basis functions and parameter optimisation in high order iterative learning control in this paper a new parameter optimal high order iterative learning control ilc algorithms is proposed to extend the work of owens and feng parameter optimisation in iterative learning control international journal of control 14 11 1059 1069 if the original plant is positive this new algorithm will result in convergent learning where the convergence is monotonic to zero tracking error if the original plant is not positive it can be shown that by adding a suitable set of basis functions into the algorithm the tracking error will again converge monotonically to zero this provides a considerable improvement to earlier work on parameter optimal ilc as it opens up the possibility of globally convergent algorithms for any linear plant g the number of parameters needed to ensure convergence could however become large the paper shows that the use of low order parameterisations is capable of achieving much of the benefit achieved in the ideal case

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

objective priors for the bivariate normal model summary objective bayesian inference for the multivariate normal distribution is illustrated using dieren t types of formal objective priors jereys invariant reference and matching dieren t modes of inference bayesian and frequentist and dieren t criteria involved in selecting optimal objective priors ease of computation frequentist performance marginalization paradoxes and decision theoretic evaluation in the course of the investigation of the bivariate normal model in berger and sun 2006 a variety of surprising results were found including the availability of objective priors that yield exact frequentist inferences for many functions of the bivariate normal parameters such as the correlation coecien t certain of these results are generalized to the multivariate normal situation the prior that most frequently yields exact frequentist inference is the righthaar prior which unfortunately is not unique two natural proposals are studied for dealing with this non uniqueness rst mixing over the right haar priors second choosing the empirical bayes right haar prior that which maximizes the marginal likelihood of the data quite surprisingly we show that neither of these possibilities yields a good solution this is disturbing and sobering it is yet another indication that improper priors do not behave as do proper priors and that it can be dangerous to apply understandings from the world of proper priors to the world of improper priors

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

bayesian goodness of fit testing using infinite dimensional exponential families we develop a nonparametric bayes factor for testing the fit of a parametric model we begin with a nominal parametric family which we then embed into an infinite dimensional exponential family the new model then has a parametric and nonparametric component we give the log density of the nonparametric component a gaussian process prior an asymptotic consistency requirement puts a restriction on the form of the prior leaving us with a single hyperparameter for which we suggest a default value based on simulation experience then we construct a bayes factor to test the nominal model versus the semiparametric alternative finally we show that the bayes factor is consistent the proof of the consistency is based on approximating the model by a sequence of exponential families

an inverse model approach to multivariable norm optimal iterative learning control with auxiliary optimisation motivated by the commonly encountered problem in which tracking is only required at selected intermediate points within the time interval a general optimisation based iterative learning control ilc algorithm is derived that ensures convergence of tracking errors to zero whilst simultaneously minimising a specified quadratic objective function of the input signals and chosen auxiliary state variables in practice the proposed solutions enable a repeated tracking task to be accurately completed whilst simultaneously reducing undesirable effects such as payload spillage vibration tendencies and actuator wear the theory is developed using the well known norm optimal ilc noilc framework using general linear functional operators between real hilbert spaces solutions are derived using feedforward action convergence is proved and robustness bounds are presented using both norm bounds and positivity conditions algorithms are specified for both continuous and discrete time state space representations

finding the number of clusters in a dataset one of the most difficult problems in cluster analysis is identifying the number of groups in a dataset most previously suggested approaches to this problem are either somewhat ad hoc or require parametric assumptions and complicated calculations in this article we develop a simple yet powerful nonparametric method for choosing the number of clusters based on distortion a quantity that measures the average distance per dimension between each observation and its closest cluster center our technique is computationally efficient and straightforward to implement we demonstrate empirically its effectiveness not only for choosing the number of clusters but also for identifying underlying structure on a wide range of simulated and real world datasets in addition we give a rigorous theoretical justification for the method based on information theoretic ideas specifically results from the subfield of electrical engineering known as rate distortion theory allow us to describe the behavior of the dist

accelerated norm optimal iterative learning control algorithms using successive projection this article proposes a novel technique for accelerating the convergence of the previously published norm optimal iterative learning control noilc methodology the basis of the results is a formal proof of an observation made by d h owens namely that the noilc algorithm is equivalent to a successive projection algorithm between linear varieties in a suitable product hilbert space this leads to two proposed accelerated algorithms together with well defined convergence properties the results show that the proposed accelerated algorithms are capable of ensuring monotonic error norm reductions and can outperform noilc by more rapid reductions in error norm from iteration to iteration in particular examples indicate that the approach can improve the performance of noilc for the problematic case of non minimum phase systems realisation of the algorithms is discussed and numerical simulations are provided for comparative purposes and to demonstrate the numerical performance and effectiveness of the proposed methods

parsimonious covariance matrix estimation for longitudinal data this article proposes a data driven method to identify parsimony in the covariance matrix of longitudinal data and to exploit any such parsimony to produce a statistically efficient estimator of the covariance matrix the approach parameterizes the covariance matrix through the cholesky decomposition of its inverse for longitudinal data this is a one step ahead predictive representation and the cholesky factor is likely to have off diagonal elements that are zero or close to zero a hierarchical bayesian model is used to identify any such zeros in the cholesky factor similar to approaches that have been successful in bayesian variable selection the model is estimated using a markov chain monte carlo sampling scheme that is computationally efficient and can be applied to covariance matrices of high dimension it is demonstrated through simulations that the proposed method compares favorably in terms of statistical efficiency with a highly regarded competing approach the estimator is applied to three

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

an algebraic approach to iterative learning control in this paper discrete time iterative learning control ilc systems are analysed from an algebraic point of view the algebraic analysis shows that a linear time invariant single input single output model can always represented equivalently as a static multivariable plant due to the finiteness of the time axis furthermore in this framework the ilc synthesis problem becomes a tracking problem of a multi channel step function the internal model principle states that for asymptotic tracking i e convergent learning it is required that an ilc algorithm has to contain an integrator along the iteration axis but at the same time the resulting closed loop system should be stable the question of stability can then be answered by analysing the closed loop poles along the iteration axis using standard results from multivariable polynomial systems theory this convergence theory suggests that time varying ilc control laws should be typically used instead of time invariant control laws in order to guarantee go

reference priors when the stopping rule depends on the parameter of interest priors related to jeffreys s prior and the reference priors of berger and bernardo are developed for experiments where a stopping rule is given these noninformative priors depend on a function of the expected stopping time in a one parameter experiment with a stopping rule the jeffreys prior is shown to be improved by taking the stopping rule into account based on a criterion using frequentist coverage probabilities in a multiparameter experiment however the jeffreys prior depends on the stopping rule in an inappropriate fashion in the latter case the reference prior effectively accounts for the stopping rule the motivation for this work was in part the statistical importance of the problem and the lack of noninformative priors for experiments when a stopping rule is given and in part a wish to develop an interesting application of a recently developed method of deriving reference prior by berger and bernardo

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

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

integrated likelihood functions for non bayesian inference consider a model with parameter 0 where is the parameter of interest and let l denote the likelihood function one approach to likelihood inference for is to use an integrated likelihood function in which is eliminated from l by integrating with respect to a density function the goal of this paper is to consider the problem of selecting so that the resulting integrated likelihood function is useful for non bayesian likelihood inference the desirable properties of an integrated likelihood function are analyzed and these suggest that should be chosen by finding a nuisance parameter that is unrelated to and then taking the prior density for to be independent of such an unrelated parameter is constructed and the resulting integrated likelihood is shown to be closely related to the modified profile likelihood

noninformative priors and frequentist risks of bayesian estimators of vector autoregressive models in this study we examine posterior properties and frequentist risks of bayesian estimators based on several noninformative priors in vector autoregressive var models we prove existence of the posterior distributions and posterior moments under a general class of priors using a variety of priors in this class we conduct numerical simulations of posteriors we find that in most examples bayesian estimators with a shrinkage prior on the var coefficients and the reference prior of yang and berger ann statist 22 1994 1195 on the var covariance matrix dominate mle bayesian estimators with the diffuse prior and bayesian estimators with the prior used in rats we also examine the informative minnesota prior and find that its performance depends on the nature of the data sample and on the tightness of the minnesota prior a tightly set minnesota prior is better when the data generating processes are similar to random walks but the shrinkage prior or constant prior can be better otherwise

multivariable norm optimal iterative learning control with auxiliary optimisation the paper describes a substantial extension of norm optimal iterative learning control noilc that permits tracking of a class of finite dimensional reference signals whilst simultaneously converging to the solution of a constrained quadratic optimisation problem the theory is presented in a general functional analytical framework using operators between chosen real hilbert spaces this is applied to solve problems in continuous time where tracking is only required at selected intermediate points of the time interval but simultaneously the solution is required to minimise a specified quadratic objective function of the input signals and chosen auxiliary state variables applications to the discrete time case including the case of multi rate sampling are also summarised the algorithms are motivated by practical need and provide a methodology for reducing undesirable effects such as payload spillage vibration tendencies and actuator wear whilst maintaining the desired tracking accuracy necessary f

on the inversion of infinite moment matrices abstract motivated by 8 we study the existence of the inverse of an infinite hermitian positive definite matrix in short hpd matrix from the point of view of the asymptotic behaviour of the smallest eigenvalues of the finite sections we prove a sufficient condition to assure the inversion of an hpd matrix with square summable rows for infinite toeplitz matrices we introduce the notion of asymptotic toeplitz matrix and we show that under certain assumptions the inverse of an infinite toeplitz positive definite matrix is asymptotic toeplitz such inverses are computed in terms of the limits of the coefficients of the associated orthogonal polynomials we apply these results in the context of the theory of orthogonal polynomials in particular we show that for measures on the unit circle t verifying that the smallest eigenvalue of the finite sections of the corresponding moment matrix are away from zero in the limit we may assure the existence of all the limits of the coefficients of the orthonormal polynomials with respect to such measures

rotation invariant moment problems an important theorem of marcel riesz cf 14 states that the polynomials are dense in l2 x when x is a determinate measure on the real line in the indeterminate case riesz also characterized the measures x for which the polynomials are dense in l2 they are the so called nevanlinna extremal measures introduced in nevanlinna 11 it does not seem to be known whether the polynomials are dense in l2 g when x is a determinate measure on r d d 1 cf the expository paper by fuglede 7 as well as the research problems book 8 p 529 where devinatz poses the problem as question 1 and ascribes it to the physicist john challifour 1978 in this paper we shall settle the question in the negative there exist rotation invariant measures on r d d 1 which are determinate but for which the polynomials are not dense in l2 p such measures x are necessarily of the following very special form

the index of determinacy for measures and the norm of orthonormal polynomials for determinate measures having moments of every order we define and study an index of determinacy which checks the stability of determinacy under multiplication by even powers of it z for z a complex number using this index of determinacy we solve the problem of determining for which z e c the sequence p m z n m e n belongs to t2 where pn n is the sequence of orthonormal polynomials associated with the measure u

information optimality and bayesian modelling the general approach of treating a statistical problem as one of information processing led to the bayesian method of moments reference priors minimal information likelihoods and stochastic complexity these techniques rest on quantities that have physical interpretations from information theory current work includes the role of prediction the emergence of data dependent priors the role of information measures in model selection and the use of conditional mutual information to incorporate partial information

bayesian inference with specified prior marginals abstract we show how to find bounds on posterior expectations of arbitrary functions of the parameters when the prior marginals are specified but when the complete joint prior is unspecified we also give a theorem that is useful for finding posterior bounds in a wide range of bayesian robustness problems we apply these techniques to two examples the first example involves a recent clinical trial for extracorporeal membrane oxygenation ecmo our analysis may be regarded as a follow up to a detailed bayesian analysis given by kass and greenhouse who concluded that the posterior probability that the treatment is superior to the control is about 95 their analysis however assumed a priori independence of the parameters we consider other prior distributions with the same marginals as kass and greenhouse but in which the parameters are not independent and conclude that as long as a priori independence is at least approximately tenable then ecmo seems superior to the control the second example is th

orthogonal polynomials and analytic functions associated to positive definite matrices abstract for a positive definite infinite matrix a we study the relationship between its associated sequence of orthonormal polynomials and the asymptotic behaviour of the smallest eigenvalue of its truncation a n of size n n for the particular case of a being a hankel or a hankel block matrix our results lead to a characterization of positive measures with finite index of determinacy and of completely indeterminate matrix moment problems respectively

a matrix exponential spatial specification we introduce the matrix exponential as a way of modelling spatially dependent data the matrix exponential spatial specification mess simplifies the log likelihood allowing a closed form solution to the problem of maximum likelihood estimation and greatly simplifies the bayesian estimation of the model the mess can produce estimates and inferences similar to those from conventional spatial autoregressive models but has analytical computational and interpretive advantages we present maximum likelihood and bayesian approaches to the estimation of this spatial model specification along with methods of model comparisons over different explanatory variables and spatial specifications

bayesian duality and risk analysis on the statistical manifold of exponential family with censored data abstract information geometry has been attracted wide attentions in the past few decades this paper focuses on the bayesian duality on a statistical manifold derived from the exponential family with data from life tests based on life testing data the statistical manifold is constructed with a new cumulant generating function the bregman divergence between two parameter points is studied the dual coordinate system and dual function are obtained then the dualistic structure on the manifold is discussed the results show that the maximum likelihood estimate can be obtained by minimizing the bregman divergence induced from the dual function the bayesian analysis and prediction are investigated based on informative and non informative priors consider the gamma distribution as an example the closed form representations of the dual coordinate system and dual function are obtained a real data set is employed to illustrate the methodologies and experimental designs developed in this paper

iterative learning control for constrained linear systems this article considers iterative learning control ilc for linear systems with convex control input constraints first the constrained ilc problem is formulated in a novel successive projection framework then based on this projection method two algorithms are proposed to solve this constrained ilc problem the results show that when perfect tracking is possible both algorithms can achieve perfect tracking the two algorithms differ however in that one algorithm needs much less computation than the other when perfect tracking is not possible both algorithms can exhibit a form of practical convergence to a best approximation the effect of weighting matrices on the performance of the algorithms is also discussed and finally numerical simulations are given to demonstrate the effectiveness of the proposed methods

bayesian inference for the common location parameter of several shifted exponential populations abstract consider k random samples which are independently drawn from k shifted exponential distributions with respective scale parameters 1 2 k and common location parameter on the basis of the given samples and in a bayesian framework we address the problem of point and interval estimation of the location parameter under the conjugate priors which are usually proper priors moreover we also address the problem of testing the equality of the location parameters we propose bayesian hypothesis testing procedures for the equality of the location parameters under the noninformative prior the noninformative prior is usually improper which yields a calibration problem that makes the bayes factor to be defined up to a multiplicative constant so we propose the default bayesian hypothesis testing procedures based on the fractional bayes factor and the intrinsic bayes factors under the reference priors our proposed bayesian procedures are compared and contrasted via a comparison study a simulation study and a real world data analysis to the existing classical exact procedures proposed by tippett tippett s method tippett 1931 fisher fisher s method fisher 1932 stouffer inverse normal method stouffer et al 1949 and george logit method george 1977 and to the generalized variable procedures proposed by tsui and weerahandi generalized p value method tsui and weerahandi 1989

default bayesian analysis with global local shrinkage priors we provide a framework for assessing the default nature of a prior distribution using the property of regular variation which we study for global local shrinkage priors in particular we show that the horseshoe priors originally designed to handle sparsity are regularly varying and thus are appropriate for default bayesian analysis to illustrate our methodology we discuss four problems of noninformative priors that have been shown to be highly informative for nonlinear functions in each case we show that global local horseshoe priors perform as required global local shrinkage priors can separate a low dimensional signal from high dimensional noise even for nonlinear functions

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

the formal definition of reference priors reference analysis produces objective bayesian inference in the sense that inferential statements depend only on the assumed model and the available data and the prior distribution used to make an inference is least informative in a certain information theoretic sense reference priors have been rigorously defined in specific contexts and heuristically defined in general but a rigorous general definition has been lacking we produce a rigorous general definition here and then show how an explicit expression for the reference prior can be obtained under very weak regularity conditions the explicit expression can be used to derive new reference priors both analytically and numerically

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

fully bayesian spline smoothing and intrinsic autoregressive priors summary there is a well known bayesian interpretation for function estimation by spline smoothing using a limit of proper normal priors the limiting prior and the conditional and intrinsic autoregressive priors popular for spatial modelling have a common form which we call partially informative normal we derive necessary and sufficient conditions for the propriety of the posterior for this class of partially informative normal priors with noninformative priors on the variance components a condition crucial for successful implementation of the gibbs sampler the results apply for fully bayesian smoothing

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

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

iterative learning control using optimal feedback and feedforward actions an algorithm for iterative learning control is developed on the basis of an optimization principle which has been used previously to derive gradient type algorithms the new algorithm has numerous benefits which include realization in terms of riccati feedback and feedforward components this realization also has the advantage of implicitly ensuring automatic step size selection and hence guaranteeing convergence without the need for empirical choice of parameters the algorithm is expressed as a very general norm optimization problem in a hilbert space setting and hence in principle can be used for both continuous and discrete time systems a basic relationship with almost singular optimal control is outlined the theoretical results are illustrated by simulation studies which highlight the dependence of the speed of convergence on parameters chosen to represent the norm of the signals appearing in the optimization problem

predictive optimal iterative learning control a new optimization based iterative learning control algorithm is proposed and its properties derived an important characteristic of this algorithm is that it uses present and future predicted errors to compute the current control in a similar manner to model based predictive control using a receding horizon in particular it enables the algorithm designer to achieve good control over convergence rate the actual implementation has a multimodel structure but uses standard linear quadratic regulator methods for a causal formulation in the iterative learning sense of what is originally a non causal algorithm the results are illustrated by simulations

small eigenvalues of large hermitian moment matrices abstract we consider an infinite hermitian positive definite matrix m which is the moment matrix associated with a measure with infinite and compact support on the complex plane we prove that if the polynomials are dense in l 2 then the smallest eigenvalue n of the truncated matrix m n of m of size n 1 n 1 tends to zero when n tends to infinity in the case of measures in the closed unit disk we obtain some related results

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

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

estimation of the common mean from heterogeneous normal observations with unknown variances summary to determine the common mean of heterogeneous normal observations the bayes procedures and the invariant maximum likelihood estimators of the weights forming the weighted means statistic are obtained when there are no variance estimates the bayes statistic is based on the reference geisser cornfield prior distribution which makes the posterior discrete distribution of the mean to be supported by the observed data with probabilities determined via the geometric means of the distances between data points the maximum likelihood estimator coincides with the observation which has the maximal posterior probability these procedures can be useful when measurement uncertainties are not reported or are misspecified

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

orthogonal matrix polynomials and higher order recurrence relations abstract it is well known that orthogonal polynomials on the real line satisfy a three term recurrence relation and conversely every system of polynomials satisfying a three term recurrence relation is orthogonal with respect to some positive borel measure on the real line we extend this result and show that every system of polynomials satisfying some 2 n 1 term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are n n matrices we apply this result to polynomials orthogonal with respect to a discrete sobolev inner product and other inner products in the linear space of polynomials as an application we give a short proof of krein s characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points

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

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

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