yamabe pde is compact in the c2 topology say a well known conjecture states that this should be true unless m p is conformally equivalent to the round sphere see 17 18 19 this conjecture has been verified in low dimensions and in the locally conformally flat case if m g is locally conformally flat compactness follows from work of r schoen 17 18 moreover schoen proposed a strategy for proving the conjecture in the non locally conformally flat case based on the pohozaev identity

the real positive semidefinite completion problem for series parallel graphs abstract we consider the partial real symmetric matrices x whose diagonal entries are equal to 1 and whose off diagonal entries are specified only on a subset of the positions the question is to determine whether x can be completed to a positive semidefinite matrix extending a result of barrett et al 3 we give a set of necessary conditions for x to be completable and show that these conditions are also sufficient if and only if the graph corresponding to the positions of the specified entries is series parallel i e has no k 4 minor

clustering and variable selection in the presence of mixed variable types and missing data we consider the problem of model based clustering in the presence of many correlated mixed continuous and discrete variables some of which may have missing values discrete variables are treated with a latent continuous variable approach and the dirichlet process is used to construct a mixture model with an unknown number of components variable selection is also performed to identify the variables that are most influential for determining cluster membership the work is motivated by the need to cluster patients thought to potentially have autism spectrum disorder on the basis of many cognitive and or behavioral test scores there are a modest number of patients 486 in the data set along with many 55 test score variables many of which are discrete valued and or missing the goal of the work is to 1 cluster these patients into similar groups to help identify those with similar clinical presentation and 2 identify a sparse subset of tests that inform the clusters in order to eliminate unnecessary testing the proposed approach compares very favorably with other methods via simulation of problems of this type the results of the autism spectrum disorder analysis suggested 3 clusters to be most likely while only 4 test scores had high 0 5 posterior probability of being informative this will result in much more efficient and informative testing the need to cluster observations on the basis of many correlated continuous discrete variables with missing values is a common problem in the health sciences as well as in many other disciplines

efficient bayesian inference for gaussian copula regression models a gaussian copula regression model gives a tractable way of handling a multivariate regression when some of the marginal distributions are non gaussian our paper presents a general bayesian approach for estimating a gaussian copula model that can handle any combination of discrete and continuous marginals and generalises gaussian graphical models to the gaussian copula framework posterior inference is carried out using a novel and efficient simulation method the methods in the paper are applied to simulated and real data

efficient estimation of covariance selection models summary a bayesian method is proposed for estimating an inverse covariance matrix from gaussian data the method is based on a prior that allows the off diagonal elements of the inverse covariance matrix to be zero and in many applications results in a parsimonious parameterisation of the covariance matrix no assumption is made about the structure of the corresponding graphical model so the method applies to both nondecomposable and decomposable graphs all the parameters are estimated by model averaging using an efficient metropolis hastings sampling scheme a simulation study demonstrates that the method produces statistically efficient estimators of the covariance matrix when the inverse covariance matrix is sparse the methodology is illustrated by applying it to three examples that are high dimensional relative to the sample size

the linear quadratic control problem for infinite dimensional systems with unbounded input and output operators this paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators

brief paper h2 norm of linear time periodic systems a perturbation analysis we consider a class of linear time periodic systems in which the dynamical generator a t represents the sum of a stable time invariant operator a 0 and a small amplitude zero mean t periodic operator ea p t we employ a perturbation analysis to develop a computationally efficient method for determination of the h 2 norm up to second order in the perturbation parameter e we show that a the h 2 norm can be obtained from a conveniently coupled system of lyapunov and sylvester equations that are of the same dimension as a 0 b there is no coupling between different harmonics of a p t in the expression for the h 2 norm these two properties do not hold for arbitrary values of e and their derivation would not be possible if we tried to determine the h 2 norm directly without resorting to perturbation analysis our method is well suited for identification of the values of period t that lead to the largest increase reduction of the h 2 norm two examples are provided to motivate the developments and illustrate the procedure

simulation of hyper inverse wishart distributions in graphical models summary we introduce and exemplify an efficient method for direct sampling from hyperinverse 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

bayesian correlation estimation we propose prior probability models for variance covariance matrices in order to address two important issues first the models allow a researcher to represent substantive prior information about the strength of correlations among a set of variables secondly even in the absence of such information the increased flexibility of the models mitigates dependence on strict parametric assumptions in standard prior models for example the model allows a posteriori different levels of uncertainty about correlations among different subsets of variables we achieve this by including a clustering mechanism in the prior probability model clustering is with respect to variables and pairs of variables our approach leads to shrinkage towards a mixture structure implied by the clustering we discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models including the evaluation of normalising constants that are functions of parameters of interest the normalising constants result from the restriction that the correlation matrix be positive definite we discuss examples based on simulated data a stock return dataset and a population genetics dataset

a new approach to cholesky based covariance regularization in high dimensions summary in this paper we propose a new regression interpretation of the cholesky factor of the covariance matrix as opposed to the well known regression interpretation of the cholesky factor of the inverse covariance which leads to a new class of regularized covariance estimators suitable for high dimensional problems regularizing the cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator in particular one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estimator of bickel levina 2008b which is not guaranteed to be positive definite we also establish theoretical connections between banding cholesky factors of the covariance matrix and its inverse and constrained maximum likelihood estimation under the banding constraint and compare the numerical performance of several methods in simulations and on a sonar data example

on l _ q optimization and sparse inverse covariance selection graphical models are well established in providing meaningful conditional probability descriptions of complex multivariable interactions in the gaussian case the conditional independencies between different variables correspond to zero entries in the precision inverse covariance matrix hence there has been much recent interest in sparse precision matrix estimation in areas such as statistics machine learning computer vision pattern recognition and signal processing a popular estimation method involves optimizing a penalized log likelihood problem the penalty is responsible for inducing sparsity and a common choice is the convex l1 norm even though the l0 penalty is the natural choice guaranteeing maximum sparsity it has been avoided due to lack of convexity as a result in this paper we bridge the gap between these two penalties and propose the non concave lq penalized log likelihood problem for sparse precision matrix estimation where 0 q 1 a novel algorithm is developed for the optimization and we provide some of its theoretic properties that are useful in sparse linear regression we illustrate on synthetic and real data showing reconstruction quality comparisons of sparsity inducing penalties l0 lq with 0 q 1 l1 and scad

bayesian semiparametric regression models for evaluating pathway effects on continuous and binary clinical outcomes many statistical methods for microarray data analysis consider one gene at a time and they may miss subtle changes at the single gene level this limitation may be overcome by considering a set of genes simultaneously where the gene sets are derived from prior biological knowledge limited work has been carried out in the regression setting to study the effects of clinical covariates and expression levels of genes in a pathway either on a continuous or on a binary clinical outcome hence we propose a bayesian approach for identifying pathways related to both types of outcomes we compare our bayesian approaches with a likelihood based approach that was developed by relating a least squares kernel machine for nonparametric pathway effect with a restricted maximum likelihood for variance components unlike the likelihood based approach the bayesian approach allows us to directly estimate all parameters and pathway effects it can incorporate prior knowledge into bayesian hierarchical model formulation and makes inference by using the posterior samples without asymptotic theory we consider several kernels gaussian polynomial and neural network kernels to characterize gene expression effects in a pathway on clinical outcomes our simulation results suggest that the bayesian approach has more accurate coverage probability than the likelihood based approach and this is especially so when the sample size is small compared with the number of genes being studied in a pathway we demonstrate the usefulness of our approaches through its applications to a type ii diabetes mellitus data set our approaches can also be applied to other settings where a large number of strongly correlated predictors are present

solving log determinant optimization problems by a newton cg primal proximal point algorithm we propose a newton cg primal proximal point algorithm ppa for solving large scale log determinant optimization problems our algorithm employs the essential ideas of ppa the newton method and the preconditioned cg solver when applying the newton method to solve the inner subproblem we find that the log determinant term plays the role of a smoothing term as in the traditional smoothing newton technique focusing on the problem of maximum likelihood sparse estimation of a gaussian graphical model we demonstrate that our algorithm performs favorably compared to existing state of the art algorithms and is much preferred when a high quality solution is required for problems with many equality constraints

a robust procedure for gaussian graphical model search from microarray data with p larger than n learning of large scale networks of interactions from microarray data is an important and challenging problem in bioinformatics a widely used approach is to assume that the available data constitute a random sample from a multivariate distribution belonging to a gaussian graphical model as a consequence the prime objects of inference are full order partial correlations which are partial correlations between two variables given the remaining ones in the context of microarray data the number of variables exceed the sample size and this precludes the application of traditional structure learning procedures because a sampling version of full order partial correlations does not exist in this paper we consider limited order partial correlations these are partial correlations computed on marginal distributions of manageable size and provide a set of rules that allow one to assess the usefulness of these quantities to derive the independence structure of the underlying gaussian graphical model furthermore we introduce a novel structure learning procedure based on a quantity obtained from limited order partial correlations that we call the non rejection rate the applicability and usefulness of the procedure are demonstrated by both simulated and real data

the sensitivity of optimal control problems to time delay a general class of time delayed optimal control problems is studied the goal is to characterize in terms of the multipliers occurring in the necessary conditions the rate of change of the value of the problem with respect to change in the delay in particular criteria for this dependence to be differentiable are given as well as a formula for the derivative these appear to be the first results of this type

on the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3 manifolds in this note we study constant mean curvature surfaces in asymptotically flat 3 manifolds we prove that outside a given compact subset in an asymptotically flat 3 manifold with positive mass stable spheres of given constant mean curvature are unique therefore we are able to conclude that the foliation of stable spheres of constant mean curvature in an asymptotically flat 3 manifold with positive mass outside a given compact subset is unique

the linear quadratic optimal control problem with delays in state and control variables a state space approach this paper generalizes the results of vinter and kwong to a control operator with a more general delay structure than the one in a ichikawa and to delay systems with both finite and infinite memories the linear quadratic optimal control problem over both finite and infinite time horizons is treated several forms of associated abstract riccati differential and algebraic equations are presented the properties of the kernel of the solution are studied and a coupled set of matrix ordinary and partial differential equations is obtained

maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix the positive definiteness constraint is the most awkward stumbling block in modelling the covariance matrix pourahmadi s 1999 unconstrained parameterisation models covariance using covariates in a similar manner to mean modelling in generalised linear models the new covariance parameters have statistical interpretation as the regression coefficients and logarithms of prediction error variances corresponding to regressing a response on its predecessors in this paper the maximum likelihood estimators of the parameters of a generalised linear model for the covariance matrix their consistency and their asymptotic normality are studied when the observations are normally distributed these results along with the likelihood ratio test and penalised likelihood criteria such as bic for model and variable selection are illustrated using a real dataset

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

finite horizon h sub spl infin state feedback control of continuous time systems with state delays the finite horizon h sub spl infin control of time invariant linear systems with a finite number of point and distributed time delays is considered the controller is obtained by solving coupled riccati type partial differential equations the solutions to these equations and the resulting controllers are approximated by series expansions in powers of the largest delay unlike the infinite horizon case these approximations possess both regular and boundary layer terms the performance of the closed loop system under a memoryless zero approximation controller is analyzed

delay sensitivity of quadratic controllers a singular perturbation approach we study the variations of the quadratic performance associated to a linear differential system of retarded type for small values of the delays from an interpretation of delays as singular perturbations of abstract evolution operators we revisit the usual theory of representation and optimal control of retarded systems this leads to a new parameterization of associated riccati operators for which insight is gained in the dependence on the delays this explicit parameterization of riccati operators by the delays enables us to prove differentiability at zero for performance viewed as a function of the delays in the lq optimal or cal h infty suboptimal control in each case the gradient is explicitly computed in terms of the nonnegative solution of the finite dimensional riccati equation associated to the nondelay control problem a thorough treatment is stated for the linear quadratic optimal case and the cal h infty suboptimal control is presented as an application

bayesian precision and covariance matrix estimation for graphical gaussian models with edge and vertex symmetries summarygraphical gaussian models with edge and vertex symmetries were introduced by hojsgaard lauritzen 2008 who gave an algorithm for computing the maximum likelihood estimate of the precision matrix for such models in this paper we take a bayesian approach to its estimation we consider only models with symmetry constraints and which thus form a natural exponential family with the precision matrix as the canonical parameter we identify the diaconis ylvisaker conjugate prior for these models develop a scheme to sample from the prior and posterior distributions and thus obtain estimates of the posterior mean of the precision and covariance matrices such a sampling scheme is essential for model selection in coloured graphical gaussian models in order to verify the precision of our estimates we derive an analytic expression for the expected value of the precision matrix when the graph underlying our model is a tree a complete graph on three vertices or a decomposable graph on four vertices with various symmetries and we compare our estimates with the posterior mean of the precision matrix and the expected mean of the coloured graphical gaussian model that is of the covariance matrix we also verify the accuracy of our estimates on simulated data

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

large isoperimetric regions in asymptotically hyperbolic manifolds we show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds furthermore we show that large coordinate spheres are uniquely isoperimetric in manifolds that are schwarzschild anti desitter at infinity these results have important repercussions for our understanding of spacelike hypersurfaces in lorentzian space times which are asymptotic to null infinity in fact as an application of our results we verify the asymptotically hyperbolic penrose inequality in the special case of the existence of connected isoperimetric regions of all volumes

a block coordinate gradient descent method for regularized convex separable optimization and covariance selection we consider a class of unconstrained nonsmooth convex optimization problems in which the objective function is the sum of a convex smooth function on an open subset of matrices and a separable convex function on a set of matrices this problem includes the covariance selection problem that can be expressed as an l 1 penalized maximum likelihood estimation problem in this paper we propose a block coordinate gradient descent method abbreviated as bcgd for solving this class of nonsmooth separable problems with the coordinate block chosen by a gauss seidel rule the method is simple highly parallelizable and suited for large scale problems we establish global convergence and under a local lipschizian error bound assumption linear rate of convergence for this method for the covariance selection problem the method can terminate in o n 3 epsilon iterations with an epsilon optimal solution we compare the performance of the bcgd method with the first order methods proposed by lu siam j optim 19 1807 1827 2009 siam j matrix anal appl 31 2000 2016 2010 for solving the covariance selection problem on randomly generated instances our numerical experience suggests that the bcgd method can be efficient for large scale covariance selection problems with constraints

effective versions of the positive mass theorem the study of stable minimal surfaces in riemannian 3 manifolds m g with non negative scalar curvature has a rich history in this paper we prove rigidity of such surfaces when m g is asymptotically flat and has horizon boundary as a consequence we obtain an effective version of the positive mass theorem in terms of isoperimetric or more generally closed volume preserving stable cmc surfaces that is appealing from both a physical and a purely geometric point of view we also include a proof of the following conjecture of schoen an asymptotically flat riemannian 3 manifold with non negative scalar curvature that contains an unbounded area minimizing surface is isometric to flat mathbb r 3

large outlying stable constant mean curvature spheres in initial data sets we give examples of asymptotically flat three manifolds m g which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold this resolves a question raised by huisken and yau invent math 124 281 311 1996 on the other hand we show that such surfaces cannot exist when m g has nonnegative scalar curvature this result depends on an intricate relationship between the scalar curvature of the initial data set and the isoperimetric ratio of large stable constant mean curvature surfaces

cholesky decomposition of a hyper inverse wishart matrix the canonical parameter of a covariance selection model is the inverse covariance matrix 1 whose zero pattern gives the conditional independence structure characterising the model in this paper we consider the upper triangular matrix obtained by the cholesky decomposition 1 t this provides an interesting alternative parameterisation of decomposable models since its upper triangle has the same zero structure as 1 and its elements have an interpretation as parameters of certain conditional distributions for a distribution for the strong hyper markov property is shown to be characterised by the mutual independence of the rows of 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

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

objective bayesian model selection in gaussian graphical models summary this paper presents a default model selection procedure for gaussian graphical models that involves two new developments 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 second we apply a class of priors that automatically handles the problem of multiple hypothesis testing we demonstrate our methods on a variety of simulated examples concluding with a real example analyzing covariation in mutual fund returns these studies reveal that the combined use of a multiplicity correction prior on graphs and fractional bayes factors for computing marginal likelihoods yields better performance than existing bayesian methods

on the sparsity of signals in a random sample this article proposes a method of moments technique for estimating the sparsity of signals in a random sample this involves estimating the largest eigenvalue of a large hermitian trigonometric matrix under mild conditions as illustration the method is applied to two well known problems the first focuses on the sparsity of a large covariance matrix and the second investigates the sparsity of a sequence of signals observed with stationary weakly dependent noise simulation shows that the proposed estimators can have significantly smaller mean absolute errors than their main competitors copyright 2012 oxford university press

on large volume preserving stable cmc surfaces in initial data sets let m g be a complete 3 dimensional asymptotically flat manifold with everywhere positive scalar curvature we prove that given a compact subset k subset m all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid k this complements the results of g huisken and s t yau and of j qing and g tian on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to schwarzschild with mass m gt 0 the analysis in g huisken and s t yau and in j qing and g tian takes place in the asymptotic regime of m here we adapt ideas from the minimal surface proof of the positive mass theorem by r schoen and s t yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of m that is far from euclidean

extensions of positive definite functions on free groups abstract an analogue of krein s extension theorem is proved for operator valued positive definite functions on free groups the proof gives also the parametrization of all extensions by means of a generalized type of szego parameters one singles out a distinguished completion called central which is related to quasi multiplicative positive definite functions an application is given to factorization of noncommutative polynomials