lower bounds for the condition number of vandermonde matrices we derive lower bounds for the condition number of then n vandermonde matrixvn x in the cases where the node vectorxt x1 x2 xn has positive elements or real elements located symmetrically with respect to the origin the bounds obtained grow exponentially inn witho 2n ando 2n 2 respectively we also compute the optimal spectral condition numbers ofvn x for the two node configurations including the optimal nodes and compare them with the bounds obtained

the condition of vandermonde like matrices involving orthogonal polynomials the condition number relative to the frobenius norm of the n x n matrix p pi ltxj ly j l is investigated where p p dx are orthogonal polynomials with respect to some weight distribution dx and xi are pairwise distinct real numbers if the nodes x j are the zeros of p the condition number is either expressed or estimated from below and above in terms of the christoffel numbers for dh depending on whether the p are normalized or not for arbitrary real xi and normalized p a lower bound of the condition number is obtained in terms of the christoffel function evaluated at the nodes numerical results are given for minimizing the condition number as a function of the nodes for selected classical distributions dx

image system for stokes flow singularity between two parallel planar walls using a recently developed image representation for stokes flow in a half space bounded by a planar wall cichocki and jones physica a 258 273 1998 the image system is constructed for the flow field produced by a force multipole in the space bounded by two parallel walls the image singularities are expressed in terms of products of double reflection matrices and the expansion is simplified using symmetries of the double reflection operation our analysis yields recurrence relations for the strengths of the image multipoles the relations are solved explicitly and a complete image system is obtained for an arbitrary source force multipole applications of our image representation for evaluating the hydrodynamic friction and mobility matrices of particles interacting with two parallel planar walls are indicated

a note on the existence of the posterior distribution for a class of mixed models for binomial responses summary necessary and sufficient conditions are given for the existence of the posterior distribution of the variance components in a class of mixed models for binomial responses the implications of our results are illustrated through an example

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

vandermonde systems on gauss lobatto chebyshev nodes abstract this paper deals with vandermonde matrices v n whose nodes are the gauss lobatto chebyshev nodes also called extrema chebyshev nodes we give an analytic factorization and explicit formula for the entries of their inverse and explore its computational issues we also give asymptotic estimates of the frobenius norm of both v n and its inverse and present an explicit formula for the determinant of v n

de finetti s coherence and statistical inference on donne la definition d une inference coherente en conformite avec une theorie des probabilites conditionnelles derivee du principe de coherence de de finetti et on effectue une comparaison critique d une telle definition avec celle proposee par heath lane et sudderth

vandermonde matrices on integer nodes the rectangular case we consider a rectangular vandermonde matrix v on integer nodes using combinatorial identities we give an explicit cholesky factorization of the matrix b v rm t v a factorization of the pseudo inverse of v and asymptotic estimates of the extremal eigenvalues of b we also discuss the numerical properties of the proposed formulas

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

data translated likelihood and jeifreys s rules according to the definition used by box tiao for a likelihood to be data translated it must have location form in terms of a sufficient statistic their definition is easily modified to cover cases such as the cauchy through the introduction of an ancillary statistic and their argument in favour of a uniform prior then becomes group theoretic their concept of approximate data translated likelihood may also be modified to produce a sharper local approximation

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

on the choice of prior distribution for the box cox transformed linear model summary the noninformative prior distribution for the parameters of the linear model transformed following box cox 1964 has the non bayesian property of depending to some extent on the data an alternative choice of prior which is not outcome dependent was suggested by pericchi 1981 but it is argued here that this prior has some undesirable features an alternative family of non outcome dependent priors is suggested leading to a noninformative prior which is closer in spirit to that proposed by box cox the posterior consequences of adopting this prior are fully explored and an example discussed

sure wins separating probabilities and the representation of linear functionals abstract we discuss conditions under which a convex cone k r admits a finitely additive probability m such that sup k k m k 0 based on these we characterise those linear functionals that are representable as finitely additive expectations a version of riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions

the method of images for regularized stokeslets the image system for the method of regularized stokeslets is developed and implemented the method uses smooth localized functions to approximate a delta distribution in the derivation of the fluid flow due to a concentrated force in order to satisfy zero flow boundary conditions at a plane wall the method of images derived for a standard singular stokeslet is extended to give exact cancellation of the regularized flow at the wall as the regularization parameter vanishes the expressions reduce to the known images for singular stokeslets the advantage of the regularized method is that it gives bounded velocity fields even for isolated forces or for distributions of forces along curves these are useful in the simulation of ciliary beats flagellar motion and particle suspensions the expression relating force and velocity can be inverted to find the forces that generate a given velocity boundary condition the latter is exemplified by modeling a cilium as a filament moving in a three dimensional flow the cilium velocity at various times is constructed from known data and used to determine the force field along the filament those forces can then reproduce the flow everywhere the validity of the method is evaluated by computing the drag on a sphere moving near a wall comparisons with known expressions for the drag show that the method gives accurate results for spheres even within a distance from the wall equal to the surface discretization size

far field approximation for hydrodynamic interactions in parallel wall geometry a complete analysis is presented for the far field creeping flow produced by a multipolar force distribution in a fluid confined between two parallel planar walls we show that at distances larger than several wall separations the flow field assumes the hele shaw form i e it is parallel to the walls and varies quadratically in the transverse direction the associated pressure field is a two dimensional harmonic function that is characterized by the same multipolar number m as the original force multipole using these results we derive asymptotic expressions for the green s matrix that represents stokes flow in the wall bounded fluid in terms of a multipolar spherical basis this green s matrix plays a central role in our recently proposed algorithm physica a 356 2005 294 for evaluating many body hydrodynamic interactions in a suspension of spherical particles in the parallel wall geometry implementation of our asymptotic expressions in this algorithm increases its efficiency substantially because the numerically expensive evaluation of the exact matrix elements is needed only for the neighboring particles our asymptotic analysis will also be useful in developing hydrodynamic algorithms for wall bounded periodic systems and implementing acceleration methods by using corresponding results for the two dimensional scalar potential

conditioning likelihood and coherence a review of some foundational concepts 2000 conditioning likelihood and coherence a review of some foundational concepts journal of the american statistical association vol 95 no 452 pp 1340 1346

how bad are hankel matrices considered are hankel vandermonde and krylov basis matrices it is proved that for any real positive definite hankel matrix of order n its spectral condition number is bounded from below by 3 cdot 2 n 6 also proved is that the spectral condition number of a krylov basis matrix is bounded from below by 3 frac 1 2 cdot 2 frac n 2 3 for v v x 1 ldots x n a vandermonde matrix with arbitrary but pairwise distinct nodes x 1 ldots x n we show that mbox rm cond 2 v geq 2 n 2 n frac 1 2 if either x j leq 1 or x j geq 1 for all j then mbox rm cond 2 v geq 2 n 2

finitely maxitive conditional possibilities bayesian like inference disintegrability and conglomerability the aim of the paper is to study bayesian like inference processes involving coherent finitely maxitive t conditional possibilities assessed on infinite sets of conditional events coherence of an assessment consisting of an arbitrary possibilistic prior and an arbitrary possibilistic likelihood function is proved thus a closed form expression for the envelopes of the relevant joint and posterior possibilities is given when t is the minimum or a strict t norm the notions of disintegrability and conglomerability are also studied and their relevance in the infinite version of the possibilistic bayes formula is highlighted

inner statistical inference ii for any linear model there is according to an invariance argument a uniquely defined prior representing ignorance which will be called the inner prior it is shown that the corresponding posterior probabilities are weighted averages of conditional confidence levels this frequency interpretation compares favorably with the usual interpretation of confidence levels as unconditional probabilities of coverage of particular interest to users of statistical methods is that inner posterior intervals are always shorter than the corresponding confidence intervals and sometimes dramatically so

bayesian interval estimates which are also confidence intervals let y 1 y n denote independent observations each distributed according to a distribution depending on a scalar parameter suppose that we are interested in constructing an interval estimate for one approach is to use bayesian inference for a given prior density we can construct an interval such that the posterior probability that lies in the interval is some specified value in this paper a method is given for constructing a bayesian interval estimate such that the coverage probability or the interval is approximately equal to the posterior probability or the interval

a new lubrication theory to derive far field axial pressure difference due to force singularities in cylindrical or annular vessels in simulations of creeping flow the effect of suspended particles on a fluid is often represented by force singularities responsible for singular stokesian solutions which are infinite at the sphere center and decay far from the particle in this article we consider such singular fields centered at a point inside a cylindrical or annular conduit containing highly viscous medium across an unbounded infinite domain these singular flows cannot produce a finite pressure difference as they decay to zero far from the center however in the presence of bounding cylindrical surfaces the reflected flow from the walls creates a finite pressure difference between the far fields across the force singularity along the axial direction to quantify the effect of the reflected flow we present a new lubrication analysis which on the one hand identifies the specific singular fields capable of producing axial pressure difference and on the other hand provides explicit expressions for the far field pressure tho

merging and testing opinions we study the merging and the testing of opinions in the context of a prediction model in the absence of incentive problems opinions can be tested and rejected regardless of whether or not data produces consensus among bayesian agents in contrast in the presence of incentive problems opinions can only be tested and rejected when data produces consensus among bayesian agents these results show a strong connection between the testing and the merging of opinions they also relate the literature on bayesian learning and the literature on testing strategic experts 1 introduction data can produce consensus among bayesian agents who initially disagree it can also test and reject opinions we relate these two critical uses of data in a model where agents may strategically misrepresent what they know in each period either 0 or 1 is observed let p and q be two probability measures on 0 1 1 such that q is absolutely continuous with respect to p if p and q are additive then as shown by blackwell and dubins 1962 the conditional probabilities of p and q merge in the sense that the two posteriors become uniformly close as the amount of observations increases q almost surely so repeated applications of bayes rule lead to consensus among bayesian agents provided that their opinions were initially compatible now consider savage s axiomatization of subjective probability he proposed postulates that characterize a preference relation over bets in terms of a nonatomic finitely additive probability p call such p for short an opinion savage s framework allows for finitely additive probabilities that are not additive in particular the conclusions of the blackwell and dubins theorem hold for some but not all opinions this flexibility makes savage s

vandermonde matrices on integer nodes this paper deals with vandermonde matrices widetilde v whose nodes are the first n integer numbers we give an analytic factorization of such matrices and explicit formulas for the entries of their inverses and explore their computational issues we also give asymptotic estimates of the frobenius norm of both widetilde v and its inverse

on the expected amount of information from a non linear model an approach to quantifying the amount of information that an experimenter expects to learn from a non linear model is given an expected utility for an experiment e motivated by the asymptotic form of shannon information gain between prior and posterior is defined this leads to a characterization of the experimenter who expects to learn the most from a non linear model such an experimenter has the design dependent jeffreys prior sufficient regularity conditions for the equivalence with asymptotic shannon information gain are given

the extent of non conglomerability of finitely additive probabilities an arbitrary finitely additive probability can be decomposed uniquely into a convex combination of a countably additive probability and a purely finitely additive pfa one the coefficient of the pfa probability is an upper bound on the extent to which conglomerability may fail in a finitely additive probability with that decomposition if the probability is defined on a field the bound is sharp hence non conglomerability or equivalently non disintegrability characterizes finitely as opposed to countably additive probability nonetheless there exists a pfa probability which is simultaneously conglomerable over an arbitrary finite set of partitions neither conglomerability nor non conglomerability in a given partition is closed under convex combinations but the convex combination of pfa ultrafilter probabilities each of which cannot be made conglomerable in a common margin is singular with respect to any finitely additive probability that is conglomerable in that margin

convergence in distribution of nonmeasurable random elements a notion of convergence in distribution for non necessarily measurable random elements due to hoffmann jorgensen is characterized in terms of weak convergence of finitely additive probability measures a similar characterization is given for a strengthened version of such a notion further it is shown that the empirical process for an exchangeable sequence can fail to converge due to the nonexistence of any measurable limit although it converges for an i i d sequence because of phenomena of this type hoffmann jorgensen s definition is extended to the case of a nonmeasurable limit in the extended definition naturally suggested by the main results the limit is a finitely additive probability measure

a reference bayesian test for nested hypotheses and its relationship to the schwarz criterion abstract to compute a bayes factor for testing h 0 0 in the presence of a nuisance parameter priors under the null and alternative hypotheses must be chosen as in bayesian estimation an important problem has been to define automatic or reference methods for determining priors based only on the structure of the model in this article we apply the heuristic device of taking the amount of information in the prior on equal to the amount of information in a single observation then after transforming to be null orthogonal to we take the marginal priors on to be equal under the null and alternative hypotheses doing so and taking the prior on to be normal we find that the log of the bayes factor may be approximated by the schwarz criterion with an error of order o p n rather than the usual error of order o p 1 this result suggests the schwarz criterion should provide sensible approximate solutions to bayesian testing problems at least when the hypotheses are nested when

vandermonde matrices on chebyshev points abstract this paper deals with vandermonde matrices on chebyshev points hereafter denoted by v we present simple formulas for the determinant of v and the frobenius norm of both v and v 1 and derive an algorithm for solving the linear systems v p and vtq g numerical experiments to asses the stability of this algorithm are also reported

conditional inference and cauchy models summary many computations associated with the two parameter cauchy model are shown to be greatly simplified if the parameter space is represented by the complex plane rather than the real plane with this convention we show that the family is closed under m6bius transformation of the sample space there is a similar induced transformation on the parameter space the chief raison d etre of the paper however is that the two parameter cauchy model provides an example of a nonunique configuration ancillary in the sense of fisher 1934 such that the maximum likelihood estimate together with either ancillary is minimal sufficient some consequences for bayesian inference and non bayesian conditional inference are explored in particular it is shown that conditional coverage probability assessments depend on the choice of ancillary for moderate deviations the effect occurs in the op n 1 term for large deviations the relative effect is op n2

on the normal matrix of the polynomial ls problem over the chebyshev points this note deals with the properties of the normal matrix of the polynomial ls problem over the chebyshev points and presents an explicit cholesky factorization of that matrix

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

reference priors for the orbit in a group model for a group model in which the group mathbf g acts freely on the parameter space mathbf omega this paper considers a prior which is a product of right haar measure on mathbf g and a limiting form of jeffreys prior for the maximal invariant when the parameter of interest is the orbit of mathbf g in mathbf omega it is shown that such a prior is the reference prior defined by bernardo a method of calculating this reference prior is given which avoids the necessity of working in a parameterization of mathbf omega which expresses mathbf omega as a product of mathbf g and a cross section examples of the multivariate normal distribution with the parameter of interest being the correlation matrix or the eigenvalues of the covariance matrix are discussed

reference prior bayesian analysis for normal mean products abstract two reference priors for the product of means of n normal distributions with common known variance are developed one of them induces an improper posterior distribution and therefore is not of much interest the other is a generalized form of the n 2 case derived by berger and bernardo the latter is compared with the uniform prior the jeffreys prior in posterior inference and the optimal frequentist coverage criterion the reference prior is shown to be better than the uniform prior in the sense of correct frequentist coverage probability of the posterior quantile by numerical computation the computation was performed by gibbs sampling for n 3 and n 10 furthermore it is shown that the reference prior is among the asymptotic optimal frequentist coverage probability priors under a transformation of the parameter space such that the parameter of interest and the nuisance parameters are orthogonal in an example the bayesian credible interval for the product of normal means using the re

fast qr factorization of vandermonde matrices abstract we introduce a fast algorithm for computing the qr factors of a complex column vandermonde matrix v the complexity of the algorithm is 5 mn 7 n 2 2 o m where m is the number of rows in v and n is the number of columns we assume that m n the matrices q and r may be computed independently if desired two special cases for the row vandermonde matrix with real elements or unit magnitude elements are also studied and similar results are obtained

on maximizing missing information about a hypothesis on revise le test bayesien d une moyenne normale h0 0 et on constraste plusieurs interpretations de l indifference a priori

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

proper posteriors from improper priors for an unidentified errors in variables model the problem considered is inference in a simple errors in variables model where consistent estimation is impossible without introducing additional exact prior information the probabilistic prior information required for bayesian analysis is found to be surprisingly light despirte the model s lack of identification a proper posterior is guaranteed for any bounded prior density including those representing improper priors this result is illustrated with the improper uniform prior which implies marginal posterior densities obtainable by integrating the likelihood function

a bayesian approach to transformations to normality summary the analysis of transformation of observations in the linear model with normal errors proposed by box cox 1964 is considered a different choice of noninformative unnormed prior is advocated which is not outcome dependent this new selection of prior leads to a formal identity between likelihood and bayesian inference both for the estimation of the best transformation to normality and for the presence of homoscedasticity and additivity under this transformation extension to a related problem is mentioned

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

frequentist and bayesian bartlett correction of test statistics based on adjusted profile likelihoods summary for inference about a multidimensional parameter b in the presence of nuisance param

parametric models for an splitting processes and mixtures a class of parametric models called splitting processes is defined by using de finetti s concept of adherent mass such splitting processes give rise to complex mixtures of distributions it is proved that the nonparametric bayesian predictive procedure a n of hill holds exactly for a member of this class called a nested splitting process the connection between a n and the dirichlet process is stated and proved a multivariate version of a n based on splitting processes is proposed

the unfathomable influence of kolmogorov a n kolmogorov contributed much to mathematical statistics both through his own statistical works and through the researches of his students and disciples in statistics but the case will be made that he influenced statistics still more through his general mathematical concepts and approaches this paper gives some examples of unexpected applications of kolmogorov s ideas in topics of nonparametric statistics

rectangular vandermonde matrices on chebyshev nodes abstract we consider a rectangular vandermonde matrix v on chebyshev nodes using combinatorial identities we give an explicit factorization of the matrix v a factorization of thepseudo inverse of v and prove that the condition number 2 v does not depend onthe dimension of nodes set we also discuss the numerical properties of the proposed formulas

ordered group reference priors with application to the multinomial problem summary noninformative priors are developed using the reference prior approach for multiparameter problems in which there may be parameters of interest and nuisance parameters for a given grouping of parameters and ordering of the groups intuitively according to inferential importance an algorithm for determining the associated reference prior is presented the algorithm is illustrated on the multinomial problem with discussion of the variety and success of various groupings and ordering strategies

bayesianly justifiable and relevant frequency calculations for the applied statistician a common reaction among applied statisticians is that the bayesian statistician s energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying bayes s theorem to calculate conditional distributions of unknowns given knowns i feel however that the applied bayesian statistician s tool kit should be more extensive and include tools that may be usefully labeled frequency calculations three types of bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician

bayesian intervals with good frequentist behaviour in the presence of nuisance parameters given a random sample from a distribution with density function that depends on an unknown parameter 1 p we are concerned with the problem of setting confidence intervals for a particular component of it say 1 treating the remaining components as a nuisance parameter adopting an objective bayesian approach we show that the bayes intervals with a certain conditional prior density on the parameter of interest 1 are confidence intervals as well having nearly the correct frequency of coverage the frequentist performance of the proposed intervals is tested in a simulation study for gamma mean and shape parameters

an alternative to the standard bayesian procedure for discrimination between normal linear models summary we consider the standard bayesian procedure for discrimination focusing on its tendency to give low posterior probabilities to some correct models even in conditions of moderate to large sampling sizes furthermore in some situations that could well occur in practice the standard procedure is even asymptotically inconsistent we claim that the explanation is that the standard procedure inflates the posterior probability of the model that has the smallest expected increase in information about its parameters we then propose to modify it by an amount related to the expected gain in information and show that the lindley paradox does not occur finally we obtain the limiting alternative criterion that results by letting the prior densities become noninformative which enables us to measure how sensitive the discrimation is to the informative prior densities used

on the scoring approach to admissibility of uncertainty measures in expert systems abstract this paper arose from our need to rigorize clarify and address fully the results of lindley s paper scoring rules and the inevitability of probability internat statist rev 50 1982 1 26 herein we develop a calculus of admissibility in a game theoretic setting in the case of an additive aggregation function it is shown that decomposable measures such as those used in fuzzy logics are admissible also the problem of the admissibility of the dempster shafer belief functions is investigated via the concept of random sets it is shown that the class of admissible measures in a scoring framework depends on the assumptions concerning the aggregation function in use in particular for nonadditive aggregation functions an admissible measure may not be transformable to a finitely additive probability measure

jeffreys prior for logit models abstract this paper investigates jeffreys prior for logit models with covariates in conditional logit models it is not recommended and in multinomial logit models it has properties similar to a neutral natural conjugate prior but none of the computational advantages

on the existence of inferences which are consistent with a given model if p is a additive statistical model and a finitely additive prior then any statistic t is sufficient with respect to a suitable inference consistent with p and provided only that p t t 0 for all and t here sufficiency is to be intended in the bayesian sense and consistency in the sense of lane and sudderth as a corollary if p is additive and diffuse then whatever the prior there is an inference which is consistent with p and two versions of the main result are also obtained for predictive problems

some remarks on the use of improper priors for the analysis of exponential regression models summary we consider bayesian inference on the exponential regression model it is known that standard improper priors on the parameters of this model lead to marginal posterior distributions on one of the parameters involved which are also improper on the interval 0 1 this has sometimes been interpreted as meaning that the observations are irrelevant we re analyse this problem using a finitely additive approach and show that the above conclusion is not generally correct this result emphasizes once again the dangers of a routine approach to bayesian inference using improper priors

noninformative priors and nuisance parameters abstract we study the conflict between priors that are noninformative for a parameter of interest versus priors that are noninformative for the whole parameter our investigation leads us to maximize a functional that has two terms an asymptotic approximation to a standardized expected kullback leibler distance between the marginal prior and marginal posterior for a parameter of interest and a penalty term measuring the distance of the prior from the jeffreys prior a positive constant multiplying the second terms determines the tradeoff between noninformativity for the parameter of interest and noninformativity for the entire parameter as the constant increases the prior tends to the jeffreys prior when the constant tends to 0 the prior becomes degenerate except in special cases this prior does not have a closed form solution but we present a simple numerical algorithm for finding the prior we compare this prior to the berger bernardo prior

further thoughts on testing for causality with econometric models abstract the ontological basis for causality testing must be some empirically interpretable system leading one to specify a logically valid a priori law that can be shown to exist such paradigms as conventional linear and non linear models be they structural or time series tend to embody contradictory assumptions that unfortunately make causal interpretability problematic the paper demonstrates how these difficulties can be avoided using a stochastic coefficient approach the final part of this article is devoted to a discussion of probabilistic logic as a valid tool for scientific analysis and interpretation of causal relationships

explicit solution of the polynomial least squares approximation problem on chebyshev extrema nodes in this paper we propose an explicit solution to the polynomial least squares approximation problem on chebyshev extrema nodes we also show that the inverse of the normal matrix on this set of nodes can be represented as the sum of two symmetric matrices a full rank matrix which admits a cholesky factorization and a 2 rank matrix finally we discuss the numerical properties of the proposed formulas

noninformative priors for inferences in exponential regression models summary in the exponential regression model inference concerning the regression parameter is notoriously difficult even when using the bayesian noninformative prior approach the reference prior approach bernardo 1979 berger bernardo 1989 is considered and argued to yield very satisfactory inferences estimation and credible sets are considered in a specific example