statistical methods for eliciting probability distributions elicitation is a key task for subjectivist bayesians although skeptics hold that elicitation cannot or perhaps should not be done in practice it brings statisticians closer to their clients and subject matter expert colleagues this article reviews the state of the art reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively and how they respond to questions about that information in a discussion of the elicitation process the first issue to address is what it means for an elicitation to be successful that is what criteria should be used our answer is that a successful elicitation faithfully represents the opinion of the person being elicited it is not necessarily true in some objectivistic sense and cannot be judged in that way we see that elicitation as simply part of the process of statistical modeling indeed in a hierarchical model at which point the likelihood ends and the pr

uncertainty in prior elicitations a nonparametric approach summary a key task in the elicitation of expert knowledge is to construct a distribution from the finite and usually small number of statements that have been elicited from the expert these statements typically specify some quantiles or moments of the distribution such statements are not enough to identify the expert s probability distribution uniquely and the usual approach is to fit some member of a convenient parametric family there are two clear deficiencies in this solution first the expert s beliefs are forced to fit the parametric family secondly no account is then taken of the many other possible distributions that might have fitted the elicited statements equally well we present a nonparametric approach which tackles both of these deficiencies we also consider the issue of the imprecision in the elicited probability judgements

non conjugate prior distribution assessment for multivariate normal sampling elicitation methods are proposed for quantifying expert opinion about a multivariate normal sampling model the natural conjugate prior family imposes a relationship between the mean vector and the covariance matrix that can portray an expert s opinion poorly instead we assume that opinions about the mean and the covariance are independent and suggest innovative forms of question which enable the expert to quantify separately his or her opinion about each of these parameters prior opinion about the mean vector is modelled by a multivariate normal distribution and about the covariance matrix by both an inverse wishart distribution and a generalized inverse wishart giw distribution to construct the latter results are developed that give insight into the giw parameters and their interrelationships certain of the elicitation methods exploit unconditional assessments as fully as possible since these can reflect an expert s beliefs more accurately than conditional assessments methods are illustrated through an example

elicitation of prior distributions for variable selection problems in regression this paper addresses the problem of quantifying expert opinion about a normal linear regression model when there is uncertainty as to which independent variables should be included in the model opinion is modeled as a mixture of natural conjugate prior distributions with each distribution in the mixture corresponding to a different subset of the independent variables it is shown that for certain values of the independent variables the predictive distribution of the dependent variable simplifies from a mixture of t distributions to a single t distribution using this result a method of eliciting the conjugate distributions of the mixture is developed the method is illustrated in an example

a predictive approach to the analysis of designed experiments abstract viewing the analysis of designed experiments as a model selection problem we introduce the use of a predictive bayesian criterion in this context based on the predictive density of a replicate experiment pdre a calibration of the criterion is provided to assist in the model choice the relationships of the proposed criterion to other prevalent criteria such as aic bic and mallows s c p are given an information theoretic criterion based on the pdre s of two competing models is also introduced and compared with the usual f statistic for two nested models examples are given to illustrate the proposed methodology

fourier or bessel transformations of highly oscillatory functions an algorithm for integral transformation of highly oscillatory functions is presented it is a generalization of the standard gaussian quadrature method but has the virtue that the usual tedious tests of convergence of the numerical results is not necessary the effectiveness and accuracy of the algorithm is tested for both large arguments and higher orders of the bessel function

marginalization and linear opinion pools abstract suppose a decision maker has asked a group of experts to assess subjective probability distributions over some space and he wishes to find a consensus probability distribution as a function of these the assumption that finding the consensus distribution commutes with marginalization of the distributions implies that the consensus must be found using a linear opinion pool weighted average provided the space being considered contains three or more points some of the consequences of this result are discussed

evaluation of fourier transforms by gauss laguerre quadratures numerical schemes for the evaluation of fourier transforms with gauss laguerre quadratures are presented these schemes which incorporate functional values at complex arguments are tested for efficacy and accuracy the authors illustrate their effectiveness on a simple model system by computing the form factor and its laplacian for the neon atom

ranges of posterior probabilities for quasiunimodal priors with specified quantiles abstract suppose several quantiles of the prior distribution for are specified or equivalently the prior probabilities of a partitioning collection of intervals ii are given in addition suppose that the prior distribution is assumed to be unimodal rather than selecting a single prior distribution to perform a bayesian analysis it is of interest to consider the class of all prior distributions compatible with these inputs for this class and unimodal likelihood functions the ranges of the posterior probabilities of the ii and the ranges of the posterior cdf at the specified prior quantiles were determined in berger and o hagan in press unfortunately calculations with this class can be difficult here a similar much more easily analyzed class of quasiunimodal prior distributions is considered and compared with other classes

priors for unit root models abstract a method of assessing an economist s subjective prior for a unit root model is given and applied the methods extend previous work by allowing a family of piecewise conjugate prior distribution that permit different opinions when 1 this larger family is still closed under sampling so it retains the simplicity that is the principal advantage of conjugate analysis

modeling expert judgments for bayesian updating this paper examines how a bayesian decision maker would update his her probability p for the occurrence of an event a in the light of a number of expert opinions expressed as probabilities q 1 cdots q n of a it is seen among other things that the linear opinion pool lambda 0p sum n i 1 lambda iq i corresponds to an application of bayes theorem when the decision maker has specified only the mean of the marginal distribution for q 1 cdots q n and requires his her formula for the posterior probability of a to satisfy a certain consistency condition a product formula similar to that of bordley 1982 is also derived in the case where the experts are deemed to be conditionally independent given a and given its complement

specifying a prior distribution in structured regression problems abstract the problem of assessing the prior mean and covariance matrix of the vector of regression coefficients in the normal multiple linear regression model is considered arguments are advanced for using expectations regarding the form of the design matrix in specifying the prior for problems with a structure that can aid in specifying these expectations this is illustrated with models having second order terms an example is given

a posteriori estimation of dimension reduction errors for elliptic problems on thin domains a new a posteriori error estimator is presented for the verification of the dimensionally reduced models stemming from the elliptic problems on thin domains the original problem is considered in a general setting without any specific assumptions on the domain geometry coefficients and the right hand sides for the energy norm of the error of the zero order dimension reduction method the proposed estimator is shown to always provide a guaranteed upper bound in the case when the original domain has constant thickness but possibly nonplane upper and lower faces the estimator demonstrates the optimal convergence rate as the thickness tends to zero it is also flexible enough to successfully cope with infinitely growing right hand sides in the equation when the domain thickness tends to zero the numerical tests indicate the efficiency of the estimator and its ability to accurately represent the local error distribution needed for an adaptive improvement of the reduced model

a new perspective on priors for generalized linear models abstract this article deals with specifications of informative prior distributions for generalized linear models our emphasis is on specifying distributions for selected points on the regression surface the prior distribution on regression coefficients is induced from this specification we believe that it is inherently easier to think about conditional means of observables given the regression variables than it is to think about model dependent regression coefficients previous use of conditional means priors seems to be restricted to logistic regression with one predictor variable and to normal theory regression we expand on the idea of conditional means priors and extend these to arbitrary generalized linear models we also consider data augmentation priors where the prior is of the same form as the likelihood we show that data augmentation priors are special cases of conditional means priors with current monte carlo methodology such as importance sampling and gibbs sampling our priors result in

a comparison of two elicitation methods for a prior distribution for a binomial parameter this paper compares two methods of elicitation for the hyperparameters of the conjugate beta distribution for a binomial sampling model the pm posterior mode method of chaloner and duncan appears to be more sensitive to elicitation errors than does a new method proposed here this new method uses the device of imaginary results due to i j good good i j 1967 a bayesian significance test for multinomial distributions jrssb29 399 431 with discussion an explicit model for elicitation errors is used in a simulation

prior elicitation variable selection and bayesian computation for logistic regression models summary bayesian selection of variables is often difficult to carry out because of the challenge in specifying prior distributions for the regression parameters for all possible models specifying a prior distribution on the model space and computations we address these three issues for the logistic regression model for the first we propose an informative prior distribution for variable selection several theoretical and computational properties of the prior are derived and illustrated with several examples for the second we propose a method for specifying an informative prior on the model space and for the third we propose novel methods for computing the marginal distribution of the data the new computational algorithms only require gibbs samples from the full model to facilitate the computation of the prior and posterior model probabilities for all possible models several properties of the algorithms are also derived the prior specification for the first challenge focuses on the observables in that the elicitation is based on a prior prediction yo for the response vector and a quantity ao quantifying the uncertainty in yo then yo and ao are used to specify a prior for the regression coefficients semi automatically examples using real data are given to demonstrate the methodology

quantifying expert opinion in linear regression problems summary this paper describes a method for choosing a natural conjugate prior distribution for a normal linear sampling model a person using the method to quantify his her opinions performs specified elicitation tasks the hyperparameters of the conjugate distribution are estimated from the elicited values the method is designed to require elicitation tasks that people can perform competently and introduces a type of task not previously reported a property of the method is that the assessed variance matrices are certain to be positive definite the method is sufficiently simple to implement with an interactive computer program on a microcomputer bayesian statistical methods provide a formal mechanism for taking into account prior knowledge meaning information available prior to the statistical data to be analysed in many circumstances prior knowledge is based on historical data that are only recorded in the form of the personal experience of experts to use the information the expert must quantify his her opinions by answering comprehensible questions concerning unknown but definite quantities of direct interest his set of answers should enable a probability distribution to be determined this distribution must satisfy the usual laws of probability so that for example any variance covariance matrix must be positive definite in addition the distribution should be accurate in some sense for example the distribution might be required to describe the expert s knowledge closely or perhaps to predict subsequent events with relative success the accuracy of the assessed distribution will depend in part on the method of elicitation in this paper we present a method of assessing a subjective prior distribution for a normal linear sampling model this is an important task owing to the wide applicability of such models and the many situations where expert personal opinion could be used more efficiently communicated more accurately and judged more critically if it were available in a suitable form we denote a particular setting of the independent variables by x and the dependent variable by y referring to the former as a design point and the latter as the response the model specifies that the objective or sampling distribution of y conditional on x and the sampling model parameters

separating probability elicitation from utilities abstract this article deals with the separation of probability elicitation from utilities we show that elicited probabilities can be related to utilities not just through the explicit or implicit payoffs related to the elicitation process but also through other stakes the expert may have in the events of interest we study three elicitation procedures lotteries scoring rules and promissory notes and show how the expert s utility function and stakes in the events can influence the resulting probabilities particularly extreme results are obtained in an example involving a market at equilibrium the applicability of a no stakes condition and some implications for probability elicitation are discussed let represent an expert s probability for an event a and let p denote the elicited probability from some elicitation procedure we determine the value of p that maximizes the expert s expected utility when utility is linear in money p for all of the procedures studied here under nonlinear utility

quadrature formulas for oscillatory integral transforms quadrature formulas are obtained for the fourier and bessel transforms which correspond to the well known gauss laguerre formula for the laplace transform these formulas provide effective asymptotic approximations complete with error bounds comparison is also made between the quadrature formulas and the asymptotic expansions of these transforms