Interactive Elicitation of Opinion for a Normal Linear Model

Kadane, Joseph B.; Dickey, James; Winkler, Robert L.; Smith, Wayne S.; Peters, Stephen C.

doi:10.2307/2287171

Cited by 114 publications

(138 citation statements)

References 11 publications

(18 reference statements)

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“…We consider the p + 1 individual parameter vectors and 2 to be unknown. Where possible, informative prior distributions for and 2 should be elicited and incorporated into the analysis|see Kadane et al 1980 andGarthwaite andDickey 1992. In the absence of expert opinion we seek to choose prior distributions which re ect uncertainty about the parameters and also embody reasonable a priori constraints.…”

mentioning

confidence: 99%

Bayesian Model Averaging for Linear Regression Models

Raftery

Madigan

Hoeting

1997

Journal of the American Statistical Association

525

572

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We consider the problem of accounting for model uncertainty in linear regression models. Conditioning on a single selected model ignores model uncertainty, and thus leads to the underestimation of uncertainty when making inferences about quantities of interest. A Bayesian solution to this problem involves averaging over all possible models i.e., combinations of predictors when making inferences about quantities of interest. This approach is often not practical. In this paper we o er two alternative approaches. First we describe an ad hoc procedure called Occam's Window" which indicates a small set of models over which a model average can be computed. Second, we describe a Markov c hain Monte Carlo approach which directly approximates the exact solution. In the presence of model uncertainty, both these model averaging procedures provide better predictive performance than any single model which might reasonably have been selected.In the extreme case where there are many candidate predictors but no relationship between any of them and the response, standard variable selection procedures often choose some subset of variables that yields a high R 2 and a highly signi cant o verall F value. In this situation, Occam's Window usually indicates the null model as the only one to be considered, or else a small number of models including the null model, thus largely resolving the problem of selecting signi cant models when there is no signal in the data.Software to implement our methods is available from StatLib.

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confidence: 99%

Bayesian Model Averaging for Linear Regression Models

Raftery

Madigan

Hoeting

1997

Journal of the American Statistical Association

525

572

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“…The aim is to choose R so as to model the expert's opinion about the dependence between the p i s. In Sect. 4.3, we introduce a method, inspired by Kadane et al (1980), to encode the correlation matrix R. Although the density in (11) is neither a multivariate normal…”

Section: Constructing a Gaussian Copula Prior Distributionmentioning

confidence: 99%

“…However, none of the methods is guaranteed to yield a positive-definite correlation matrix. On the other hand, the elicitation method of Kadane et al (1980) has been designed to encode a positive-definite covariance matrix of a multivariate t-distribution as a conjugate prior for the hyperparameters of a normal multiple linear regression model. Their method can be useful in a variety of multivariate elicitation problems that require the assessment of positive-definite matrices (Garthwaite et al 2005.…”

Section: Introductionmentioning

confidence: 99%

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Eliciting Dirichlet and Gaussian copula prior distributions for multinomial models

Elfadaly

Garthwaite

2016

Stat Comput

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In this paper, we propose novel methods of quantifying expert opinion about prior distributions for multinomial models. Two different multivariate priors are elicited using median and quartile assessments of the multinomial probabilities. First, we start by eliciting a univariate beta distribution for the probability of each category. Then we elicit the hyperparameters of the Dirichlet distribution, as a tractable conjugate prior, from those of the univariate betas through various forms of reconciliation using least-squares techniques. However, a multivariate copula function will give a more flexible correlation structure between multinomial parameters if it is used as their multivariate prior distribution. So, second, we use beta marginal distributions to construct a Gaussian copula as a multivariate normal distribution function that binds these marginals and expresses the dependence structure between them. The proposed method elicits a positive-definite correlation matrix of this Gaussian copula. The two proposed methods are designed to be used through interactive graphical software written in Java.

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“…A number of procedures for arriving at these values have been devised (Winkler 1967(Winkler , 1977Kadane et al 1980;Zellner 1985). A major advance toward rendering the assessment of natural conjugate priors operational is represented by Zellner's g-priors (Zellner 1982(Zellner , 1986.…”

Section: Background: Bayesian Regressionmentioning

confidence: 99%

Bayesian Regression and the Expansion Method

Casetti¹

1992

Geographical Analysis

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This paper shows that within a Bayesian frame of reference the propensity to regard a model as invariant can be formulized into a prior probability density function of the parameters of an expanded formulation in which this model is encapsulated. The Bayesian approach allows to bring to the sugace the implications of alternative levels of commitment to invariance assumptions on the results of empirical analyses, and can quantijiy the comparative strength of alternative dri3 specijications. The "expanded Bayesian regressions are demonstrated by an example focusing upon the effectiveness of family planning. INTRODUCTIONThe mathematical models in the social sciences and elsewhere are very often estimated under implicit or explicit assumptions of "parametric stability" or "invariance." At the very least, they are presumed valid (read, stable) over the data sets from which they are estimated, and in fact the conventional estimation of population parameters presupposes that a model "holds" over the data employed to estimate it. The stability assumptions have an even deeper foundation and reach whenever a mathematical model is viewed as the expression of a "law, " or a candidate for law status. Invariance and universal validity are the distinguishing traits of laws.The alternative to stability is "instability" or "drift." Let us say that a model drifts across a context whenever it holds with different parameter values at different points of the context. As a research philosophy, the expansion methodology suggests that the stability of mathematical models should never be assumed. On the contrary, the contextual drift of models should be expected, hypothesized, searched for, and tested, and theories concerning the nexus between models and contexts should both guide and follow the investigations of contextual drift.However, the focus of this paper is not on the comparative merits of assuming contextual stability or contextual drift, but rather on how to investigate the impacts of these alternative assumptions on regression results, which brings us to the central issue of this presentation. This paper will show that, within a Bayesian frame of reference, the propensity to regard a model as invariant can be formalized Emilio Casetti I 59 into a prior probability density function of the parameters of an expanded formulation in which this model is encapsulated. This prior combines with a likelihood function reflecting sample information to produce a posterior probability density function on the basis of which hypotheses of parametric drift can be investigated and tested.In the specific Bayesian approach employed here, the intensity of commitment to assumptions of parametric invariance is represented by a parameter, g. By changing the value of this g, we can investigate the impact of alternative levels of commitment to invariance on the results of an empirical analysis, and we can determine at which level of commitment to invariance "drift hypotheses" are rejected. The notion of parametric stability is supported when the rejec...

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Interactive Elicitation of Opinion for a Normal Linear Model

Cited by 114 publications

References 11 publications

Bayesian Model Averaging for Linear Regression Models

Bayesian Model Averaging for Linear Regression Models

Eliciting Dirichlet and Gaussian copula prior distributions for multinomial models

Bayesian Regression and the Expansion Method

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