discrete riemann surfaces and the ising model we define a new theory of discrete riemann surfaces and present its basic results the key idea is to consider not only a cellular decomposition of a surface but the union with its dual discrete holomorphy is defined by a straightforward discretisation of the cauchy riemann equation a lot of classical results in riemann theory have a discrete counterpart hodge star harmonicity hodge theorem weyl s lemma cauchy integral formula existence of holomorphic forms with prescribed holonomies giving a geometrical meaning to the construction on a riemann surface we define a notion of criticality on which we prove a continuous limit theorem we investigate its connection with criticality in the ising model we set up a dirac equation on a discrete universal spin structure and we prove that the existence of a dirac spinor is equivalent to criticality

bayesian inference for gaussian graphical models beyond decomposable graphs bayesian inference for graphical models has received much attention in the literature in recent years it is well known that when the graph g is decomposable bayesian inference is significantly more tractable than in the general non decomposable setting penalized likelihood inference in contrast has made tremendous gains in the past few years in terms of scalability and tractability bayesian inference however has not had the same level of success though a scalable bayesian approach has its strengths especially in terms of quantifying uncertainty to address this gap we propose a scalable and flexible novel bayesian approach for estimation and model selection in gaussian undirected graphical models we first develop a class of generalized g wishart distributions with multiple shape parameters for an arbitrary underlying graph this class contains the g wishart distribution as a special case we then introduce the class of generalized bartlett graphs and derive an efficient gibbs sampling algorithm to obtain posterior draws from generalized g wishart distributions corresponding to a generalized bartlett graph the class of generalized bartlett graphs contains the class of decomposable graphs as a special case but is substantially larger than the class of decomposable graphs we proceed to derive theoretical properties of the proposed gibbs sampler we then demonstrate that the proposed gibbs sampler is scalable to significantly higher dimensional problems compared with using an accept reject or a metropolis hasting algorithm finally we show the efficacy of the proposed approach on simulated and real data in particular we demonstrate that our generalized bartlett methodology can be used for efficient model selection by reducing the graph search space by using penalized likelihood and pseudolikelihood methods

total positivity in markov structures we discuss properties of distributions that are multivariate totally positive of order two mtp2 related to conditional independence in particular we show that any independence model generated by an mtp2 distribution is a compositional semi graphoid which is upward stable and singletontransitive in addition we prove that any mtp2 distribution satisfying an appropriate support condition is faithful to its concentration graph finally we analyze factorization properties of mtp2 distributions and discuss ways of constructing mtp2 distributions in particular we give conditions on the log linear parameters of a discrete distribution which ensure mtp2 and characterize conditional gaussian distributions which satisfy mtp2

a characterization of markov equivalence classes for acyclic digraphs undirected graphs and acyclic digraphs adgs as well as their mutual extension to chain graphs are widely used to describe dependencies among variables in multivariate distributions in particular the likelihood functions of adg models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building bayesian networks for expert systems whereas the undirected graph associated with a dependence model is uniquely determined there may however be many adgs that determine the same dependence markov model thus the family of all adgs with a given set of vertices is naturally partitioned into markov equivalence classes each class being associated with a unique statistical model statistical procedures such as model selection or model averaging that fail to take into account these equivalence classes may incur substantial computational or other inefficiencies here it is shown that each markov equivalence class is uniquely determined by a single chain graph the essential graph that is itself simultaneously markov equivalent to all adgs in the equivalence class essential graphs are characterized a polynomial time algorithm for their construction is given and their applications to model selection and other statistical questions are described

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

invariant causal prediction for sequential data we investigate the problem of inferring the causal predictors of a response y from a set of d explanatory variables x1 xd classical ordinary least square regression includes all predictors t

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

group identities on units of group algebras abstract let u be the group of units of the group algebra fg of a group g over a field f suppose that either f is infinite or g has an element of infinite order we characterize groups g so that u satisfies a group identity under the assumption that g modulo the torsion elements is nilpotent this gives a complete classification of such groups for torsion groups this problem has already been settled in recent years

fast bayes and the dynamic junction forest abstract it has been shown that junction tree algorithms can provide a quick and efficient method for propagating probabilities in complex multivariate problems when they can be described by a fixed conditional independence structure in this paper we formalise and illustrate with two practical examples how these probabilistic propagation algorithms can be applied to high dimensional processes whose conditional independence structure as well as their underlying distributions are augmented through the passage of time

a numerical adaptation of self avoiding walk identities from the honeycomb to other 2d lattices recently duminil copin and smirnov proved a long standing conjecture of nienhuis that the connective constant of self avoiding walks saws on the honeycomb lattice is

a conjugate prior for discrete hierarchical log linear models in the bayesian analysis of contingency table data the selection of a prior distribution for either the log linear parameters or the cell probabilities parameter is a major challenge though the conjugate prior on cell probabilities has been defined by dawid and lauritzen 1993 for decomposable graphical models it has not been identified for the larger class of graphical models markov with respect to an arbitrary undirected graph or for the even wider class of hierarchical log linear models in this paper working with the log linear parameters used by glim we first define the conjugate prior for these parameters and then derive the induced prior for the cell probabilities this is done for the general class of hierarchical log linear models we show that the conjugate prior has all the properties that one expects from a prior notational simplicity ability to reflect either no prior knowledge or a priori expert knowledge a moderate number of hyperparameters and mathematical convenience it also has the strong hyper markov property which allows for local updates within prime components for graphical models

sparse multi dimensional graphical models a unified bayesian framework abstractmulti dimensional data constituted by measurements along multiple axes have emerged across many scientific areas such as genomics and cancer surveillance a common objective is to investigate the conditional dependencies among the variables along each axes taking into account multi dimensional structure of the data traditional multivariate approaches are unsuitable for such highly structured data due to inefficiency loss of power and lack of interpretability in this article we propose a novel class of multi dimensional graphical models based on matrix decompositions of the precision matrices along each dimension our approach is a unified framework applicable to both directed and undirected decomposable graphs as well as arbitrary combinations of these exploiting the marginalization of the likelihood we develop efficient posterior sampling schemes based on partially collapsed gibbs samplers empirically through simulation studies we show the superior performance of our approach in compari

asymptotic accuracy of distribution based estimation of latent variables hierarchical statistical models are widely employed in information science and data engineering the models consist of two types of variables observable variables that represent the given data and latent variables for the unobservable labels an asymptotic analysis of the models plays an important role in evaluating the learning process the result of the analysis is applied not only to theoretical but also to practical situations such as optimal model selection and active learning there are many studies of generalization errors which measure the prediction accuracy of the observable variables however the accuracy of estimating the latent variables has not yet been elucidated for a quantitative evaluation of this the present paper formulates distribution based functions for the errors in the estimation of the latent variables the asymptotic behavior is analyzed for both the maximum likelihood and the bayes methods

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

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

the analysis of contingency tables by graphical models summary some aspects of the analysis of multidimensional contingency tables in practice are considered the class of graphical models defined by darroch lauritzen speed 1980 is described strategies for model selection based on this class are considered and an example is given

bayesian inference for graphical factor analysis models we generalize factor analysis models by allowing the concentration matrix of the residuals to have nonzero off diagonal elements the resulting model is named graphical factor analysis model allowing a structure of associations gives information about the correlation left unexplained by the unobserved variables which can be used both in the confirmatory and exploratory context we first present a sufficient condition for global identifiability of this class of models with a generic number of factors thereby extending the results in stanghellini 1997 and vicard 2000 we then consider the issue of model comparison and show that fast local computations are possible for this purpose if the conditional independence graphs on the residuals are restricted to be decomposable and a bayesian approach is adopted to achieve this aim we propose a new reversible jump mcmc method to approximate the posterior probabilities of the considered models we then study the evolution of political democracy in 75 developing countries based on eight measures of democracy in two different years

retrospective prospective symmetry in the likelihood and bayesian analysis of case control studies summary prentice pyke 1979 established that the maximum likelihood estimate of an odds ratio in a case control study is the same as would be found by fitting a logistic regression in other words for this specific target the incorrect prospective model is inferentially equivalent to the correct retrospective model similar results have been obtained for other models and conditions have also been identified under which the corresponding bayesian property holds namely that the posterior distribution of the odds ratio is the same whether it is computed using the prospective or the retrospective likelihood in this article we demonstrate how these results follow directly from certain parameter independence properties of the models and priors and identify prior laws that support such reverse analysis for both standard and stratified designs somekeywords case controlstudy conditionalindependence hypermarkovlaw logisticregression retrospective likelihood

parameter priors for directed acyclic graphical models and the characterization of several probability distributions we develop simple methods for constructing parameter priors for model choice among directed acyclic graphical dag models in particular we introduce several assumptions that permit the construction of parameter priors for a large number of dag models from a small set of assessments we then present a method for directly computing the marginal likelihood of every dag model given a random sample with no missing observations we apply this methodology to gaussian dag models which consist of a recursive set of linear regression models we show that the only parameter prior for complete gaussian dag models that satisfies our assumptions is the normal wishart distribution our analysis is based on the following new characterization of the wishart distribution let w be an n times n n ge 3 positive definite symmetric matrix of random variables and f w be a pdf of w then f w is a wishart distribution if and only if w 11 w 12 w 22 1 w 12 is independent of w 12 w 22 for every block partitioning w 11 w 12 w 12 w 22 of w similar characterizations of the normal and normal wishart distributions are provided as well

the isserlis matrix and its application to non decomposable graphical gaussian models the operations of matrix completion and isserlis matrix construction arise in the statistical analysis of graphical gaussian models in both the bayesian and frequentist approaches in this paper we study the properties of the isserlis matrix of the completion of a positive definite matrix and propose an edge set indexing notation which highlights the symmetry existing between and its isserlis matrix in this way well known properties of can be exploited to give an easy proof of certain asymptotic results for decomposable graphical gaussian models as well as to extend such results to the non decomposable case in particular we consider the asymptotic variance of maximum likelihood estimators the non informative jeffreys prior the laplace approximation to the bayes factor the asymptotic distribution of the maximum likelihood estimators and the asymptotic posterior distribution of the parameters in a bayesian conjugate analysis in dealing with distributions over non decomposable models an extension of the hyper markov property to models with non chordal graphs is required our proposed extension is justified by an analysis of the conditional independence structure of the sufficient statistic

markov properties of nonrecursive causal models this paper aims to solve an often noted incompatibility between graphical chain models which elucidate the conditional independence structure of a set of random variables and simultaneous equations systems which focus on direct linear interactions and correlations between random variables various authors have argued that the incompatibility arises mainly from the fact that in a simultaneous equations system e g a lisrel model reciprocal causality is possible whereas this is not so in the case of graphical chain models in this article it is shown that this view is not correct in fact the definition of the markov property embodied in a graph can be generalized to a wider class of graphs which includes certain nonrecursive graphs the resulting class of reciprocal graph probability models strictly includes the class of chain graph probability models the class of lattice conditional independence probability models is also strictly included it is shown that the resulting methodology is directly applicable to quite general simultaneous equations systems that are subject to mild restrictions only provided some adjustments are made general simultaneous equations systems can be handled as well in all cases consistency with the lisrel methodology is maintained

marginalization and collapsibility in graphical interaction models the behaviour of a graphical interaction model under marginalization is discussed a graphical interaction model is called collapsible onto a set of variables if the class of marginal distributions is the same as that implied by the related subgraph the necessary and sufficient condition for collapsibility is found and it is shown that collapsibility is equivalent to a range of other important statistical properties of the model

flexible covariance estimation in graphical gaussian models in this paper we propose a class of bayes estimators for the covariance matrix of graphical gaussian models markov with respect to a decomposable graph g working with the w pg family defined by letac and massam ann statist 35 2007 1278 1323 we derive closed form expressions for bayes estimators under the entropy and squared error losses the w pg family includes the classical inverse of the hyper inverse wishart but has many more shape parameters thus allowing for flexibility in differentially shrinking various parts of the covariance matrix moreover using this family avoids recourse to mcmc often infeasible in high dimensional problems we illustrate the performance of our estimators through a collection of numerical examples where we explore frequentist risk properties and the efficacy of graphs in the estimation of high dimensional covariance structures

discretely holomorphic parafermions and integrable loop models we define parafermionic observables in various lattice loop models including examples where no kramers wannier duality holds for a particular rhombic embedding of the lattice in the plane and a value of the parafermionic spin these variables are discretely holomorphic they satisfy a lattice version of the cauchy riemann equations as long as the boltzmann weights satisfy certain linear constraints in the cases considered the weights then also satisfy the critical yang baxter equations with the spectral parameter being related linearly to the angle of the elementary rhombus

critical and multicritical o n models soluble o n models on the square lattice are found via the yang baxter equations in a parameter space of three coupling constants a number of distinct soluble cases are found for each value of n some of these constitute critical or multicritical points in the phase diagram for all solutions the free energy per spin can be determined using the bethe ansatz by studying the finite size corrections at the critical points the critical exponents and the central charge can be found in some cases analytically and failing that numerically by extrapolation of exact results for finite size systems

discretely holomorphic parafermions and integrable boundary conditions in two dimensional statistical models possessing a discretely holomorphic parafermion we introduce a modified discrete cauchy riemann equation on the boundary of the domain and we show that the solution of this equation yields integrable boundary boltzmann weights this approach is applied to i the square lattice o n loop model where the exact locations of the special and ordinary transitions are recovered and ii the fateev zamolodchikov spin model where a new rotation invariant integrable boundary condition is discovered for generic n

discrete holomorphicity at two dimensional critical points after a brief review of the historical role of analyticity in the study of critical phenomena an account is given of recent discoveries of discretely holomorphic observables in critical two dimensional lattice models these are objects whose correlation functions satisfy a discrete version of the cauchy riemann relations their existence appears to have a deep relation with the integrability of the model and they are presumably the lattice versions of the truly holomorphic observables appearing in the conformal field theory cft describing the continuum limit this hypothesis sheds light on the connection between cft and integrability and if verified can also be used to prove that the scaling limit of certain discrete curves in these models is described by schramm loewner evolution sle

shrinkage estimation for multivariate hidden markov models abstractmotivated from a changing market environment over time we consider high dimensional data such as financial returns generated by a hidden markov model that allows for switching between different regimes or states to get more stable estimates of the covariance matrices of the different states potentially driven by a number of observations that are small compared to the dimension we modify the expectation maximization em algorithm so that it yields the shrinkage estimators for the covariance matrices the final algorithm turns out to reproduce better estimates not only for the covariance matrices but also for the transition matrix it results into a more stable and reliable filter that allows for reconstructing the values of the hidden markov chain in addition to a simulation study performed in this article we also present a series of theoretical results that include dimensionality asymptotics and provide the motivation for certain techniques used in the algorithm supplementary materials fo

reference priors for discrete graphical models summary the combination of graphical models and reference analysis represents a powerful tool for bayesian inference in highly multivariate settings it is typically difficult to derive reference priors in complex problems in this paper we present a suitable mixed para meterisation for a discrete decomposable graphical model and derive the corresponding reference prior

nonmatrix varieties and nil generated algebras whose units satisfy a group identity abstract letr denote the group of units of an associative algebrarover an infinite fieldf we prove that ifris unitarily generated by its nilpotent elements thenr satisfies a group identity precisely whenrsatisfies a nonmatrix polynomial identity as an application we examine the group algebrafgof a torsion groupgand the restricted enveloping algebrau l of ap nil restricted lie algebral giambruno sehgal and valenti recently proved that if the group of units fg satisfies a group identity thenfgsatisfies a polynomial identity thus confirming a conjecture of brian hartley we show that in fact fg satisfies a group identity if and only iffgsatisfies a nonmatrix polynomial identity in the case of restricted enveloping algebras we prove thatu l satisfies a group identity if and only ifu l satisfies the engel condition

symmetric units satisfying a group identity abstract let k be an infinite field of characteristic p 2 g a locally finite group and kg its group algebra let kg kg denote the k linear extension of an involution defined on g in this paper we prove under some assumptions that if the set of symmetric units of kg satisfies a group identity then kg satisfies a polynomial identity moreover in case the prime radical of kg is nilpotent we characterize the groups for which the symmetric units satisfy a group identity

critical properties of the izergin korepin and solvable o n models and their related quantum spin chains the central charge of the izergin korepin model the corresponding quantum spin chain and the o n model is calculated analytically via the bethe ansatz the calculation extends a technique recently developed for the zamolodchikov fateev model in addition critical exponents and the central charge for these models are obtained from numerical solutions of bethe ansatz equations for finite systems as a physical application the authors find the exponents v 12 23 and gamma 53 46 for the theta transition of polymers in two dimensions

a bayesian analysis for change point problems a sequence of observations undergoes sudden changes at unknown times we model the process by supposing that there is an underlying sequence of parameters partitioned into contiguous blocks of equal parameter values the beginning of each block is said to be a change point observations are then assumed to be independent in different blocks given the sequence of parameters in a bayesian analysis it is necessary to give probability distributions to both the change points and the parameters we use product partition models barry and hartigan 1992 which assume that the probability of any partition is proportional to a product of prior cohesions one for each block in the partition and that given the blocks the parameters in different blocks have independent prior distributions given the observations a new product partition model holds with posterior cohesions for the blocks and new independent block posterior distributions for parameters the product model thus provides a convenient machinery for allo

some properties on rings with units satisfying a group identity abstract for convenience a ring with units satisfying a group identity will be called a gi ring we show that gi rings have the following properties which are also properites of pi rings 1 any gi ring is dedekind finite von neumann finite 2 nilpotent elements of a semiprimitive gi ring have bounded index 3 the kurosh problem has a positive answer for gi algebras namely any algebraic gi algebra is locally finite we also study hartley s problem for algebraic gi algebras

graphical modelling of multivariate time series we introduce graphical time series models for the analysis of dynamic relationships among variables in multivariate time series the modelling approach is based on the notion of strong granger causality and can be applied to time series with non linear dependences the models are derived from ordinary time series models by imposing constraints that are encoded by mixed graphs in these graphs each component series is represented by a single vertex and directed edges indicate possible granger causal relationships between variables while undirected edges are used to map the contemporaneous dependence structure we introduce various notions of granger causal markov properties and discuss the relationships among them and to other markov properties that can be applied in this context examples for graphical time series models include nonlinear autoregressive models and multivariate arch models

sequential imputations and bayesian missing data problems abstract for missing data problems tanner and wong have described a data augmentation procedure that approximates the actual posterior distribution of the parameter vector by a mixture of complete data posteriors their method of constructing the complete data sets is closely related to the gibbs sampler both required iterations and similar to the em algorithm convergence can be slow we introduce in this article an alternative procedure that involves imputing the missing data sequentially and computing appropriate importance sampling weights in many applications this new procedure works very well without the need for iterations sensitivity analysis influence analysis and updating with new data can be performed cheaply bayesian prediction and model selection can also be incorporated examples taken from a wide range of applications are used for illustration

the parafermionic observable in sle the parafermionic observable has recently been used by number of authors to study discrete models believed to be conformally invariant and to prove convergence results for these processes to sle beffara and duminil copin in arxiv 1010 0526v2 2011 duminil copin and smirnov in arxiv 1007 0575v2 2011 hongler and smirnov in arxiv 1008 2645v3 2011 ikhlef and cardy in j phys a 42 102001 2009 lawler in preprint 2011 rajabpour and cardy in j phys a 40 14703 2007 riva and cardy in j stat mech theory exp 2006 smirnov in international congress of mathematicians vol ii pp 1421 1451 2006 smirnov in ann math 172 2 1435 1467 2010 smirnov in proceedings of the international congress of mathematicians hyderabad 2010 vol i pp 595 621 2010 we provide a definition for a one parameter family of continuum versions of the parafermionic observable for sle which takes the form of a normalized limit of expressions identical to the discrete definition we then show the limit defining the observable exists compute the value of the observable up to a finite multiplicative constant and prove this constant is non zero for a wide range of finally we show our observable for sle becomes a holomorphic function for a particular choice of the parameter which provides a new point of view on a fundamental property of the discrete observable

extended conditional independence and applications in causal inference the goal of this paper is to integrate the notions of stochastic conditional independence and variation conditional independence under a more general notion of extended conditional independence we show that under appropriate assumptions the calculus that applies for the two cases separately axioms of a separoid still applies for the extended case these results provide a rigorous basis for a wide range of statistical concepts including ancillarity and sufficiency and in particular the decision theoretic framework for statistical causality which uses the language and calculus of conditional independence in order to express causal properties and make causal inferences

prior induction in log linear models for general contingency table analysis log linear modelling plays an important role in many statistical applications particularly in the analysis of contingency table data with the advent of powerful new computational techniques such as reversible jump mcmc bayesian analyses of these models and in particular model selection and averaging have become feasible coupled with this is the desire to construct and use suitably flexible prior structures which allow efficient computation while facilitating prior elicitation the latter is greatly improved in the case where priors can be specified on interpretable parameters about which relevant experts can express their beliefs in this paper we show how the specification of a general multivariate normal prior on the log linear parameters induces a multivariate lognormal prior on the corresponding cell counts of a contingency table we derive the parameters of this distribution in an explicit practical form and state the corresponding mean and covariances of the cell counts we discuss the importance of these results in terms of applying both uninformative and informative priors to the model parameters and provide an illustration in the context of the analysis of a 2 3 contingency table 1 introduction the analysis of general k way contingency table data is of interest in a wide variety of areas of statistical application model selection is notoriously difficult in suchsituations since th e number of models rises doubly exponentially withdimension obviously exhaustive comparisons are impossible but various computational techniques have been proposed for obtaining posterior model probabilities for problems of this sort depending upon the parameters upon which the analyst wishes to express prior opinions with the introduction of powerful new computational techniques these forms of analysis have been greatly simplified and are becoming increasingly common in the applied literature in undertaking a bayesian analysis of contingency table data it is necessary to specify priors either for the cell counts which could alternatively be expressed in terms of the cell probabilities and total cell count or equivalently the log linear parameters madigan and york 1997 choose to choose to place hyper dirichlet priors dawid and lauritzen 1993 on the cell probabilities which has the advantage that priors of this form allow a factorization of the likelihood through the identification of cliques within the corresponding model graph giudici green and tarantola 1999 illustrate how this decomposition

conditional independence graph for nonlinear time series and its application to international financial markets abstract conditional independence graphs are proposed for describing the dependence structure of multivariate nonlinear time series which extend the graphical modeling approach based on partial correlation the vertexes represent the components of a multivariate time series and edges denote direct dependence between corresponding series the conditional independence relations between component series are tested efficiently and consistently using conditional mutual information statistics and a bootstrap procedure furthermore a method combining information theory with surrogate data is applied to test the linearity of the conditional dependence the efficiency of the methods is approved through simulation time series with different linear and nonlinear dependence relations finally we show how the method can be applied to international financial markets to investigate the nonlinear independence structure

scaling dimensions and conformal anomaly in anisotropic lattice spin models the effect of anisotropic interactions on the eigenvalue spectrum of the row to row transfer matrix of critical lattice spin models is investigated it is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta ay ax sin theta where ay and ax are lattice spacings and theta is an angle describing the anisotropy for exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices this is done for the eight vertex model hard hexagons and interacting hard squares and it is found that theta pi u lambda 10u 3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter for each of these models the amplitude of the finite size corrections to the free energy at criticality is found to be of the form pi zeta c 6n2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c 1 4 5 and 7 10 respectively this is an analytic result for the eight vertex model for the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to n 48 finally the authors argue that the anisotropy angle for magnetic hard squares and the q state potts models is also given by theta pi u lambda so this result is quite general

solvable lattice models labelled by dynkin diagrams an equivalence between generalized restricted solid on solid models associated with sets of graphs and multi colour loop models is established as an application the authors consider solvable loop models and in this way obtain new solvable families of critical rsos models these families can all be classified by the dynkin diagrams of the simply laced lie algebras for one of the rsos models labelled by the lie algebra pair al al and related to the c2 1 vertex model they give an off critical extension which breaks the z2 symmetry of the dynkin diagrams

model based inference for categorical survey data subject to non ignorable non response we consider non response models for a single categorical response with categorical covariates whose values are always observed we present bayesian methods for ignorable models and a particular non ignorable model and we argue that standard methods of model comparison are inappropriate for comparing ignorable and non ignorable models uncertainty about ignorability of non response is incorporated by introducing parameters describing the extent of non ignorability into a pattern mixture specification and integrating over the prior uncertainty associated with these parameters our approach is illustrated using polling data from the 1992 british general election panel survey we suggest sample size adjustments for surveys when non ignorable non response is expected

group identities on the units of algebraic algebras with applications to restricted enveloping algebras abstract an algebra a is called a gi algebra if its group of units a satisfies a group identity we provide positive support for the following two open problems 1 does every algebraic gi algebra satisfy a polynomial identity 2 is every algebraically generated gi algebra locally finite

hyperbolicity of semigroup algebras abstract let a be a finite dimensional q algebra and a a z order we classify those a with the property that z 2 u and refer to this as the hyperbolic property we apply this in case a k s is a semigroup algebra with k q or k q d a complete classification is given when ks is semi simple and also when s is a non semi simple semigroup

the collapsed gibbs sampler in bayesian computations with applications to a gene regulation problem abstract this article describes a method of grouping and collapsing in using the gibbs sampler and proves from an operator theory viewpoint that the method is in general beneficial the norms of the forward operators associated with the corresponding nonreversible markov chains are used to discriminate among different simulation schemes when applied to bayesian missing data problems the idea of collapsing suggests skipping the steps of sampling parameter s values in standard data augmentation by doing this we obtain a predictive update version of the gibbs sampler a procedure of calculating the posterior odds ratio via the collapsed gibbs sampler when incomplete observations are involved is presented as an illustration of possible applications three examples along with a bayesian treatment for identifying common protein binding sites in unaligned dna sequences are provided

fractional bayes factors for model comparison bayesian comparison of models is achieved simply by calculation of posterior probabilities of the models themselves however there are difficulties with this approach when prior information about the parameters of the various models is weak partial bayes factors offer a resoiution of the problem by setting aside part of the data as a training sampie the training sampie is used to obtain an initiai informative posterior distribution of the parameters in each model model comparison is then based on a bayes factor calculated from the remaining data properties of partial bayes factors are discussed particularly in the context of weak prior information and they are found to have advantages over other proposed methods of model comparison a new variant of the partial bayes factor the fractional bayes factor is advocated on grounds of consistency simplicity robustness and coherence

decomposition of maximum likelihood in mixed graphical interaction models summary we show how decompositions of marked graphs induce corresponding decompositions of maximum likelihood estimates in mixed graphical interaction models we use this result to obtain explicit formulae for the estimates in decomposable models and to obtain factorizations of likelihood ratio statistics

bayesian models for sparse probability tables we wish to make inferences about the conditional probabilities p y x many of which are zero when the distribution of x is unknown and one observes only a multinomial sample of the y variates to do this fixed likelihood ratio models and quasi incremental distributions are defined it is shown that quasi incremental distributions are intimately linked to decomposable graphs and that these graphs can guide us to transformations of x and y which admit a conjugate bayesian analysis on a reparametrization of the conditional probabilities of interest

elliptical graphical modelling we propose elliptical graphical models based on conditional uncorrelatedness as a gen eralization of gaussian graphical models by letting the population distribution be elliptical instead of normal allowing the tting of data with arbitrarily heavy tails we study the class of propor tionally ane equivariant scatter estimators and show how they can be used to perform elliptical graphical modelling leading to a new class of partial correlation estimators and analogues of the classical deviance test general expressions for the asymptotic variance of partial correlation esti mators unconstrained and under decomposable models are given and the asymptotic chi square approximation of the pseudo deviance test statistic is proved the feasibility of our approach is demonstrated by a simulation study using among others tyler s scatter estimator which is distribution free within the elliptical model our approach provides a robustication of gauss ian graphical modelling the latter is likelihood based and known to be very sensitive to model misspecication and outlying observations 1 introduction and notation the statistical theory of undirected graphical models for continuous variables is usually based on the assumption of multivariate normality in practice data may deviate from the normal model in various ways outliers and heavy tails pose a problem of particular gravity they frequently occur and the normal likelihood methods such as the sample covariance matrix are very susceptible to them our objective is to deal with heavy tailed data and to safeguard graphical modelling against the impact of faulty outliers we restrict our attention to the case where we have only continuous variables and only undirected edges joint multivariate normality is often assumed in this situation and the statistical methodology is called gaussian graphical modelling we propose the class of elliptical distributions as a more general model and call our approach elliptical graphical modelling the lack of robustness of gaussian graphical modelling has been noted by several authors four proposals of robust approaches to gaussian graphical modelling are known to us becker 2005 and gottard and pacillo 2010 suggest replacing the sample covariance matrix by the reweighted minimum covariance determinant estimator miyamura and kano 2006 propose an alternative m type estimation and finegold drton arxiv 1009 3669 consider robustied versions of the graphical lasso by friedman et al 2008 this article delivers a systematic treatment of the plug in approach used in the rst two ref erences we show that the sample covariance matrix may be replaced by any ane equivariant

critical behaviour and conformal anomaly of the o n model on the square lattice finite size scaling and transfer matrix techniques are used to determine the conformal anomaly and critical exponents of o n models on the square lattice these calculations were performed on five branches of critical points parametrised by n the results for two of the branches agree with the known universal properties of the o n model as derived for the honeycomb lattice finite size convergence is poor for most of the third branch so that in this case the classification in terms of known models is uncertain the authors data show that the fourth branch can be interpreted in terms of a combination of a low temperature o n and a critical ising model finally the numerical results for the o n model on the remaining branch agree accurately with the known results for the low temperature o n 1 model they prove this equivalence by means of an exact mapping

tensor decompositions and sparse log linear models contingency table analysis routinely relies on log linear models with latent structure analysis providing a common alternative latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data while log linear models achieve dimensionality reduction through sparsity little is known about the relationship between these notions of dimensionality reduction in the two paradigms we derive several results relating the support of a log linear model to nonnegative ranks of the associated probability tensor motivated by these findings we propose a new collapsed tucker class of tensor decompositions which bridge existing parafac and tucker decompositions providing a more flexible framework for parsimoniously characterizing multivariate categorical data taking a bayesian approach to inference we illustrate empirical advantages of the new decompositions

partial factor modeling predictor dependent shrinkage for linear regression we develop a modified gaussian factor model for the purpose of inducing predictor dependent shrinkage for linear regression the new model predicts well across a wide range of covariance structures on real and simulated data furthermore the new model facilitates variable selection in the case of correlated predictor variables which often stymies other methods

on group identities for the unit group of algebras and semigroup algebras over an infinite field let k be an infinite field we fully describe when the unit group of a semigroup algebra k s of a semigroup s generated by finitely many periodic elements satisfies a group identity this and some other recent results are proved by first showing that semiprime k algebras generated by units are necessarily reduced whenever their unit group satisfies a group identity

integrability as a consequence of discrete holomorphicity loop models it has recently been established that imposing the condition of discrete holomorphicity on a lattice parafermionic observable leads to the critical boltzmann weights in a number of lattice models remarkably the solutions of these linear equations also solve the yang baxter equations we extend this analysis for the zn model by explicitly considering the condition of discrete holomorphicity on two and three adjacent rhombi for two rhombi this leads to a quadratic equation in the boltzmann weights and for three rhombi a cubic equation the two rhombus equation implies the inversion relations the star triangle relation follows from the three rhombus equation we also show that these weights are self dual as a consequence of discrete holomorphicity this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

collapsibility and response variables in contingency tables summary various definitions of the collapsibility of a hierarchical log linear model for a multidimensional contingency table are considered and shown to be equivalent necessary and sufficient conditions for collapsibility are found in terms of the generating class it is shown that log linear models are appropriate for tables with response and explanatory variables if and only if they are collapsible onto the explanatory variables some key word8 collapsibility contingency table graphical model interaction graph log linear model response variable s sufficiency shown to be closely related some models have the property that relations between a set of the classifying factors may be studied by examination of the table of marginal totals formed by summing over the remaining factors such models are said to be collapsible onto the given set of factors collapsibility has important consequences for hypothesis testing and model selection and can be useful in data reduction we consider various definitions of collapsibility and show their equivalence furthermore necessary and sufficient conditions for collapsi bility are found in terms of the generating class many tables analysed in practice involve response variables simple examples one of which is given in 3 suffice to show the importance of distinguishing between response and explanatory variables first that inappropriate models may be avoided and second that natural and relevant models that are not log linear may be considered this paper characterizes appropriate and inappropriate log linear models for tables with response variables and some alternative approaches for the analysis of such tables are briefly considered we consider a multidimensional contingency table n based on a set of classifying factors f for a given subset a of f we are interested in the table of marginal totals na that is to say the table of cell counts summed over the remaining factors ac that is the complement of a in f we identify a hierarchical log linear model l that is the set of probabilities p e l with its generating class whose elements generators are given in square brackets thus for example the model ab bcd for a 4 way table corresponds

beyond independent components trees and clusters we present a generalization of independent component analysis ica where instead of looking for a linear transform that makes the data components independent we look for a transform that makes the data components well fit by a tree structured graphical model this tree dependent component analysis tca provides a tractable and flexible approach to weakening the assumption of independence in ica in particular tca allows the underlying graph to have multiple connected components and thus the method is able to find clusters of components such that components are dependent within a cluster and independent between clusters finally we make use of a notion of graphical models for time series due to brillinger 1996 to extend these ideas to the temporal setting in particular we are able to fit models that incorporate tree structured dependencies among multiple time series

wishart distributions for decomposable graphs when considering a graphical gaussian model n g markov with respect to a decomposable graph g the parameter space of interest for the precision parameter is the cone p g of positive definite matrices with fixed zeros corresponding to the missing edges of g the parameter space for the scale parameter of n g is the cone q g dual to p g of incomplete matrices with submatrices corresponding to the cliques of g being positive definite in this paper we construct on the cones q g and p g two families of wishart distributions namely the type i and type ii wisharts 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 statist 21 1993 1272 1317 we show that the type i and ii wisharts have properties similar to those of the hyper and hyper inverse wishart 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 moreover the inverse type ii wishart as a conjugate family presents the advantage of having a multidimensional shape parameter thus offering flexibility for the choice of a prior both type i and ii wishart distributions depend on multivariate shape parameters a shape parameter is acceptable if and only if it satisfies a certain eigenvalue property we show that the sets of acceptable shape parameters for a noncomplete g have dimension equal to at least one plus the number of cliques in g these families as conjugate families are richer than the traditional diaconis ylvisaker conjugate families which all have a shape parameter set of dimension one a decomposable graph which does not contain a three link chain as an induced subgraph is said to be homogeneous in this case our wisharts are particular cases of the wisharts on homogeneous cones as defined by andersson and wojnar j theoret probab 17 2004 781 818 and the dimension of the shape parameter set is even larger than in the nonhomogeneous case it is indeed equal to the number of cliques plus the number of distinct minimal separators using the model where g is a three link chain we show by computing a 7 tuple integral that in general we cannot expect the shape parameter sets to have dimension larger than the number of cliques plus one

some matrix variate distribution theory notational considerations and a bayesian application summary we introduce and justify a convenient notation for certain matrix variate distributions which by its emphasis on the important underlying parameters and the theory on which it is based eases greatly the task of manipulating such distributions important examples include the matrix variate normal t f and beta and the wishart and inverse wishart distributions the theory is applied to compound matrix distributions and to bayesian prediction in the multivariate linear model

joint high dimensional bayesian variable and covariance selection with an application to eqtl analysis summary we describe a bayesian technique to a perform a sparse joint selection of significant predictor variables and significant inverse covariance matrix elements of the response variables in a high dimensional linear gaussian sparse seemingly unrelated regression ssur setting and b perform an association analysis between the high dimensional sets of predictors and responses in such a setting to search the high dimensional model space where both the number of predictors and the number of possibly correlated responses can be larger than the sample size we demonstrate that a marginalization based collapsed gibbs sampler in combination with spike and slab type of priors offers a computationally feasible and efficient solution as an example we apply our method to an expression quantitative trait loci eqtl analysis on publicly available single nucleotide polymorphism snp and gene expression data for humans where the primary interest lies in finding the significant associations between the sets of snps and possibly correlated genetic transcripts our method also allows for inference on the sparse interaction network of the transcripts response variables after accounting for the effect of the snps predictor variables we exploit properties of gaussian graphical models to make statements concerning conditional independence of the responses our method compares favorably to existing bayesian approaches developed for this purpose

structural learning with time varying components tracking the cross section of financial time series summary when modelling multivariate financial data the problem of structural learning is compounded by the fact that the covariance structure changes with time previous work has focused on modelling those changes by using multivariate stochastic volatility models we present an alternative to these models that focuses instead on the latent graphical structure that is related to the precision matrix we develop a graphical model for sequences of gaussian random vectors when changes in the underlying graph occur at random times and a new block of data is created with the addition or deletion of an edge we show how a bayesian hierarchical model incorporates both the uncertainty about that graph and the time variation thereof

model selection and accounting for model uncertainty in graphical models using occam s window abstract we consider the problem of model selection and accounting for model uncertainty in high dimensional contingency tables motivated by expert system applications the approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic p values leading to the selection of a single model inference is then conditional on the selected model the sampling properties of such a strategy are complex and the failure to take account of model uncertainty leads to underestimation of uncertainty about quantities of interest in principle a panacea is provided by the standard bayesian formalism that averages the posterior distributions of the quantity of interest under each of the models weighted by their posterior model probabilities furthermore this approach is optimal in the sense of maximizing predictive ability but this has not been used in practice because computing the posterior model probabilities is hard and the number of models is very large often greater than 1

on the bayesian analysis of population size summary we consider the problem of estimating the total size of a population from a series of incomplete census data we observe that inference is typically highly sensitive to the choice of model and we demonstrate how bayesian model averaging techniques easily overcome this problem we combine and extend the work of madigan york 1997 and dellaportas forster 1999 using reversible jump markov chain monte carlo simulation to calculate posterior model probabilities which can then be used to estimate modelaveraged statistics of interest we provide a detailed description of the simulation procedures involved and consider a wide variety of modelling issues such as the range of models considered their parameterisation both prior choice and sensitivity and computational efficiency we consider a detailed example concerning adolescent injuries in pennsylvania on the basis of medical school and survey data in the context of this example we discuss the relationship between posterior model probabilities and the associated information criteria values for model selection we also discuss cost efficiency issues with particular reference to inclusion and exclusion of sources on the grounds of cost we consider a decision theoretic approach which balances the cost and accuracy of different combinations of data sources to guide future decisions on data collection

on substantive research hypotheses conditional independence graphs and graphical chain models summary graphs consisting of points and lines or arrows as connections between selected pairs of points are used to formulate hypotheses about relations between variables points stand for variables connections represent associations when a missing connection is interpreted as a conditional independence the graph characterizes a conditional independence structure as well statistical models called graphical chain models correspond to special types of graphs which are interpreted in this fashion examples are used to illustrate how conditional independences are reflected in summary statistics derived from the models and how the graphs help to identify analogies and equivalences between different models graphical chain models are shown to provide a unifying concept for many statistical techniques that in the past have proven to be useful in analyses of data they also provide tools for new types of analysis

bayesian selection of decomposable models with incomplete data this article describes a new approach to bayesian selection of decomposable models with incomplete data this approach requires the characterization of new ignorability conditions for the missing data mechanism and the development of new computational methods both issues are considered and solutions are proposed theory and methods are assessed in controlled experiments and in the analysis of one real life incomplete dataset

a nonparametric approach for multiple change point analysis of multivariate data change point analysis has applications in a wide variety of fields the general problem concerns the inference of a change in distribution for a set of time ordered observations sequential detection is an online version in which new data are continually arriving and are analyzed adaptively we are concerned with the related but distinct offline version in which retrospective analysis of an entire sequence is performed for a set of multivariate observations of arbitrary dimension we consider nonparametric estimation of both the number of change points and the positions at which they occur we do not make any assumptions regarding the nature of the change in distribution or any distribution assumptions beyond the existence of the th absolute moment for some 0 2 estimation is based on hierarchical clustering and we propose both divisive and agglomerative algorithms the divisive method is shown to provide consistent estimates of both the number and the location of change points under standard

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

an approximation to maximum likelihood estimates in reduced models summary an approximation to the maximum likelihood estimates of the parameters in a model can be obtained from the corresponding estimates and information matrices in an extended model i e a model with additional parameters the approximation is close provided that the data are consistent with the first model applications are described to log linear models for discrete data to models for multivariate normal distributions with special covariance matrices and to mixed discrete continuous models

boundary conformal field theories and loop models we propose a systematic method to extract conformal loop models for rational conformal field theories cft the method is based on defining an ade model for boundary primary operators by using the fusion matrices of these operators as adjacency matrices these loop models respect the conformal boundary conditions we discuss the loop models that can be extracted by this method for minimal cfts and then we give dilute o n loop models on the square lattice as examples for these loop models we also give some proposals for wzw su 2 models

graphical and recursive models for contingency tables summary we discuss two classes of models for contingency tables graphical and recursive models both of which arise from restrictions that are expressible as conditional independencies of variable pairs the first of these is a subclass of hierarchical log linear models each of its models can be represented by an undirected graph in the second class each model corresponds to a particular kind of a directed graph instead and can be characterized by a nontrivial factorization of the joint distribution in terms of response variables we derive decomposable or multiplicative models as the intersecting class this result has useful consequences for exploratory types of analysis as well as for the model interpretation we can give an aid for detecting well fitting decomposable models in a transformation of the observed contingency table and each decomposable model may be interpreted with the help of an undirected or directed graph

finite size left passage probability in percolation we obtain an exact finite size expression for the probability that a percolation hull will touch the boundary on a strip of finite width in terms of clusters this corresponds to the one arm probability our calculation is based on the q deformed knizhnik zamolodchikov approach and the results are expressed in terms of symplectic characters in the large size limit we recover the scaling behaviour predicted by schramm s left passage formula we also derive a general relation between the left passage probability in the fortuin kasteleyn cluster model and the magnetisation profile in the open xxz chain with diagonal complex boundary terms

conserved currents in the six vertex and trigonometric solid on solid models we construct quasi local conserved currents in the six vertex model with anisotropy parameter by making use of the quantum group approach of bernard and felder from these currents we construct parafermionic operators with spin that obey a discrete integral condition around lattice plaquettes embedded into the complex plane these operators are identified with primary fields in a c 1 compactified free boson conformal field theory we then consider a vertex face correspondence that takes the six vertex model to a trigonometric sos model and construct sos operators that are the image of the six vertex currents under this correspondence we define corresponding sos parafermionic operators with spins s 1 and that obey discrete integral conditions around sos plaquettes embedded into the complex plane we consider in detail the cyclic sos case corresponding to the choice with coprime we identify our sos parafermionic operators in terms of the screening operators and primary fields of the associated conformal field theory

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