m m 1 queue in two alternating environments and its heavy traffic approximation abstract we investigate an m m 1 queue operating in two switching environments where the switch is governed by a two state time homogeneous markov chain this model allows to describe a system that is subject to regular operating phases alternating with anomalous working phases or random repairing periods we first obtain the steady state distribution of the process in terms of a generalized mixture of two geometric distributions in the special case when only one kind of switch is allowed we analyze the transient distribution and investigate the busy period problem the analysis is also performed by means of a suitable heavy traffic approximation which leads to a continuous random process its distribution satisfies a partial differential equation with randomly alternating infinitesimal moments for the approximating process we determine the steady state distribution the transient distribution and a first passage time density

analysis of the linearization around a critical point of an infinite dimensional hamiltonian system partial differential equations that conserve energy can often be written as infinite dimensional hamiltonian systems of the following general form du dt je u where j x x is a symplectic matrix i e jj 1 and e x r is a c 2 functional defined on some hilbert space x

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

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

geometry moments and conditional independence trees with hidden variables we study the geometry of the parameter space for bayesian directed graphical models with hidden variables that have a tree structure and where all the nodes are binary we show that the conditional independence statements implicit in such models can be expressed in terms of polynomial relationships among the central moments this algebraic structure will enable us to identify the inequality constraints on the space of the manifest variables that are induced by the conditional independence assumptions as well as determine the degree of unidentifiability of the parameters associated with the hidden variables by understanding the geometry of the sample space under this class of models we shall propose and discuss simple diagnostic methods 1 introduction graphical models have proved to be a powerful tool for building bayesian models to analyze multivariate problems where all variables are observed e g spiegelhalter dawid lauritzen and cowell 1993 in particular it is possible to estimate all the conditional probabilities that parameterize such models by using a conjugate analysis however when all the data on certain variables in an explanatory model are missing conjugacy usually disappears estimates of these conditional probabilities become highly dependent on one another and they often cannot be determined from data no matter how extensive that data is see e g settimi and smith 1998 in this paper we propose a geometrical approach to analyze such difficulties we first observe that conditional independence assumptions implicit in directed graphical models induce some constraints on the model space that can be expressed as polynomial equations among the central moments we then exploit such an algebraic structure to explore the geometry and the irregularities of the parameter space for bayesian directed graphical models with hidden variables defined over a set of binary variables and such that the conditional independence assumptions are represented via a directed tree understanding the geometry and the singularities of the parameter spaces will enable us to investigate practical statistical issues such as parameter identifiability model dimension and diagnostic methods in the statistical analyses of problems with missing data it has been common practice either to use various methods of approximation to calculate the posterior probabilities of the model parameters see e g spiegelhalter and

graphical krein signature theory and evans krein functions two concepts evidently very different in nature have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory i the krein signature of an eigenvalue a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half plane leading to dynamical instability under perturbation of the system and ii the evans function an analytic function detecting the location of eigenvalues one might expect these two concepts to be related but unfortunately examples demonstrate that there is no way in general to deduce the krein signature of an eigenvalue from the evans function for example by studying derivatives of the latter the purpose of this paper is to recall and popularize a simple graphical interpretation of the krein signature well known in the spectral theory of polynomial operator pencils once esta

a fast procedure for model search in multidimensional contingency tables summary a procedure to select the simplest acceptable models for a multidimensional contingency table is proposed it is based on two rules first that if a model is accepted then all models that include it are considered to be accepted and secondly that if a model is rejected then all its submodels are considered to be rejected two versions are described one for the class of graphical models and the other for the class of hierarchical log linear models application of both versions to a six dimensional table is illustrated the procedure can be regarded as an alternative to fitting all possible models made computationally feasible by application of the two rules it is a generalization of the procedure proposed by havr anek 1984 but is in many cases considerably faster

gaussian influence diagrams an influence diagram is a network representation of probabilistic inference and decision analysis models the nodes correspond to variables that can be either constants uncertain quantities decisions or objectives the arcs reveal probabilistic dependence of the uncertain quantities and information available at the time of the decisions the influence diagram focuses attention on relationships among the variables as a result it is increasingly popular for eliciting and communicating the structure of a decision or probabilistic model this paper develops the framework for assessment and analysis of linear quadratic gaussian models within the influence diagram representation the gaussian influence diagram exploits conditional independence in a model to simplify elicitation of parameters for the multivariate normal distribution it is straightforward to assess and maintain a positive semi definite covariance matrix problems of inference and decision making can be analyzed using simple transformations to the assessed model and these procedures have attractive numerical properties algorithms are also provided to translate between the gaussian influence diagram and covariance matrix representations for the normal distribution

a continuous time markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes abstract we consider a continuous time markov chain n t t 0 with state space s a point process x t t 0 using a function a 0 s in the following way influences the evolution of n t whenever an event of x t occurs at time t the markov chain of interest goes immediately at state a t and evolves according to the dynamics of n t until the next event of x t and so on we study the transient and the limiting distribution of the resulting process and prove some stochastic comparison facts several examples that demonstrate the applicability of the model are also included

a bayesian information criterion for singular models summary we consider approximate bayesian model choice for model selection problems that involve models whose fisher information matrices may fail to be invertible along other competing submodels such singular models do not obey the regularity conditions underlying the derivation of schwarz s bayesian information criterion bic and the penalty structure in bic generally does not reflect the frequentist large sample behaviour of the marginal likelihood although large sample theory for the marginal likelihood of singular models has been developed recently the resulting approximations depend on the true parameter value and lead to a paradox of circular reasoning guided by examples such as determining the number of components in mixture models the number of factors in latent factor models or the rank in reduced rank regression we propose a resolution to this paradox and give a practical extension of bic for singular model selection problems

on the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias causal discovery becomes especially challenging when the possibility of latent confounding and or selection bias is not assumed away for this task ancestral graph models are particularly useful in that they can represent the presence of latent confounding and selection effect without explicitly invoking unobserved variables based on the machinery of ancestral graphs there is a provably sound causal discovery algorithm known as the fci algorithm that allows the possibility of latent confounders and selection bias however the orientation rules used in the algorithm are not complete in this paper we provide additional orientation rules augmented by which the fci algorithm is shown to be complete in the sense that it can under standard assumptions discover all aspects of the causal structure that are uniquely determined by facts of probabilistic dependence and independence the result is useful for developing any causal discovery and reasoning system based on ancestral graph models

a state dependent queueing system with asymptotic logarithmic distribution abstract a markovian single server queueing model with poisson arrivals and state dependent service rates characterized by a logarithmic steady state distribution is considered the laplace transforms of the transition probabilities and of the densities of the first passage time to zero are explicitly evaluated the performance measures are compared with those ones of the well known m m 1 queueing system finally the effect of catastrophes is introduced in the model and the steady state distribution the asymptotic moments and the first visit time density to zero state are determined

strong limit theorems for the bayesian scoring criterion in bayesian networks in the machine learning community the bayesian scoring criterion is widely used for model selection problems one of the fundamental theoretical properties justifying the usage of the bayesian scoring criterion is its consistency in this paper we refine this property for the case of binomial bayesian network models as a by product of our derivations we establish strong consistency and obtain the law of iterated logarithm for the bayesian scoring criterion

dynamics of optical pulses in randomly birefringent fibers abstract a nonlinear optical fiber with random birefringence is modeled by a system of nonlinear schrodinger nls equations with stochastic coupling an average variational principle is used to obtain the approximate evolution of the parameters describing pulses with nls soliton like shapes then in the diffusion limit the stochastic problem is reduced to a partial differential equation of the poincare sphere solutions to this equation give probability densities for the polarization state of a propagating pulse

on the toric algebra of graphical models we formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model or a log linear model or other more general exponential models for decomposable graphical models these conditions are equivalent to a set of conditional independence statements similar to the hammersley clifford theorem however we show that for nondecomposable graphical models they are not we also show that nondecomposable models can have nonrational maximum likelihood estimates these results are used to give several novel characterizations of decomposable graphical models

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

structural stability of non ground state traveling waves of coupled nonlinear schrodinger equations abstract we consider the linear stability and structural stability of non ground state traveling waves of a pair of coupled nonlinear schrodinger equations cnls which describe the evolution of co propagating polarized pulses in the presence of birefringence viewing the cnls equations as a hamiltonian perturbation of the manakov equations we find parameter regimes in which there are two stable families of traveling waves the usual variational methods for stability analysis of ground states do not apply instead we employ a liapunov schmidt type reduction to detect eigenvalues bifurcating from the imaginary axis we also demonstrate the instability of a family of vector solitons

stability of multi parameter solitons asymptotic approach a general asymptotic approach to the stability problem of multi parameter solitons in hamiltonian systems has been developed the presented approach gives an analytical criterion for oscillatory instability and also predicts novel stationary instability of solitons higher order approximations allow one to calculate corresponding eigenvalues with an arbitrary accuracy it is also shown that asymptotic study of the soliton stability reduces to the calculation of a certain sequence of the determinants where the famous determinant of the matrix consisting from the derivatives of the system invariants is just the first in the series

stratified exponential families graphical models and model selection we describe a hierarchy of exponential families which is useful for distinguishing types of graphical models undirected graphical models with no hidden variables are linear exponential families lefs directed acyclic graphical dag models and chain graphs with no hidden variables including dag models with several families of local distributions are curved exponential families cefs graphical models with hidden variables are what we term stratified exponential families sefs a sef is a finite union of cefs of various dimensions satisfying some regularity conditions we also show that this hierarchy of exponential families is noncollapsing with respect to graphical models by providing a graphical model which is a cef but not a lef and a graphical model that is a sef but not a cef finally we show how to compute the dimension of a stratified exponential family these results are discussed in the context of model selection of graphical models

optimal structure identification with greedy search in this paper we prove the so called meek conjecture in particular we show that if a dag h is an independence map of another dag g then there exists a finite sequence of edge additions and covered edge reversals in g such that 1 after each edge modification h remains an independence map of g and 2 after all modifications g h as shown by meek 1997 this result has an important consequence for bayesian approaches to learning bayesian networks from data in the limit of large sample size there exists a two phase greedy search algorithm that when applied to a particular sparsely connected search space provably identifies a perfect map of the generative distribution if that perfect map is a dag we provide a new implementation of the search space using equivalence classes as states for which all operators used in the greedy search can be scored efficiently using local functions of the nodes in the domain finally using both synthetic and real world datasets we demonstrate that the two phase greedy approach leads to good solutions when learning with finite sample sizes

stochastic factorizations sandwiched simplices and the topology of the space of explanations we study the space of stochastic factorizations of a stochastic matrix v motivated by the statistical problem of hidden random variables we show that this space is homeomorphic to the space of simplices sandwiched between two nested convex polyhedra and use this geometrical model to gain some insight into its structure and topology we prove theorems describing its homotopy type and in the case where the rank of v is 2 we give a complete description including bounds on the number of connected components and examples in which these bounds are attained we attempt to make the notions of topology accessible and relevant to statisticians

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

asymptotic model selection for naive bayesian networks we develop a closed form asymptotic formula to compute the marginal likelihood of data given a naive bayesian network model with two hidden states and binary features this formula deviates from the standard bic score our work provides a concrete example that the bic score is generally incorrect for statistical models that belong to stratified exponential families this claim stands in contrast to linear and curved exponential families where the bic score has been proven to provide a correct asymptotic approximation for the marginal likelihood

algebraic factor analysis tetrads pentads and beyond factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables known as factors and all the random variables follow a multivariate normal distribution the parameter space of a factor analysis model is a subset of the cone of positive definite matrices this parameter space is studied from the perspective of computational algebraic geometry grobner bases and resultants are applied to compute the ideal of all polynomial functions that vanish on the parameter space these polynomials known as model invariants arise from rank conditions on a symmetric matrix under elimination of the diagonal entries of the matrix besides revealing the geometry of the factor analysis model the model invariants also furnish useful statistics for testing goodness of fit

on the choice of a model to fit data from an exponential family let x 1 cdots x n be iid observations coming from an exponential family the problem of interest is this given a finite number of models m j smoothly curved manifolds in mathbb r k choose the best model to fit the observations with some penalty for choosing models with dimensions which are too large a result of schwarz is made more specific and is extended to the case where the models are curved manifolds if s y n j is up to a constant c n independent of the model the log of the posterior probability of the j th model where the sample mean y n 1 n sum n i 1 x i has been replaced by y schwarz suggested an asymptotic expansion of s y n j whose leading terms are gamma y n j n sup psi in m j cap theta y psi b psi frac 1 2 k j log n in the case where the models are affine subspaces of mathbb r k we establish a similar asymptotic expansion including the next term with uniform bounds for y in a compact neighborhood of nabla b theta where theta is the true value of the parameter we suggest a criterion for the choice of the best model that consists of maximizing the three leading terms in the expansion s y n j we show that the criterion gives the correct model with probabilities p n theta rightarrow 1 as n rightarrow infty

the krein matrix general theory and concrete applications in atomic bose einstein condensates when finding the nonzero eigenvalues for hamiltonian eigenvalue problems it is especially important to locate not only the unstable eigenvalues i e those with positive real part but also those which are purely imaginary but have negative krein signature these latter eigenvalues have the property that they can become unstable upon collision with other purely imaginary eigenvalues i e they are a necessary building block in the mechanism leading to the so called hamiltonian hopf bifurcation in this paper we review a general theory for constructing a meromorphic matrix valued function the so called krein matrix which has the property of not only locating the unstable eigenvalues but also those with negative krein signature these eigenvalues are realized as zeros of the determinant the resulting finite dimensional problem obtained by setting the determinant of the krein matrix to zero presents a valuable simplification in this paper the usefulness of the technique is illustrated through prototypi

on a class of birth death processes with time varying intensity functions abstract in this paper we investigate on a class of time inhomogeneous birth death chains obtained by applying the composition method to two time inhomogeneous double ended chains then we consider the corresponding restricted birth death process with zero reflecting boundary finally starting from the restricted process we construct a time inhomogeneous bd chain symmetric with respect to zero state we obtain closed form expressions for the transition probabilities and for the conditional moments furthermore the first passage time problem is also taken in consideration finally various numerical computations are performed for periodic intensity functions

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

a normal form for nonlinear resonance of embedded solitons a normal form for nonlinear resonance of embedded solitons is derived for a coupled two wave system that generalizes the second harmonic generating model this wave system is non hamiltonian in general an embedded soliton is a localized mode of the nonlinear system that coexists with the linear wave spectrum it occurs as a result of a codimension one bifurcation of non local wave solutions nonlinearity couples the embedded soliton and the linear wave spectrum and induces a one sided radiation driven decay of embedded solitons the normal form shows that the embedded soliton is semi stable i e it survives under perturbations of one sign but is destroyed by perturbations of the opposite sign when a perturbed embedded soliton sheds continuous wave radiation the radiation amplitude is generally not minimal even if the wave system is hamiltonian the results of the analytical theory are confirmed by numerical computations

transient solution of an m m 1 queue with catastrophes abstract a transient solution for the system size in the m m 1 queueing model with the possibility of catastrophes at the service station is derived in the direct way asymptotic behavior of the probability of the server being idle and mean queue size are discussed steady state probabilities are also obtained

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

ancestral graph markov models this paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all dag independence models this class of graphs called maximal ancestral graphs has two attractive features there is at most one edge between each pair of vertices every missing edge corresponds to an independence relation these features lead to a simple parameterization of the corresponding set of distributions in the gaussian case

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

markov processes on time like graphs we study markov processes where the time parameter is replaced by paths in a directed graph from an initial vertex to a terminal one along each directed path the process is markov and has the same distribution as the one along any other directed path if two directed paths do not interact in a suitable sense then the distributions of the processes on the two paths are conditionally independent given their values at the common endpoint of the two paths conditions on graphs that support such processes e g hexagonal lattice are established next we analyze a particularly suitable family of markov processes called harnesses which includes brownian motion and other l evy processes on such time like graphs finally we investigate continuum limits of harnesses on a sequence of time like graphs that admits a limit in a suitable sense

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

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples