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

on the notion of concept ii it is well known that classical set theory is not expressive enough to adequately model categorization and prototype theory recent work on compositionality and concept determination showed that the quantitative solution initially offered by classical fuzzy logic also led to important drawbacks several qualitative approaches were thereafter tempted that aimed at modeling membership through ordinal scales or lattice fuzzy sets most of the solutions obtained by these theoretical constructions however are of difficult use in categorization theory we propose a simple qualitative model in which membership relative to a given concept f is represented by a function that takes its value in a finite abstract set a f equipped with a total order this function is recursively built through a stratification of the set of concepts at hand based on a notion of complexity similarly the typicality associated with a concept f will be described using an ordering that takes into account the characteristic features of f once the basic notions of membership and typicality are set the study of compound concepts is possible and leads to interesting results in particular we investigate the internal structure of concepts and obtain the characterization of all smooth subconcepts of a given concept

propagation of probabilities means and variances in mixed graphical association models abstract a scheme is presented for modeling and local computation of exact probabilities means and variances for mixed qualitative and quantitative variables the models assume that the conditional distribution of the quantitative variables given the qualitative is multivariate gaussian the computational architecture is set up by forming a tree of belief universes and the calculations are then performed by local message passing between universes the asymmetry between the quantitative and qualitative variables sets some additional limitations for the specification and propagation structure approximate methods when these are not appropriately fulfilled are sketched it has earlier been shown how to exploit the local structure in the specification of a discrete probability model for fast and efficient computation thereby paving the way for exploiting probability based models as parts of realistic systems for planning and decision support the purpose of this article is to extend this computational s

spaces of matrices of constant rank and uniform vector bundles abstract we consider the problem of determining l r a the maximal dimension of a subspace of a a matrices of rank r we first review in the language of vector bundles the known results then using known facts on uniform bundles we prove some new results and make a conjecture finally we determine l r a for every r 1 r a when a 10 showing that our conjecture holds true in this range

distance between populations using mixed continuous and categorical variables summary a general form of the location model is considered for mixed continuous and categorical variables observed in a number of different populations and some special cases of practical interest are cited the distance between any two populations is derived for each of these models estimation of parameters in these distance measures is discussed and the methods are illustrated by application to previously published data

reformulation of the theory of conceptual spaces this paper presents a conceptual system based on two independently developed extensions of gardenfors formulation of conceptual spaces the new approach continues to emphasize the role of properties and their associations in conceptual representation and to recognize the importance of similarity judgments in reasoning tasks in the new theory domains are sets equipped with a measure and a property is a measurable membership function on a domain concepts are sets of properties and their binary associations a new notion of the degree of ambiguity of properties for a given reasoning task is used to focus attention upon properties having the greatest discrimination power the theory is demonstrated on a maritime application involving the categorization of vessel routes

moishezon s theorem and degeneration abstract let l be a tensor product of two very ample line bundles on the smooth projective complex threefold x under the hypothesis that h 0 x k x l 0 we show that the restriction map r pic x pic y is an isomorphism for very general y in the linear system l for such l this result recovers the noether lefschetz theorem of moishezon who extended the original topological and hodge theoretic arguments of lefschetz we give a degeneration argument which is almost completely algebraic in nature

the tm algorithm for maximising a conditional likelihood function summary this paper describes an algorithm for maximising a conditional likelihood function when the corresponding unconditional likelihood function is more easily maximised the algorithm is similar to the em algorithm but different as the parameters rather than the data are augmented and the conditional rather than the marginal likelihood function is maximised in exponential families the algorithm takes a particular simple form which is computationally very close to the em algorithm the algorithm alternates between a t step which calculates a tilted version of the unconditional likelihood function and an m step which maximises it the algorithm applies to mixed graphical chain models lauritzen wermuth 1989 and their generalisations edwards 1990 and was developed with these in mind but it may have applications beyond these the algorithm has been implemented in the most recent version of the mim software edwards 2000 where it was named the me algorithm the name has been changed to avoid confusion with the algorithm described by marschner 2001

chain graph models and their causal interpretations chain graphs are a natural generalization of directed acyclic graphs and undirected graphs however the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent there are many simple and apparently plausible but ultimately fallacious interpretations of chain graphs that are often invoked implicitly or explicitly these interpretations also lead to flawed methods for applying background knowledge to model selection we present a valid interpretation by showing how the distribution corresponding to a chain graph may be generated from the equilibrium distributions of dynamic models with feed back these dynamic interpretations lead to a simple theory of intervention extending the theory developed for directed acyclic graphs finally we contrast chain graph models under this interpretation with simultaneous equation models which have traditionally been used to model feed back in econometrics

characterizing markov equivalence classes for amp chain graph models chain graphs cg adicyclic graphs use undirected and directed edges to represent both structural and associative dependences like acyclic directed graphs adgs the cg associated with a statistical markov model may not be unique so cgs fall into markov equivalence classes which may be superexponentially large leading to unidentifiability and computational inefficiency in model search and selection it is shown here that under the andersson madigan perlman amp interpretation of a cg each markov equivalence class can be uniquely represented by a single distinguished cg the amp essential graph that is itself simultaneously markov equivalent to all cgs in the amp markov equivalence class a complete characterization of amp essential graphs is obtained like the essential graph previously introduced for adgs the amp essential graph will play a fundamental role for inference and model search and selection for amp cg models

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

limit linear series and ranks of multiplication maps we develop a new technique for studying ranks of multiplication maps for linear series via limit linear series and degenerations to chains of genus 1 curves we use this approach to prove a purely elementary criterion for proving cases of the maximal rank conjecture and then apply the criterion to several ranges of cases giving a new proof of the case of quadrics and also treating several families in the case of cubics our proofs do not require restrictions on direction of approach so we recover new information on the locus in the moduli space of curves on which the maximal rank condition fails

response models for mixed binary and quantitative variables summary a number of special representations are considered for the joint distribution of qualitative mostly binary and quantitative variables in addition to the conditional gaussian models and to conditional gaussian regression chain models some emphasis is placed on models derived from an underlying multivariate normal distribution and on models in which discrete probabilities are specified linearly in terms of unknown parameters the possibilities for choosing between the models empirically are examined as well as the testing of independence and conditional independence and the estimation of parameters often the testing of independence is exactly or nearly the same for a number of different models

a graphical representation of equivalence classes of amp chain graphs this paper deals with chain graph models under alternative amp interpretation a new representative of an amp markov equivalence class called the largest deflagged graph is proposed the representative is based on revealed internal structure of the amp markov equivalence class more specifically the amp markov equivalence class decomposes into finer strong equivalence classes and there exists a distinguished strong equivalence class among those forming the amp markov equivalence class the largest deflagged graph is the largest chain graph in that distinguished strong equivalence class a composed graphical procedure to get the largest deflagged graph on the basis of any amp markov equivalent chain graph is presented in general the largest deflagged graph differs from the amp essential graph which is another representative of the amp markov equivalence class

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 conditioning for lwf chain graphs supported by grant fa9550 12 1 0392 from the u s air force office of scientific research afosr and the defense advanced research projects agency darpa

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

strong collapsibility of association measures in linear models a measure of association in linear models is strongly collapsible over a discrete background variable if it remains unchanged no matter how the background variable is partially pooled and if it also coincides with the corresponding marginal measure of association strong collapsibility implies that the measure of association can be studied no matter how the background variable is categorized and no matter whether or not it is recorded in this paper necessary and sufficient conditions are given for strong collapsibility of the measure of association in linear models

ideal parent structure learning for continuous variable bayesian networks bayesian networks in general and continuous variable networks in particular have become increasingly popular in recent years largely due to advances in methods that facilitate automatic learning from data yet despite these advances the key task of learning the structure of such models remains a computationally intensive procedure which limits most applications to parameter learning this problem is even more acute when learning networks in the presence of missing values or hidden variables a scenario that is part of many real life problems in this work we present a general method for speeding structure search for continuous variable networks with common parametric distributions we efficiently evaluate the approximate merit of candidate structure modifications and apply time consuming exact computations only to the most promising ones thereby achieving significant improvement in the running time of the search algorithm our method also naturally and efficiently facilitates the addition of useful new hidden variables into the network structure a task that is typically considered both conceptually difficult and computationally prohibitive we demonstrate our method on synthetic and real life data sets both for learning structure on fully and partially observable data and for introducing new hidden variables during structure search

estimating equations for parameters in means and covariances of multivariate discrete and continuous responses generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response these equations are to be solved jointly with score equations from a generalized linear model for mean parameters a class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion and proposals for the use of such equations are developed comments on the relative merits of the ad hoc and model based approaches to estimation are given and a regression illustration with a bivariate response is provided

distortion of effects caused by indirect confounding summary undetected confounding may severely distort the effect of an explanatory variable on a response variable as defined by a stepwise data generating process the best known type of distortion which we call direct confounding arises from an unobserved explanatory variable common to a response and its main explanatory variable of interest it is relevant mainly for observational studies since it is avoided by successful randomization by contrast indirect confounding which we identify in this paper is an issue also for intervention studies for general stepwise generating processes we provide matrix and graphical criteria to decide which types of distortion may be present when they are absent and how they are avoided we then turn to linear systems without other types of distortion but with indirect confounding for such systems the magnitude of distortion in a least squares regression coefficient is derived and shown to be estimable so that it becomes possible to recover the effect of the generating process from the distorted coefficient

local propagation in conditional gaussian bayesian networks this paper describes a scheme for local computation in conditional gaussian bayesian networks that combines the approach of lauritzen and jensen 2001 with some elements of shachter and kenley 1989 message passing takes place on an elimination tree structure rather than the more compact and usual junction tree of cliques this yields a local computation scheme in which all calculations involving the continuous variables are performed by manipulating univariate regressions and hence matrix operations are avoided

limit linear series basic theory in this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves curves of compact type the technically much simpler special case of 1 dimensional series was developed by beauville 2 knudsen 21 23 harris and mumford 17 in the guise of admissible covers it has proved very useful for studying the moduli space of curves the above papers and harris 16 and the simplest sorts of weierstrass points diaz 4 with our extended tools we are able to prove for example that in this paper we present the basic theory of limit linear series necessary for proving these results the results themselves will be taken up in our forthcoming papers 8 12 simpler applications not requiring the tools developed in this paper but perhaps clarified by them have already been given in our papers 5 7

dimensionality reduction in unsupervised learning of conditional gaussian networks this paper introduces a novel enhancement for unsupervised learning of conditional gaussian networks that benefits from feature selection our proposal is based on the assumption that in the absence of labels reflecting the cluster membership of each case of the database those features that exhibit low correlation with the rest of the features can be considered irrelevant for the learning process thus we suggest performing this process using only the relevant features then every irrelevant feature is added to the learned model to obtain an explanatory model for the original database which is our primary goal a simple and thus efficient measure to assess the relevance of the features for the learning process is presented additionally the form of this measure allows us to calculate a relevance threshold to automatically identify the relevant features the experimental results reported for synthetic and real world databases show the ability of our proposal to distinguish between relevant and irrelevant features and to accelerate learning while still obtaining good explanatory models for the original database

hierarchical interaction models lauritzen and wermuth have proposed a class of models for mixed qualitative and continuous data defined by two properties that the continuous variables are normally distributed given the qualitative variables and that a set of conditional independence relations hold between specified pairs of variables the present paper examines an extension to this class called hierarchical interaction models a compact form for model representation is described and an estimation algorithm is given some properties of the models a compact form for model representation is described and an estimation algorithm is given some properties of the models concerning marginalization and conditioning are examined the class includes and generalizes hierarchical log linear models standard fixed effect anova multivariate anova and multivariate regression models

a general dynamical statistical model with causal interpretation summary we develop a general dynamical model as a framework for causal interpretation we first state a criterion of local independence in terms of measurability of processes that are involved in the doob meyer decomposition of stochastic processes then we define direct and indirect influence we propose a definition of causal influence using the concepts of a physical system this framework makes it possible to link descriptive and explicative statistical models and encompasses quantitative processes and events one of the features of the paper is the clear distinction between the model for the system and the model for the observation we give a dynamical representation of a conventional joint model for human immunodeficiency virus load and cd4 cell counts we show its inadequacy to capture causal influences whereas in contrast known mechanisms of infection by the human immunodeficiency virus can be expressed directly through a system of differential equations

a general formulation of conceptual spaces as a meso level representation representing cognitive processes remains one of the great research challenges many important application areas such as clinical diagnosis operate in an environment of relative magnitudes counts shapes colours etc which are not well captured by current representational approaches this paper presents conceptual spaces as a meso level representation for cognitive systems between the high level symbolic representations and the subconceptual connectionist representations which have dominated ai conceptual spaces emphasize orders and measures and therefore naturally represent counts magnitudes and volumes taking grdenfors decade long investigation of conceptual spaces grdenfors conceptual spaces the geometry of thought mit press 2000 as start point the paper presents a formal foundation for conceptual spaces shows how they are theoretically and practically linked to higher and lower representational levels and develops dynamics which allow the orbits of states in the space to solve appropriate meso level reasoning tasks interpretations of conceptual spaces are given to illustrate the formal definitions and show the flexibility of the representation

simple linear time algorithms to test chordality of graphs test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs chordal graphs arise naturally in the study of gaussian elimination on sparse symmetric matrices acyclic hypergraphs arise in the study of relational data bases rose tarjan and lueker siam j c

hierarchical conceptual spaces for concept combination we introduce a hierarchical framework for conjunctive concept combination based on conceptual spaces and random set theory the model has the flexibility to account for composition of concepts at various levels of complexity we show that the conjunctive model includes linear combination as a special case and that the more general model can account for non compositional behaviours such as overextension non commutativity preservation of necessity and impossibility of attributes and to some extent attribute loss or emergence we investigate two further aspects of human concept use the conjunction fallacy and the guppy effect

graphical regression models for polytomous variables when modeling the relationship between two nominal categorical variables it is often desirable to include covariates to understand how individuals differ in their response behavior typically however not all the relevant covariates are available with the result that the measured variables cannot fully account for the associations between the nominal variables under the assumption that the observed and unobserved variables follow a homogeneous conditional gaussian distribution this paper proposesrc m regression models to decompose the residual associations between the polytomous variables based on goodman s 1979 1985 rc m association model a distinctive feature ofrc m regression models is that they facilitate the joint estimation of effects due to manifest and omitted continuous variables without requiring numerical integration therc m regression models are illustrated using data from the high school and beyond study tatsuoka lohnes 1988

log multiplicative association models as item response models log multiplicative association lma models which are special cases of log linear models have interpretations in terms of latent continuous variables two theoretical derivations of lma models based on item response theory irt arguments are presented first we show that anderson and colleagues anderson vermunt 2000 anderson bockenholt 2000 anderson 2002 who derived lma models from statistical graphical models made the equivalent assumptions as holland 1990 when deriving models for the manifest probabilities of response patterns based on an irt approach we also present a second derivation of lma models where item response functions are specified as functions of rest scores these various connections provide insights into the behavior of lma models as item response models and point out philosophical issues with the use of lma models as item response models we show that even for short tests lma and standard irt models yield very similar to nearly identical results when data arise from standard irt models log multiplicative association models can be used as item response models and do not require numerical integration for estimation

regression models for a bivariate discrete and continuous outcome with clustering abstract developmental toxicity studies of laboratory animals play a crucial role in the testing and regulation of chemicals and pharmaceutical compounds exposure to developmental toxicants typically causes a variety of adverse effects such as fetal malformations and reduced fetal weight at term in this article we discuss regression methods for jointly analyzing bivariate discrete and continuous outcomes that are motivated by the statistical problems that arise in analyzing data from developmental toxicity studies we focus on marginal regression models that is models in which the marginal expectation of the bivariate response vector is related to a set of covariates by some known link functions in these models the regression parameters for the marginal expectation are of primary scientific interest whereas the association between the binary and continuous response is considered to be a nuisance characteristic of the data we describe a likelihood based approach based on the general location mode

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

markov properties for linear causal models with correlated errors a linear causal model with correlated errors represented by a dag with bi directed edges can be tested by the set of conditional independence relations implied by the model a global markov property specifies by the d separation criterion the set of all conditional independence relations holding in any model associated with a graph a local markov property specifies a much smaller set of conditional independence relations which will imply all other conditional independence relations which hold under the global markov property for dags with bi directed edges associated with arbitrary probability distributions a local markov property is given in richardson 2003 which may invoke an exponential number of conditional independencies in this paper we show that for a class of linear structural equation models with correlated errors there is a local markov property which will invoke only a linear number of conditional independence relations for general linear models we provide a local markov property that often invokes far fewer conditional independencies than that in richardson 2003 the results have applications in testing linear structural equation models with correlated errors

joint response graphs and separation induced by triangular systems summary we consider joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions the independences are captured by missing edges in a directed graph a matrix form of such a graph called the generating edge matrix is triangular so the distributions that are generated over such graphs are called triangular systems we study consequences of triangular systems after grouping or reordering of the variables for analyses as chain graph models i e for alternative recursive factorizations of the given density using joint conditional distributions for this we introduce families of linear triangular equations which do not require assumptions of distributional form the strength of the associations that are implied by such linear families for chain graph models is derived the edge matrices of chain graphs that are implied by any triangular system are obtained by appropriately transforming the generating edge matrix it is shown how induced independences and dependences can be studied by graphs by edge matrix calculations and via the properties of densities some ways of using the results are illustrated

uncertainty modelling for vague concepts a prototype theory approach an epistemic model of the uncertainty associated with vague concepts is introduced label semantics theory is proposed as a framework for quantifying an agent s uncertainty concerning what labels are appropriate to describe a given example an interpretation of label semantics is then proposed which incorporates prototype theory by introducing uncertain thresholds on the distance between elements and prototypes for description labels this interpretation naturally generates a functional calculus for appropriateness measures a more general model with distinct threshold variables for different labels is discussed and we show how different kinds of semantic dependence can be captured in this model

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

moderating effects of subgroups in linear models summary possibilities for moderating effects of a subgrouping variable on strength or direction of an association have been much discussed by social scientists but have not been given satisfactory statistical formulations the results concern directed measures of associations in linear models containing just three variables some key words analysis of covariance analysis of variance cg distribution conditional independence graphical chain model parallel regressions yule simpson paradox 1 introduction linear models are commonly used as a framework to estimate and test how a continuous response variable depends on potential influencing variables this paper is concerned with the situation in which two influences are random variables as well which may be discrete or continuous the continuous random variables are to stand for quantitative properties of observational units while the discrete variables either capture qualitative properties or they represent subgroups of the population to which the observational units belong to think of all variables as random variables is usually appropriate for data obtained from observational studies where one cannot control which levels or values of the influencing variables are to be observed the linear model permits an analysis conditional on fixed levels and values of the influences but frequently the association structure among the influences is in social science applications of interest itself and as it will turn out it is important for a correct interpretation of how the response depends on the influences awareness of such problems in interpretation is widespread among social scientists as is documented by extensive discussions of moderating effects see for example saunders 1956 zedeck 1971 and baron kenny 1986 there a variable is called a moderator if its presence changes the strength or direction of an association however methods recommended to social scientists for identifying moderating effects given in the above literature or by cohen cohen 1983 pp 310 4 have been shown to be seriously deficient wermuth 1988 thus there is need for clarification we treat the case of a potential discrete moderator variable and call its categories or levels the subgroups in the linear model a moderating effect is closely tied to the notion of consistent results on an association wermuth 1987 in the present context results are said to be weakly consistent if the associations within subgroups coincide in direction and strongly consistent if the associ ations within subgroups coincide in direction and strength this notion depends typically on the chosen measure of association in linear models we look at directed measures of

symmetry and lattice conditional independence in a multivariate normal distribution a class of multivariate normal models with symmetry restrictions given by a finite group and conditional independence restrictions given by a finite distributive lattice is defined and studied the statistical properties of these models including maximum likelihood inference invariance and hypothesis testing are discussed

minimal sufficient causation and directed acyclic graphs notions of minimal sufficient causation are incorporated within the directed acyclic graph causal framework doing so allows for the graphical representation of sufficient causes and minimal sufficient causes on causal directed acyclic graphs while maintaining all of the properties of causal directed acyclic graphs this in turn provides a clear theoretical link between two major conceptualizations of causality one counterfactual based and the other based on a more mechanistic understanding of causation the theory developed can be used to draw conclusions about the sign of the conditional covariances among variables

noether lefschetz locus for surfaces we generalize m green s explicit noether lefschetz theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces moreover we give a new proof of the density theorem due to c ciliberto j harris and r miranda 5

on chain graph models for description of conditional independence structures a chain graph cg is a graph admitting both directed and undirected edges with partially directed cycles forbidden it generalizes both the concept of undirected graph ug and the concept of directed acyclic graph dag a chain graph can be used to describe efficiently the conditional independence structure of a multidimensional discrete probability distribution in the form of a graphoid that is in the form of a list of statements x is independent of y given z obeying a set of five properties axioms an input list of independency statements for every cg is defined and it is shown that the classic moralization criterion for cgs embraces exactly the graphoid closure of the input list a new direct separation criterion for reading independency statements from a cg is introduced and shown to be equivalent to the moralization criterion using this new criterion it is proved that for every cg there exists a strictly positive discrete probability distribution that embodies exactly the independency statements displayed by the graph thus both criteria are shown to be complete and the use of cgs as tools for description of conditional independence structures is justified

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

nonmonotonic reasoning preferential models and cumulative logics abstract many systems that exhibit nonmonotonic behavior have been described and studied already in the literature the general notion of nonmonotonic reasoning though has almost always been described only negatively by the property it does not enjoy i e monotonicity we study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties we concentrate on a number of families of nonmonotonic consequence relations defined in the style of gentzen 13 both proof theoretic and semantic points of view are developed in parallel the former point of view was pioneered by gabbay 10 while the latter has been advocated by shoham 38 five such families are defined and characterized by representation theorems relating the two points of view one of the families of interest that of preferential relations turns out to have been studied by adams 2 the preferential models proposed here are a much stronger tool than adams probabilistic semantics the basic language used in this paper is that of propositional logic the extension of our results to first order predicate calculi and the study of the computational complexity of the decision problems described in this paper will be treated in another paper

penalized maximum likelihood estimation of multi layered gaussian graphical models analyzing multi layered graphical models provides insight into understanding the conditional relationships among nodes within layers after adjusting for and quantifying the effects of nodes from other layers we obtain the penalized maximum likelihood estimator for gaussian multi layered graphical models based on a computational approach involving screening of variables iterative estimation of the directed edges between layers and undirected edges within layers and a final refitting and stability selection step that provides improved performance in finite sample settings we establish the consistency of the estimator in a high dimensional setting to obtain this result we develop a strategy that leverages the biconvexity of the likelihood function to ensure convergence of the developed iterative algorithm to a stationary point as well as careful uniform error control of the estimates over iterations the performance of the maximum likelihood estimator is illustrated on synthetic data

hypersurfaces of the flag variety deformation theory and the theorems of kodaira spencer torelli lefschetz m noether and serre

the surface c c on jacobi varieties and 2nd order theta functions preprint enviat per a la seva publicacio en una revista cientifica acta mathematica volume 157 1986 1 22 https projecteuclid org euclid acta 1485890436

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

parameter updating in a bayes network abstract a bayes network is a directed acyclic graph in which the links are quantified by fixed conditional probabilities and the nodes represent random variables the primary use of the network is to provide an efficient method for updating conditional probabilities in the graph we consider the consequences of using the network as the computational device for updating parameter estimates in the dynamic linear model a discrete time bayesian model we show that using the network characterizes the dynamic linear model and its computations in a unified way the generality of the network permits nonsequential data collection and thereby provides a straightforward method of incorporating delayed data an on line diagnostic is offered to complement the conventional forecast error and an approximation to the posterior distribution is proposed algorithms for data propagation in multivariate gaussian causal trees are presented

maximum likelihood estimation for mixed continuous and categorical data with missing values summary maximum likelihood procedures for analysing mixed continuous and categorical data with missing values are presented the general location model of olkin tate 1961 and extensions introduced by krzanowski 1980 1982 form the basis for our methods maximum likelihood estimation with incomplete data is achieved by an application of the em algorithm dempster laird rubin 1977 special cases of the algorithm include orchard woodbury s 1972 algorithm for incomplete normal samples fuchs s 1982 algorithms for log linear modelling of partially classified contingency tables and day s 1969 algorithm for multivariate normal mixtures applications include a imputation of missing values b logistic regression and discriminant analysis with missing predictors and unclassified observations c linear regression with missing continuous and categorical predictors and d parametric cluster analysis with incomplete data methods are illustrated using data from the st louis risk research project some key word8 cluster analysis discriminant analysis em algorithm incomplete data linear regression logistic regression log linear model mixture model

on the 1 curve conjecture of friedman and morgan we will prove that every differentiably embedded sphere with self intersection 1 in a simply connected algebraic surface withp g 0 is homologous to an algebraic class if the surface has a minimal model with picard number 1 or k min contains a smooth curve and eitherp g ork min 2 is even then every such sphere is homologous to a 1 curve as conjectured by friedman and morgan

mixed multivariate generalized linear models for assessing lower limb arterial stenoses experiments and observational studies often involve gathering information on several response variables enabling us to model their dependence on observable predictor variables despite associations between the response variables they are often analysed separately using general and generalized linear models this paper investigates applications of multivariate regression analysis to improve the accuracy of predictions and decisions in the specific context of diagnosing arterial stenoses in human legs two basic models are developed for this application using i four binary responses and ii a mixture of two binary and two normal responses the results clearly demonstrate the potential advantages offered by this approach

gaussian markov distributions over finite graphs on s interesse aux modeles de selection de covariance introduites par dempster 1972 et etudies par wermuth 1976

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

log linear modelling of pairwise interobserver agreement on a categorical scale this article uses log linear models to describe pairwise agreement among several raters who classify a sample on a subjective categorical scale the models describe agreement structure simultaneously for second order marginal tables of a multidimensional cross classification of ratings practical difficulties arise in fitting the models because models refer to pairwise marginal tables of a very large and sparse table a standard analysis that treats the marginal tables as independent yields consistent estimates of model parameters but not of the covariance matrix of the estimates we estimate the covariance matrix using the jackknife we apply the models to describe agreement between evaluations made by seven pathologists of carcinoma in situ of the uterine cervix using a five level ordinal scale previous analyses showed differences among the pathologists in their pairwise levels of agreement but we observe near homogeneity in the dependence structure of their ratings

the three semantics of fuzzy sets abstract three main semantics for membership functions seem to exist in the literature similarity preference and uncertainty each semantics underlies a particular class of applications similarity notions are exploited in clustering analysis and fuzzy controllers uncertainty is captured by fuzzy sets in the framework of possibility theory the membership function of a fuzzy set is also sometimes a kind of utility function that represents flexible constraints in decision problems this paper advocates the claim that progress in operational semantics of membership functions presupposes that these distinct semantics be acknowledged and related to more basic measurement issues in terms of distance cost and frequency on which scientific traditions exist

statistics and causal inference abstract problems involving causal inference have dogged at the heels of statistics since its earliest days correlation does not imply causation and yet causal conclusions drawn from a carefully designed experiment are often valid what can a statistical model say about causation this question is addressed by using a particular model for causal inference holland and rubin 1983 rubin 1974 to critique the discussions of other writers on causation and causal inference these include selected philosophers medical researchers statisticians econometricians and proponents of causal modeling

marginal log linear parameterization of conditional independence models models defined by a set of conditional independence restrictions play an important role in statistical theory and applications especially but not only in graphical modelling in this paper we identify a subclass of these consisting of hierarchical marginal log linear models as defined by bergsma rudas 2002a such models are smooth which implies the applicability of standard asymptotic theory and simplifies interpretation furthermore we give a marginal log linear parameterization and a minimal specification of the models in the subclass which implies the applicability of standard methods to compute maximum likelihood estimates and simplifies the calculation of the degrees of freedom of chi squared statistics to test goodness of fit the utility of the results is illustrated by applying them to block recursive markov models associated with chain graphs

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

improved likelihood ratio statistics for covariance selection models summary the expected value of the likelihood ratio statistic for covariance selection models when this statistic has a closed form is derived correct to terms of order n 2 n being the sample size then a bartlett type correction factor is available for such statistics by simulation the practical use of such a correction factor is illustrated and is compared with an alternative correction factor in a special case the null density of the corrected statistic is also considered

the connectedness of symmetric and skew symmetric degeneracy loci even ranks a degeneracy locus is the set of points where a vector bundle map has rank at most a given integer such a set is symmetric or skew symmetric according as whether the vector bundle map is symmetric or skew symmetric we prove a connectedness result first conjectured by fulton and lazarsfeld for skew symmetric degeneracy loci and for symmetric degeneracy loci of even ranks

multivariate regression analyses for categorical data summary it is common to observe a vector of discrete and or continuous responses in scientific problems where the objective is to characterize the dependence of each response on explanatory variables and to account for the association between the outcomes the response vector can comprise repeated observations on one variable as in longitudinal studies or genetic studies of families or can include observations for different variables this paper discusses a class of models for the marginal expectations of each response and for pairwise associations the marginal models are contrasted with log linear models two generalized estimating equation approaches are compared for parameter estimation the first focuses on the regression parameters the second simultaneously estimates the regression and association parameters the robustness and efficiency of each is discussed the methods are illustrated with analyses of two data sets from public health research

maximum likelihood estimation in graphical models with missing values in this paper we discuss maximum likelihood estimation when some observations are missing in mixed graphical interaction models assuming a conditional gaussian distribution as introduced by lauritzen wermuth 1989 for the saturated case ml estimation with missing values via the em algorithm has been proposed by little schluchter 1985 we expand their results to the special restrictions in graphical models and indicate a more efficient way to compute the e step the main purpose of the paper is to show that for certain missing patterns the computational effort can considerably be reduced

propagating imprecise probabilities in bayesian networks abstract often experts are incapable of providing exact probabilities likewise samples on which the probabilities in networks are based must often be small and preliminary in such cases the probabilities in the networks are imprecise the imprecision can be handled by second order probability distributions it is convenient to use beta or dirichlet distributions to express the uncertainty about probabilities the problem of how to propagate point probabilities in a bayesian network now is transformed into the problem of how to propagate dirichlet distributions in bayesian networks it is shown that the propagation of dirichlet distributions in bayesian networks with incomplete data results in a system of probability mixtures of beta binomial and dirichlet distributions approximate first order probabilities and their second order probability density functions are obtained by stochastic simulation a number of properties of the propagation of imprecise probabilities are discussed by the use of examples an important property is that the imprecision of inferences increases rapidly as new premises are added to an argument the imprecision can be used as a pruning criterion in a network to keep the number of variables involved in an inferential argument small thus imprecision may be used as an ockam s razor in bayesian networks