two laplacians for the distance matrix of a graph abstract we introduce a laplacian and a signless laplacian for the distance matrix of a connected graph called the distance laplacian and distance signless laplacian respectively we show the equivalence between the distance signless laplacian distance laplacian and the distance spectra for the class of transmission regular graphs there is also an equivalence between the laplacian spectrum and the distance laplacian spectrum of any connected graph of diameter 2 similarities between n as a distance laplacian eigenvalue and the algebraic connectivity are established

the graphs with the least distance eigenvalue at least 1 172 let g be a connected graph with order n and d g be the distance matrix of g suppose that 1 d g n d g are the distance spectra of g in this paper we characterize the graphs with n d g 1 172 1 1 1 2 where is the smallest root of x3 x2 3x 1 0 and 1 172 1 1 2 furthermore we show that the graphs with n 1 172 are determined by their d spectrum

frequentist validity of posterior quantiles in the presence of a nuisance parameter higher order asymptotics given a random sample from a distribution with density function that depends on an unknown parameter 1 2 we are concerned with frequentist validity up to o n 1 of posterior quantiles of 1 treating 2 as a nuisance parameter we propose to make the best choice of the prior on by matching as far as practicable the posterior and frequentist coverage probabilities up to o n 1

probability matching priors for non regular cases we study probability matching priors in non regular cases both one parameter and multi parameter cases are considered the resulting priors are shown to satisfy certain differential equations several examples are discussed it is observed that the reference prior in the sense of bernardo is in many cases also probability matching as a necessary tool for finding such priors asymptotic expansion of posterior distributions for multi parameter non regular cases is also obtained

on the invariance of noninformative priors various noninformative priors have been proposed in the literature this article explores the invariance or lack thereof of some of these noninformative priors including the reference prior of berger and bernardo the reverse reference prior of j k ghosh and the probabilitymatching prior of peers and stein under reparameterization berger and bernardo s m group ordered reference prior is shown to remain invariant under a special type of reparameterization the reverse reference prior of j k ghosh is shown not to remain invariant under reparameterization however the probability matching prior is shown to remain invariant under any reparameterization also for spherically symmetric distributions certain noninformative priors are derived using the principle of group invariance

the aα spectral radius characterization of some digraphs abstract let d be the a spectral radius of digraph d and let g n r be the set of digraphs with order n and dichromatic number r in this paper we characterize the digraph which has the maximal a spectral radius in g n r moreover we also determine the unique digraph which attains the maximum resp minimum a spectral radius among all strongly connected bicyclic digraphs

characterizing ctl like logics on finite trees we associate a modal operator with each language belonging to a given class of regular tree languages and use the cascade product of tree automata to give an algebraic characterization of the expressive power of the resulting logic

on graphs with at least three distance eigenvalues less than 1 abstract let g be a connected graph with order n and d g be the distance matrix of g suppose that 1 d 2 d n d are the d eigenvalue of g in this paper we show that n 1 d g 1 if n 4 and n 2 d g 1 if n 7 we also characterize all connected graphs with n 1 d g 1 moreover it is shown that these graphs are determined by their distance spectra

on priors providing frequentist validity of bayesian inference for multiple parametric functions summary we characterise priors which match up to o n 1 the posterior joint cumulative distribution function of multiple parametric functions with the corresponding frequentist cumulative distribution function this work extends and unifies the work of ghosh mukerjee 1993 and datta ghosh 1995a on the topic of probability matching priors a set of necessary and sufficient conditions is obtained for the above characterisation some of these conditions depend only on the parametric functions and not on the prior examples are given where the joint probability matching is possible and where it is not possible

regular languages defined with generalized quantifiers we study an extension of first order logic obtained by adjoining quantifiers that count with respect to an integer modulus it is shown that the languages definable in this framework are precisely the regular languages whose syntactic monoids contain only solvable groups we obtain an analogous result for regular languages and establish some connections with complexity theory for fixed depth families of circuits

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

frequentist validity of posterior quantiles for a two parameter exponential family summary berger bernardo s 1989 reference priors and tibshirani 1989 priors are derived for distributions in bar lev reiser s 1982 two parameter exponential family when either the location or the scale parameter is of interest the reference prior is shown to be a tibshirani prior furthermore conditions under which a tibshirani prior is a higherorder matching prior are investigated when both the parameters are of interest it is also shown that the reference prior agrees with the prior matching the posterior and frequentist expansions based on the signed square root of log likelihood ratio the normal inverse gaussian and gamma distributions are used to illustrate the results

some remarks on noninformative priors abstract this article focuses primarily on a comparison between the reference priors of berger and bernardo and the reverse reference priors suggested by j k ghosh sufficient conditions are given that provide agreement between the two classes of priors several examples are given showing the agreement or disagreement between the two in addition these priors are compared under a criterion that requires the frequentist coverage probability of the posterior region of a real valued parametric function to match a nominal level with a remainder of o n 1 where n denotes the sample size the latter priors first introduced by welch and peers are obtained by solving a differential equation due to peers finally in the presence of several parameters of interest a general class of priors that satisfies the matching criterion separately for each parameter is constructed and examples are given to illustrate how reference or reverse reference priors fit within this class of priors

maximizing the spectral radius of k connected graphs with given diameter abstract the spectral radius of a graph is the largest eigenvalue of its adjacency matrix p hansen and d stevanovic 2008 9 determined the graphs with maximum spectral radius among all connected graphs of order n with diameter d in this paper we generalize this result to k connected graphs of order n with diameter d

the generalized distance matrix of digraphs abstract let d g and d q g d i a g t r d g be the distance matrix and distance signless laplacian matrix of a simple strongly connected digraph g respectively where d i a g t r diag d 1 d 2 d n be the diagonal matrix with vertex transmissions of the digraph g to track the gradual change of d g into d q g in this paper we propose to study the convex combinations of d g and d i a g t r defined by d g d i a g t r 1 d g 0 1 this study reduces to merging the distance spectral and distance signless laplacian spectral theories the eigenvalue with the largest modulus of d g is called the d spectral radius of g denoted by g we determine the digraph which attains the maximum or minimum d spectral radius among all strongly connected digraphs moreover we also determine the digraphs which attain the minimum d spectral radius among all strongly connected digraphs with given parameters such as dichromatic number vertex connectivity or arc connectivity

noninformative priors for one parameter of many summary we consider the problem of constructing a prior that is noninformative for a single parameter in the presence of nuisance parameters our approach is to require that the resulting marginal posterior intervals have accurate frequentist coverage stein 1985 derived nonrigorously a sufficient condition for such a prior through the use of orthogonal parameters we give a general form for the class of priors satisfying stein s condition the priors are proportional to the square root of the information element for the parameter of interest times an arbitrary function of the nuisance parameters this is in contrast to jeffreys 1946 invariant prior for the overall parameter which is proportional to the square root of the determinant of the information matrix several examples are given and comparisons are made to the reference priors of bernardo 1979

bootstrap choice of the smoothing parameter in kernel density estimation summary cross validation based on integrated squared error has already been applied to the choice of smoothing parameter in the kernel method of density estimation in this paper an alternative resampling plan based on the bootstrap is proposed in an attempt to estimate mean integrated squared error this leads to a further data based choice of smoothing parameter the two methods are compared and some simulations and examples demonstrate the relative merits for large samples the bootstrap performs better than cross validation for many distributions

some results on the distance and distance signless laplacian spectral radius of graphs and digraphs let d g denote the distance spectral radius of a graph g and g denote the distance signless laplacian spectral radius of a digraph g let g n k d be the set of all k connected graphs of order n with diameter d in this paper we first determine the unique graph with minimum distance spectral radius in g n k d we then give sharp upper and lower bounds for the distance signless laplacian spectral radius of strongly connected digraphs we also characterize the digraphs having the maximal and minimal distance signless laplacian spectral radii among all strongly connected digraphs furthermore we determine the extremal digraph with the minimal distance signless laplacian spectral radius with given dichromatic number

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

on the implementation of local probability matching priors for interest parameters probability matching priors are priors for which the posterior probabilities of certain specified sets are exactly or approximately equal to their coverage probabilities these priors arise as solutions of partial differential equations that may be difficult to solve either analytically or numerically recently levine casella 2003 presented an algorithm for the implementation of probability matching priors for an interest parameter in the presence of a single nuisance parameter in this paper we develop a local implementation that is very much more easily computed a local probability matching prior is a data dependent approximation to a probability matching prior and is such that the asymptotic order of approximation of the frequentist coverage probability is not degraded we illustrate the theory with a number of examples including three discussed in levine casella 2003

on the distance spectrum of graphs abstract let g be a connected graph and d g be the distance matrix of g suppose that 1 d 2 d n d are the d eigenvalues of g in this paper we characterize all connected graphs with n d 2 furthermore we characterize all connected graphs of diameter 2 with exactly three d eigenvalues when 1 d is not an integer we also conjecture that the complete k partite graph is determined by its d spectrum

implications of reference priors for prior information and for sample size abstract here we use posterior densities based on relative entropy reference priors for two purposes the first purpose is to identify data implicit in the use of informative priors we represent an informative prior as the posterior from an experiment with a known likelihood and a reference prior minimizing the relative entropy distance between this posterior and the informative prior over choices of data results in a data set that can be regarded as representative of the information in the informative prior the second implication from reference priors is obtained by replacing the informative prior with a class of densities from which one might wish to make inferences for each density in this class one can obtain a data set that minimizes a relative entropy the maximum of these sample sizes as the inferential density varies over its class can be used as a guess as to how much data is required for the desired inferences we bound this sample size above and below by other techniques that permit it to

recognisability for algebras of infinite trees we develop an algebraic language theory for languages of infinite trees we define a class of algebras called w hyperclones and we show that a language of infinite trees is regular if and only if it is recognised by a finitary path continuous w hyperclone

general varieties of tree languages we consider varieties of tree languages which are not restricted to a fixed ranked alphabet varieties of finite algebras that contain algebras of all finite types and a matching notion of varieties of congruences a variety theorem that yields isomorphisms between the lattices formed by these three types of varieties is proved to achieve this some basic universal algebra is suitably generalized and we define syntactic algebras so that two tree languages over any alphabets belong to the same varieties exactly in case their syntactic algebras are isomorphic many families of regular tree languages are shown to be varieties in our sense in particular we prove that every family of regular tree languages which can be characterized by syntactic monoids is such a variety but that the converse does not hold

the inertia and energy of the distance matrix of a connected graph abstract let g be a connected graph and d g be the distance matrix of g suppose that 1 d 2 d n d are the d eigenvalues of g in this paper we show that the distance matrix of a clique tree is non singular moreover we also prove that the distance matrix of a clique tree has exactly one positive d eigenvalue in addition we determine the extremal graphs with maximum and minimum distance energy among all clique trees

implementing matching priors for frequentist inference summary nuisance parameters do not pose any problems in bayesian inference as marginalisation allows for study of the posterior distribution solely in terms of the parameter of interest however no general solution is available for removing nuisance parameters under the frequentist paradigm in this paper we merge the two approaches to construct a general procedure for frequentist elimination of nuisance parameters through the use of matching priors in particular we perform bayesian marginalisation with respect to a prior distribution under which posterior inferences have approximate frequentist validity matching priors are constructed as solutions to a partial differential equation unfortunately except in simple cases these partial differential equations do not yield to analytical nor even standard numerical methods of solution we present a numerical monte carlo algorithm for obtaining the matching prior in general as a solution to the appropriate partial differential equation and draw posterior inferences to be specific we develop an automated routine through an implementation of the metropolis hastings algorithm for deriving frequentist valid inferences via the matching prior we illustrate our results in the contexts of fitting random effects models fitting logistic regression models and fitting teratological data by beta binomial models

estimating a product of means bayesian analysis with reference priors abstract suppose that we observe x n 1 and independently y n 1 and are concerned with inference mainly estimation and confidence statements about the product of means this problem arises most obviously in situations of determining area based on measurements of length and width it also arises in other practical contexts however for instance in gypsy moth studies the hatching rate of larvae per unit area can be estimated as the product of the mean of egg masses per unit area times the mean number of larvae hatching per egg mass approximately independent samples can be obtained for each mean see southwood 1978 noninformative prior bayesian approaches to the problem are considered in particular the reference prior approach of bernardo 1979 an appropriate reference prior for the problem is developed and relatively easily implementable formulas for posterior moments e g the posterior mean and variance and credible sets are derived comparisons with alternative noninf

on the distance and distance laplacian eigenvalues of graphs abstract let g v e be a simple graph with vertex set v g v 1 v 2 v n and edge set e g let d g be the distance matrix of g for a given nonnegative integer k when n is sufficiently large with respect to k we show that n k d 1 thereby solving a problem proposed by lin et al 2014 8 the distance laplacian spectral radius of a connected graph g is the spectral radius of the distance laplacian matrix of g defined as d l g tr g d g where tr g is the diagonal matrix of vertex transmissions of g aouchiche and hansen 2014 3 conjectured that m 1 d l n 2 when g k n and the equality holds if and only if either g k 1 n 1 or g k n 2 n 2 in this paper we confirm the conjecture

the distance signless laplacian spectral radius of digraphs with given arc connectivity abstract let g n k denote the set of strongly connected digraphs with order n and arc connectivity k and let g n k denote the set of digraphs in g n k with all vertices having outdegree and indegree greater than k in this paper we determine the unique digraph with the maximum signless laplacian spectral radius among all digraphs in g n k we also determine the unique one with the maximum signless laplacian spectral radius among all digraphs in g n k with k 1 2 for the general case we propose a conjecture on the maximum signless laplacian spectral radius among all digraphs in g n k moreover we characterize the extremal digraph achieving the minimum distance signless laplacian spectral radius among all digraphs in g n k we also characterize the extremal digraph achieving the minimum distance signless laplacian spectral radius among all digraphs in g n k with k 1 2 for the general case we propose a conjecture on the minimum distance signless laplacian spectral radius among all digraphs in g n k

on the distance spectral radius of some graphs abstract for a connected graph the distance spectral radius is the largest eigenvalue of its distance matrix in this paper we determine the unique graph with minimum distance spectral radius among all connected graphs of order n with a given diameter moreover we determine the unique graph with maximum distance spectral radius among the catacondensed hexagonal systems with h hexagons

characterization of extremal graphs from distance signless laplacian eigenvalues abstract let g v e be a connected graph with vertex set v g v 1 v 2 v n and edge set e g the transmission tr v i of vertex v i is defined to be the sum of distances from v i to all other vertices let tr g be the n n diagonal matrix with its i i entry equal to tr g v i the distance signless laplacian is defined as d q g tr g d g where d g is the distance matrix of g let 1 g 2 g n g denote the eigenvalues of distance signless laplacian matrix of g in this paper we first characterize all graphs with n g n 2 secondly we characterize all graphs with 2 g n 2 n when n 11 furthermore we give the lower bound on 2 g with independence number and the extremal graph is also characterized

default priors for bayesian and frequentist inference summary we investigate the choice of default priors for use with likelihood for bayesian and frequentist inference such a prior is a density or relative density that weights an observed likelihood function leading to the elimination of parameters that are not of interest and then a density type assessment for a parameter of interest for independent responses from a continuous model we develop a prior for the full parameter that is closely linked to the original bayes approach and provides an extension of the right invariant measure to general contexts we then develop a modified prior that is targeted on a component parameter of interest and by targeting avoids the marginalization paradoxes of dawid and co workers this modifies jeffreys s prior and provides extensions to the development of welch and peers these two approaches are combined to explore priors for a vector parameter of interest in the presence of a vector nuisance parameter examples are given to illustrate the computation of the priors