optimal estimation from limited noisy data abstract we discuss the following problem let a b f j dy f j e j 1 j n g3 j e p c jp given f j e j 1 j n estimate f x the idea is as follows the estimate is sought in the form a b f j dy f j j 1 j n j p c jp given f j j 1 j n estimate f x the idea is as follows the estimate is sought in the form f n x j 1 n j x f j j the problem is to find j x such that 1 e j 0 then tf n f 0 n 2 if g3 j e p c jp then d f n f 2 where 0 is a certain number we find j 1 j n optimal in a certain sense namely if e j 0 then j are found from the requirements a j 1 n j x a b j dy 1 a b l 1 n j x j y 2 x y 2 dy min note that a j 1 n j x j y n x y in an optimal way if e j 0 then j are found from a and c 2 if this problem is soluble and if not one increases 2 so that this problem becomes soluble

local tomography with nonsmooth attenuation ii we continue investigation of local tomography with nonsmooth attenuation which was proposed earlier by the author our main result is a new and simple characterization of the wave front of the local tomography function tilde f lambda left phi right which holds under fairly general conditions on the density function f and the attenuation coefficient

noninformative priors for dispersion models summary the present paper provides a bayesian analysis of two parameter dispersion models under noninformative priors the focus is on first and second order probability matching priors other noninformative priors including jeffreys and reference priors are examined in the light of these criteria for certain models some unique second order probability matching priors are found which are different from either jeffreys or reference priors

local tomography for the generalized radon transform let f be piecewise smooth with only jump discontinuities across s singsupp f the asymptotic behavior of cal b f near s is obtained here cal b is an elliptic pseudodifferential operator which satisfies certain not very restrictive assumptions an efficient numerical scheme for computing local tomography lt functions for the generalized radon transform grt is proposed the effect of mollification of the two lt functions is investigated an algorithm for finding values of jumps of f knowing local tomographic data corresponding to the grt is developed and tested on the simulated data

coverage probability bias objective bayes and the likelihood principle summary we review objective bayes procedures based on both parametric and predictive coverage probability bias and explore the extent to which such procedures contravene the likelihood principle in the case of a scalar parameter the discussion encompasses choice of objective priors objective posterior probability statements and objective predictive probability statements we conclude with some remarks concerning the future development and implementation of objective priors based on small coverage probability bias

jeffreys prior analysis of the simultaneous equations model in the case with n 1 endogenous variables this paper analyzes the behavior of posterior distributions under the jereys prior in a simul taneous equations model the case under study is that of a general limited information setup with n 1 endogenous variables the jereys prior is shown to give rise to a marginal pos terior density which has cauchy like tails similar to that exhibited by the exactnite sample distribution of the corresponding liml estimator a stronger correspondence is established in the special case of a just identied orthonormal canonical model where the posterior density under the jereys prior is shown to have the same functional form as the density of thenite sample distribution of the liml estimator the work here generalizes that of chao and phillips j econ 87 1998 49 which gave analogous results for the special case of an equation with two endogenous variables c 2002 elsevier science b v all rights reserved jel classication c11 c31

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

single observation unbiased priors this paper studies a class of default priors which we call single observation unbiased priors soup a prior for a parameter is a soup if the corresponding posterior mean of the parameter based on a single observation is an unbiased estimator of the parameter we prove that under mild regularity conditions a default prior for a convolution parameter is noninformative in the sense of yielding a posterior inference invariant under amalgamation only if it is a soup therefore when amalgamation invariance is desirable as in our motivating example of performing imputation for census undercount soup is the only possible coherent noninformative prior for bayesian predictions that will be utilized under aggregation the use of soup also mutually calibrates bayesian and frequentist inferences for aggregates of convolution parameters across many small areas we describe approaches that identify soups in many common models in particular a constructive duality method that identifies soups in pairs of distribution families we introduce o completeness a necessary and sufficient condition for a prior distribution to be uniquely characterized by the corresponding posterior mean uniqueness of the soup is determined by the o completeness of the dual family o completeness of a natural exponential family is implied by its completeness hence the diaconis ylvisaker characterization of the conjugate prior for natural exponential families via linear posterior expectation is a direct consequence of the completeness of such families for most of the examples we have examined the inverse of the variance function is the soup for the mean parameter of the corresponding family suggesting that hartigan s results on asymptotic unbiasedness can be generalized to some families with discrete parameters we also discuss a possible extension of berger s result on the inadmissibility of unbiased estimators as the nonexistence of soup can be a first step in establishing such inadmissibility

on asymptotic normality of limiting density functions with bayesian implications walker 1969 presented a straightforward approach to asymptotic normality of posterior distributions based on independent and identically distributed i i d observations dawid 1970 extended this work by providing a weaker set of conditions to cover some distributions whose ranges depend on the parameters of interest more recently heyde and johnstone 1979 simplified walker s conditions for stochastic processes in this paper we introduce three basic normality conditions for limiting density functions in the bayesian context our approach is data determined and is nonprobabilistic in nature nevertheless the results thus obtained can be easily converted to probabilistic ones when compared with those given by other investigators it is shown that our conditions as a whole have a greater generality and flexibility than those previously assumed either by dawid or heyde and johnstone in particular for the i i d case the commonly assumed wald type uniform boundedness condition for tail behaviours is shown to be unnecessary for asymptotic normality section 3 3 the advantage of our approach is further exemplified by a simple proof of the asymptotic normality of distributions conjugate to the exponential family

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

the character of the infinite wedge representation abstract we study the character of the infinite wedge projective representation of the algebra of differential operators on the circle we prove quasi modularity of this character and also compute certain generating functions for traces of differential operators which we call correlation functions these correlation functions are sums of determinants built from genus 1 theta functions and their derivatives

practical use of higher order asymptotics for multiparameter exponential families recently developed asymptotics based on saddlepoint methods provide important practical methods for multiparameter exponential families especially in generalized linear models the aim here is to clarify and explore these attention is restricted to tests and confidence intervals regarding a single parametric function which can be represented as a natural parameter of a full rank exponential family excellent approximations to exact conditional inferences are often available in terms of simple adjustments to the signed square root of the likelihood ratio statistic

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

local algorithms in exterior tomography exterior tomographic data are taken over lines outside a central region and such data occur in the industrial nondestructive evaluation of large objects such as rockets we explain using microlocal analysis which singularities are well reconstructed from exterior data and we explain how this phenomenon is reflected in the singular value decomposition for the exterior transform e t quinto singular value decompositions and inversion methods for the exterior radon transform and a spherical transform j math anal appl 95 1983 437 448 we extend lambda tomography to exterior data and to limited angle exterior data the algorithm is tested on industrial data from perceptics inc

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

vertex operators and quasimodularity of chern numbers on the hilbert scheme abstract in carlsson and okounkov preprint 7 okounkov and the author defined a family of vertex operators on the equivariant cohomology groups of the hilbert scheme of points on a smooth quasi projective surface as a characteristic class of certain canonical bundles on hilb s hilb s we then proved a bosonization formula in terms of nakajima s heisenberg operators nakajima 1997 23 in this paper we apply this operator in the special case when s c 2 with a particular action of a torus and prove that the generating functions of equivariant chern numbers on hilb n over n are quasimodular forms in the generating variable this property determines the answer up to a finite dimensional vector space of functions of the generating variable q these generating functions can be thought of as the analogous correlation functions to nekrasov s partition function in rank 1 we present a different proof of the bosonization formula which is based on the proof of a more general formula in k theory given in an upcoming paper by nekrasov okounkov and the author in preparation 6 but specialized to the surface of interest by altering a certain bundle that appears in carlsson et al in preparation 6 and specializing the surface the proof actually reduces to a much simpler self contained application of the infinite wedge representation this picture is consistent with both the original introduction of this operator in nekrasov and okounkov 2006 26 and with haiman s character theory of the bridgeland king and reid isomorphism haiman 2003 11

elimination of nuisance parameters with reference priors summary the problem of eliminating nuisance parameters is tackled from different points of view standard likelihood techniques such as profile likelihood and its modifications are compared with a bayesian analysis based on reference priors berger bernardo 1992a examples are considered to illustrate what happens in multiparameter problems in particular the frequentist coverage probabilities of the credible sets derived from the reference priors are shown to be better than those computed from a likelihood approach advantages and disadvantages of bayesian and likelihood analyses are discussed

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

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

bayesian prediction with approximate frequentist validity we characterize priors which asymptotically match the posterior coverage probability of a bayesian prediction region with the corresponding frequentist coverage probability this is done considering both posterior quantiles and highest predictive density regions with reference to a future observation the resulting priors are shown to be invariant under reparameterization the role of jeffreys prior in this regard is also investigated it is further shown that for any given prior it may be possible to choose an interval whose bayesian predictive and frequentist coverage probabilities are asymptotically matched

second order probability matching priors for a parametric function with application to bayesian tolerance limits summary conditions are obtained for the approximate frequentist validity of the posterior quantiles of any smooth parametric function an application to bayesian tolerance limits is indicated

smoothed projection methods for the moment problem we discuss the problem of reconstructing a functionf from a finite set of moments problems of this kind typically arise as discretizations of integral equations of the first kind we propose an algorithm which is based on a pointwise optimization of the pointspread function which makes it particularly suitable for local reconstructions the method is compared with known methods as backus gilbert and projection methods convergence of the method is proved and the rate of convergence is determined the influence of noisy data is examined and numerical examples show the usefulness of the method

inversion of limited angle tomographic data abstract a new method is given for analytical inversion of the radon transform given limited angle data numerical aspects of this inversion problem are discussed

wavelet based multiresolution local tomography we develop an algorithm to reconstruct the wavelet coefficients of an image from the radon transform data the proposed method uses the properties of wavelets to localize the radon transform and can be used to reconstruct a local region of the cross section of a body using almost completely local data that significantly reduces the amount of exposure and computations in x ray tomography the property that distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet bases with sufficiently many vanishing moments the ramp filtered version of the scaling function as well as the wavelet function has extremely rapid decay we show that the variance of the elements of the null space is negligible in the locally reconstructed image also we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm to reconstruct a local region 16 pixels in radius in a 256 spl times 256 image we require 22 of full exposure data

objective bayes conditional inference and the signed root likelihood ratio statistic barndorffnielsen oe 1986 biometrika v73 p307 barndorff nielsen o e 1994 inference asymptotic datta g s 2004 probability matching diciccio tj 1993 j roy stat soc b met v55 p305 diciccio tj 1993 biometrika v80 p731 doi 10 1093 biomet 80 4 731 diciccio tj 1994 j roy stat soc b met v56 p397 diciccio tj 1994 scand j stat v21 p447 diciccio tj 2010 biometrika v97 p497 doi 10 1093 biomet asq002 hall p 1992 bootstrap edgeworth lawless jf 1982 stat models methods mukerjee r 1993 biometrika v80 p499 doi 10 1093 biomet 80 3 499 mukerjee r 1997 biometrika v84 p970 doi 10 1093 biomet 84 4 970 nicolaou a 1993 j roy stat soc b met v55 p377 peers hw 1965 j roy stat soc b v27 p9 tibshirani r 1989 biometrika v76 p604 doi 10 1093 biomet 76 3 604 tierney l 1986 j am stat assoc v81 p82 doi 10 2307 2287970 welch b l 1963 j r stat soc b v62 p159

heisenberg algebra and hilbert schemes of points on projective surfaces the purpose of this paper is to throw a bridge between two seemingly unrelated subjects one is the hilbert scheme of points on projective surfaces which has been intensively studied by various people see e g i br es g61 g62 the other is the infinite dimensional heisenberg algebra which is closely related to affine lie algebras see e g k we shall construct a representation of the heisenberg algebra on the homology group of the hilbert scheme in other words the homology group will become a fock space the basic idea is to introduce certain correspondences in the product of the hilbert scheme then they define operators on the homology group by a well known procedure they give generators of the heisenberg algebra and the only thing we must check is that they satisfy the defining relation here we remark that the components of the hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components our correspondences live in the product of the different components thus it is quite essential to study all components together our construction has the same spirit as the author s construction nal na4 of representations of affine lie algebras on homology groups of moduli spaces of instantons 1 on ale spaces which are minimal resolutions of simple singularities certain correspondences called hecke correspondences were used to define operators these twist instantons along curves irreducible components of the exceptional set while ours twist ideals around points in fact the hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles or more precisely torsion free sheaves our construction should be considered as a first step to extend nal na4 to general 4 manifolds the same program was also proposed by ginzburg kapranov and vasserot gkv

a bayesian model averaging approach with non informative priors for cost effectiveness analyses we consider the problem of assessing new and existing technologies for their cost effectiveness in the case where data on both costs and effects are available from a clinical trial and we address it by means of the cost effectiveness acceptability curve the main difficulty in these analyses is that cost data usually exhibit highly skew and heavy tailed distributions so that it can be extremely difficult to produce realistic probabilistic models for the underlying population distribution here in order to integrate the uncertainty about the model into the analysis of cost data and into cost effectiveness analyses we consider an approach based on bayesian model averaging bma in the particular case of weak prior informations about the unknown parameters of the different models involved in the procedure the main consequence of this assumption is that the marginal densities required by bma are undetermined however in accordance with the theory of partial bayes factors and in particular of fractional bayes factors we suggest replacing each marginal density with a ratio of integrals that can be efficiently computed via path sampling

numerical solution of some inverse scattering problems of geophysics some results are reported of the numerical solution of the inverse problem which consists of finding an inhomogeneity v x in the 3d region r 3 x x3 0 under the surface of the earth from the knowledge of the acoustic field u x y k at low frequencies k for all positions of the source y and the receiver x on the surface p x x3 0 of the earth the inhomogeneity v x in the refraction coefficient is related to the data of u x y k on p by an integral equation of the first kind this inverse problem is extremely ill conditioned the numerical computations in this paper show how to successfully obtain v x which was a priori assumed to be a known constant in an a priori unknown region b and zero outside b an ad hoc b searching procedure together with a regularization method have been used to solve the resulting ill condioned linear system for the discretized problem a subsequent direct comparison finds b more accurately this type of computation can be used to detect piecewise constant inhomogeneity in a medium for example to detect a hole or a crack in a rock or in a construction element

agt and the segal sugawara construction the conjectures of alday et al lett math phys 91 167 197 2010 and their generalizations have been mathematically formulated as the existence of an action of a w algebra on the cohomology or k theory of the instanton moduli space together with a whitaker vector a braverman et al e print arxiv 1406 2381 2014 d maulik and a okounkov e print arxiv 1211 1287 2012 pp 1 276 o schiffmann and e vasserot publ math inst hautes etud sci 118 213 342 2013 however the original conjectures also predict intertwining properties with the natural higher rank version of the ext1 operator which was previously studied by okounkov and the author in carlsson and okounkov duke math j 161 1797 1815 2012 a result which is now sometimes referred to as agt in rank one a alba et al lett math phys 98 33 64 2011 m pedrini et al j geom phys 103 43 89 2016 physically this corresponds to incorporating matter in the nekrasov partition functions an obviously important fe

stable calculation of the legendre transform abstract given x such that x x l is derived such that formula l c the constant c depends on sup d 2 where d 2 is an arbitrary second derivative of x x r n an example is given which shows that the critical point of x cannot be calculated stably from the noisy data x the notion of the generalized legendre transform is introduced and a stable method for its calculation given the noisy data is described

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

maxentropic reconstruction of fourier and laplace transforms in this note the first of a series we present a maxentropic procedure to recover a function from the knowledge of a finite number of values of a linear integral transform and apply it to the case of the fourier or the laplace transforms also we consider the case in which the values of the transform are known to be polluted by noise

bootstrap confidence intervals for a class of parametric problems summary we consider the following class of problems having observed a multivariate normal data vector y with unknown mean vector q covariance matrix the identity find an approximate confidence interval for 0 t q a real valued function of q a simple geometric construction is given which leads to highly accurate solutions this construction shows that the standard approximation based on maximum likelihood theory 04 cz00 can be quite misleading when 0 is nonlinear in q we discuss bootstrap based confidence intervals which remove most of the error in the standard approximation at the expense of considerably more calculation the bootstrap intervals are invariant under transformation of both y and i and so they automatically produce accurate solutions in problems which can be transformed to multivariate normality without requiring knowledge of the normalizing transformation

characterization of metal artifacts in x ray computed tomography metal streak artifacts in x ray computerized tomography ct are characterized here using the notion of the wavefront set from microlocal analysis the metal artifacts are caused mainly from the mismatch of the forward model of the filtered back projection the presence of metallic subjects in an imaging subject violates the model s assumption of the ct sinogram data being the radon transform of an image the increasing use of metallic implants has increased demand for the reduction of metal artifacts in the field of dental and medical radiography however it is a challenging issue due to the serious difficulties in analyzing the x ray data which depends nonlinearly on the distribution of the metallic subject in this paper we for the first time provide a mathematical analysis to characterize the structure of metal streaking artifacts the metal streaking artifacts are produced along the line tangent to boundaries of the metal region touching at least two different boundaries we also found a sufficient condition for the nonexistence of the metal streaking artifacts 2016 wiley periodicals inc

asymptotics and the theory of inference asymptotic analysis has always been very useful for deriving distributions in statistics in cases where the exact distribution is unavailable more importantly asymptotic analysis can also provide insight into the inference process itself suggesting what information is available and how this information may be extracted the development of likelihood inference over the past twenty some years provides an illustration of the interplay between techniques of approximation and statistical theory

posterior distributions in limited information analysis of the simultaneous equations model using the jeffreys prior abstract this paper studies the use of the jeffreys prior in bayesian analysis of the simultaneous equations model sem exact representations are obtained for the posterior density of the structural coefficient in canonical sems with two endogenous variables for the general case with m endogenous variables and an unknown covariance matrix the laplace approximation is used to derive an analytic formula for the same posterior density both the exact and the approximate formulas we derive are found to exhibit cauchy like tails analogous to comparable results in the classical literature on liml estimation moreover in the special case of a two equation just identified sem in canonical form the posterior density of is shown to have the same infinite series representation as the density of the finite sample distribution of the corresponding liml estimator this paper also examines the occurrence first documented in kleibergen and van dijk 1994a of a nonintegrable asymptotic cusp in the posterior distribution of the coefficient matrix of the reduced form equations for the included endogenous regressors an explanation for this phenomenon is provided in terms of the jacobian of the mapping from the structural model to the reduced form this interpretation assists in understanding the success of the jeffreys prior in resolving this problem

incomplete data problems in x ray computerized tomography in the present paper we study truncated projections for the fanbeam geometry in computerized tomography first we derive consistency conditions for the divergent beam transform then we study a singular value decomposition for the case where only the interior rays in the fan are provided as for example in region of interest tomography we show that the high angular frequency components of the searched for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition

numerical recovery of the layered medium from the surface data abstract in this paper an inverse problem is formulated and its numerical solution is given the problem is of interest for example in geophysics numerical results show that the algorithm is efficient for noisy data practical recommendations are given the problem consists of finding n z the refraction coefficient of the medium given the surface data u x k x p x x3 0 and all k 0 k0 where k0 0 is an arbitrary small wave number the governing equation is 2 k2n z u x x x1 x2 z analytical solution to this inverse problem is given a numerical implementation is suggested and the results of the numerical experiments are reported

local tomography ii standard or global tomography involves the reconstruction of a function f from line integrals local tomography in this paper involves the reconstruction of a related function lf alpha lambda f mu lambda 1 f where lambda is the square root of the positive laplacian delta this article is a sequel to the article local tomography siam j appl math 52 1992 pp 459 484 1193 1198 by faridani ritman and smith the principal new results are 1 good bounds for lambda f and lambda 1 f outside the support of f particularly when f has 0 moments up to some order 2 identification and reduction of global effects in local tomography i e identification and reduction of the dependence of lf x on the values of f at points at an intermediate distance from x 3 an algorithm for computing approximate density jumps from lambda f when f is a linear combination of characteristic functions and a smooth background several examples are given some from real x ray da

singularities of the radon transform the singular support of the radon transform rf of a function f with a compact support is described in terms of the generalized legendre transform the asymptotic behavior of rf near the singular support modulo smooth functions is described in terms of a geometrical invariant introduced in the paper

on a property of probability matching priors matching the alternative coverage probabilities summary in frequentist inference based on confidence sets both the true coverage and the probability for a confidence set to include an alternative value of the parameter of interest are important thus if probability matching priors also match such alternative coverage probabilites there is perhaps a stronger justification for calling them noninformative considering contiguous alternatives we obtain the relevant necessary and sufficient conditions in particular with n denoting the sample size it is seen that probability matching priors up to o n 2 also match the alternative coverage probabilities up to that order while this is not necessarily the case with probability matching priors up to o n

incomplete data problems in x ray computerized tomography i singular value decomposition of the limited angle transform etude des problemes intrinseques relatifs aux donnees incompletes dans la tomographie aux rayons x assistee par ordinateur decomposition de la transformation de radon

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

integrable expansions for posterior distributions for a two parameter exponential family asymptotic expansions of posterior distributions are derived for a two dimensional exponential family which includes normal gamma inverse gamma and inverse gaussian distributions reparameterization allows us to use a data dependent transformation convert the likelihood function to the two dimensional standard normal density and apply a version of stein s identity to assess the posterior distributions applications are given to characterize optimal noninformative priors in the sense of stein to suggest the form of a high order correction to the distribution function of a sequential likelihood ratio statistic and to provide confidence intervals for one parameter in the presence of other nuisance parameters

enriched conjugate and reference priors for the wishart family on symmetric cones a general wishart family on a symmetric cone is a natural exponential family nef having a homogeneous quadratic variance function using results in the abstract theory of euclidean jordan algebras the structure of conditional reducibility is shown to hold for such a family and we identify the associated parameterization and analyze its properties the enriched standard conjugate family for and the mean parameter are defined and discussed this family is considerably more flexible than the standard conjugate one the reference priors for and are obtained and shown to belong to the enriched standard conjugate family in particular this allows us to verify that reference posteriors are always proper the above results extend those available for nefs having a simple quadratic variance function specifications of the theory to the cone of real symmetric and positive definite matrices are discussed in detail and allow us to perform bayesian inference on the covariance matrix of a multivariate normal model under the enriched standard conjugate family in particular commonly employed bayes estimates such as the posterior expectation of and 1 are provided in closed form

reference priors with partial information summary in this paper reference priors are derived for three cases where partial information is available if a subjective conditional prior is given two reasonable methods are proposed for finding the marginal reference prior if instead a subjective marginal prior is available a method for defining the conditional reference prior is proposed a sufficient condition is then given under which this conditional reference prior agrees with the conditional reference prior derived in the first stage of the reference prior algorithm of berger bernardo 1989 1992 finally under the assumption of independence a method for finding marginal reference priors is also proposed various examples are given to illustrate the methods

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

a decomposition for the likelihood ratio statistic and the bartlett correction a bayesian argument let l n 1 log p x be the log likelihood of an n dimensional x under a p dimensional let j be the mle under h j 1 1 0 j j 0 and 0 be the unrestricted mle define t j as 2n l j 1 1 2 sgn j j 1 j 0 let t t 1 t p then under regularity conditions the following theorem is proved under 0 t is asymptotically n n 1 2 a 0 n 1 a j n 1 o n 3 2 where j is the identity matrix the result is proved by first establishing an analogous result when is random and then making the prior converge to a degenerate distribution the existence of the bartlett correction to order n 3 2 follows from the theorem we show that an edgeworth expansion with error o n 2 for t involves only polynomials of degree less than or equal to 3 and hence verify rigorously lawley s 1956 result giving the order of the error in the bartlett correction as o n 2

asymptotic inference for mixture models by using data dependent priors for certain mixture models improper priors are undesirable because they yield improper posteriors however proper priors may be undesirable because they require subjective input we propose the use of specially chosen data dependent priors we show that in some cases data dependent priors are the only priors that produce intervals with second order correct frequentist coverage the resulting posterior also has another interpretation it is the product of a fixed prior and a pseudolikelihood

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

second order probability matching priors summary the paper considers priors obtained by ensuring approximate frequentist validity of a posterior quantiles and of b the posterior distribution function it is seen that at the second order of approximation the two approaches do not necessarily lead to identical conclusions examples are given to illustrate this the role of invariance in the context of probability matching is also discussed

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

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

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

prior distributions from pseudo likelihoods in the presence of nuisance parameters consider a model parameterized by where is the parameter of interest the problem of eliminating the nuisance parameter can be tackled by resorting to a pseudo likelihood function l for namely a function of only and the data y with properties similar to those of a likelihood function if one treats l as a true likelihood the posterior distribution y l for can be considered where is a prior distribution on the goal of this article is to construct probability matching priors for a scalar parameter of interest only i e priors for which bayesian and frequentist inference agree to some order of approximation to be used in y when l is a marginal a conditional or a modification of the profile likelihood we show that is simply proportional to the square root of the inverse of the asymptotic variance of the pseudo maximum likelihood estimator the proposed priors are compared with the reference or jeffreys priors in four

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

local tomography in electron microscopy we present a new local tomographic algorithm applicable to electron microscope tomography our algorithm applies to the standard data acquisition method single axis tilting as well as to more arbitrary acquisition methods including double axis and conical tilt using microlocal analysis we put the reconstructions in a mathematical context explaining which singularities are stably visible from the limited data given by the data collection protocol in the electron microscope finally we provide reconstructions of real specimens from electron tomography data

singular value decompositions and inversion methods for the exterior radon transform and a spherical transform abstract the classical radon transform r maps an integrable function in rn to its integrals over all n 1 dimensional hyperplanes and the exterior radon transform is the transform r restricted to hyperplanes that do not intersect a given disc a singular value decomposition for the exterior transform is given for spaces of square integrable functions on the exterior of the disc this decomposition in orthogonal functions explicitly produces the null space and range of the exterior transform and gives a new method for inverting the transform modulo the null space a modification of this method is given that will exactly invert functions of compact support these results generalize theorems of r m perry and the author a singular value decomposition for the radon transform that integrates over spheres in rn containing the origin is also given this follows from the singular value decomposition for r and yields the null space and a new inversion method for this transform

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

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

polynomial based algorithms for computed tomography ii this paper is concerned with two dimensional parallel beam tomography we investigate the computational effectiveness of the general approach outlined in a previous paper by one of us that approach was based on reconstructions by means of convolution with a polynomial kernel here we consider only the case of uniformly spaced views to take advantage of the symmetry in this case we give a class of algorithms and calculate several numerical examples we also obtain estimates on the expected mean square error when the data is perturbed by white noise these estimates lead to several surprising applications

on priors providing frequentist validity for bayesian inference summary we derive the differential equation that a prior must satisfy if the posterior probability of a one sided credibility interval for a parametric function and its frequentist probability agree up to o n 1 this equation turns out to be identical with stein s equation for a slightly different problem for which also our method provides a rigorous justification our method is different in details from stein s but similar in spirit to dawid 1991 and bickel ghosh 1990 some examples are provided

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