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More Aspects of Polya Tree Distributions for Statistical Modelling
 Ann. Statist
, 1994
"... : The definition and elementary properties of Polya tree distributions are reviewed. Two theorems are presented showing that Polya trees can be constructed to concentrate arbitrarily closely about any desired pdf, and that Polya tree priors can put positive mass in every relative entropy neighborhoo ..."
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Cited by 65 (1 self)
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: The definition and elementary properties of Polya tree distributions are reviewed. Two theorems are presented showing that Polya trees can be constructed to concentrate arbitrarily closely about any desired pdf, and that Polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy, thereby satisfying a consistency condition. Such theorems are false for Dirichlet processes. Models are constructed combining partially specified Polya trees with other information like monotonicity or unimodality. It is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications. A numerical example is given. A theorem of Diaconis and Freedman about Dirichlet processes is generalized to Polya trees, allowing Polya trees to be the models for errors in regression problems. Finally, empirical Bayes models using Dirichlet processes are generalized to Polya trees. An example from Berry and Chris...
Statistical Methods for Eliciting Probability Distributions
 Journal of the American Statistical Association
, 2005
"... Elicitation is a key task for subjectivist Bayesians. While skeptics hold that it cannot (or perhaps should not) be done, in practice it brings statisticians closer to their clients and subjectmatterexpert colleagues. This paper reviews the stateoftheart, reflecting the experience of statisticia ..."
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Cited by 52 (2 self)
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Elicitation is a key task for subjectivist Bayesians. While skeptics hold that it cannot (or perhaps should not) be done, in practice it brings statisticians closer to their clients and subjectmatterexpert colleagues. This paper reviews the stateoftheart, 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, i.e. what criteria should be employed? 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 that way. We see elicitation as simply part of the process of statistical modeling. Indeed in a hierarchical model it is ambiguous at which point the likelihood ends and the prior begins. Thus the same kinds of judgment that inform statistical modeling in general also inform elicitation of prior distributions.
Uncertainty in prior elicitations: a nonparametric approach
, 2005
"... A key task in the elicitation of expert knowledge is to construct a specific elicited distribution from the finite, and usually small, number of statements that the have been elicited from the expert. These statements typically specify some quantiles of the distribution, perhaps the mode and sometim ..."
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Cited by 12 (1 self)
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A key task in the elicitation of expert knowledge is to construct a specific elicited distribution from the finite, and usually small, number of statements that the have been elicited from the expert. These statements typically specify some quantiles of the distribution, perhaps the mode and sometimes the mean or other moments. 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. Second, no account is then taken of the many other possible distributions that might have fitted the elicited statements equally well. We present an approach which tackles both of these deficiencies. Our model is nonparametric, allowing the expert’s distribution to take any continuous form. It also quantifies the uncertainty in the resulting elicited distribution. Formally, the expert’s density function is treated as an unknown function, about which we make inference. The result is a posterior distribution for the expert’s density function. The posterior mean serves as a ‘best fit ’ elicited distribution, while 1 the variance around this fit expresses the uncertainty in the elicitation. When data become available, uncertainty about the expert’s posterior distribution induced by the uncertainty in their prior distribution can then be described. We also briefly consider the issue of the imprecision in any elicited probability judgment, and suggest a modification of our model to account for this.
On a Global Sensitivity Measure for Bayesian Inference
"... We define a global sensitivity measure that is useful in assessing sensitivity to deviations from a specified prior. We argue that this measure has a common interpretation irrespective of the context of the problem, or the unit of measurements, and is therefore easy to interpret. We also study the a ..."
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Cited by 2 (0 self)
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We define a global sensitivity measure that is useful in assessing sensitivity to deviations from a specified prior. We argue that this measure has a common interpretation irrespective of the context of the problem, or the unit of measurements, and is therefore easy to interpret. We also study the asymptotic behavior of this global sensitivity measure. We find that it does not always converge to 0 as the sample size goes to infinity. We also show that, under certain conditions, this measure does go to 0 as the sample size goes to infinity. Thus, unlike the usual global sensitivity measure range, this measure behaves asymptotically like the usual local sensitivity measure. 1 AMS 1991 subject classifications. Primary 62F35; secondary 62C10 Key words and phrases. Bayesian robustness, global sensitivity, asymptotics. 1 1 Introduction In a Bayesian analysis involving a subjectively elicited prior, one is usually concerned with sensitivity to deviations from the specified prior, 0 . In ...
Elicitation
, 2004
"... Elicitation is a key task for subjectivist Bayesians. While skeptics hold that it cannot (or perhaps should not) be done, in practice it brings statisticians closer to their clients and subjectmatterexpert colleagues. This paper reviews the stateoftheart, reflecting both the experience of statis ..."
Abstract
 Add to MetaCart
Elicitation is a key task for subjectivist Bayesians. While skeptics hold that it cannot (or perhaps should not) be done, in practice it brings statisticians closer to their clients and subjectmatterexpert colleagues. This paper reviews the stateoftheart, reflecting both the experience of statisticians and 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, i.e. what criteria should be employed? 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 that way. We see elicitation as simply part of the process of statistical modeling. Indeed in a hierarchical model it is ambiguous at which point the likelihood ends and the prior begins. Thus the same kinds of judgment that inform statistical modeling in general also inform elicitation of prior distributions.