Academic journal article
By Gelfand, Alan E.; Mallick, Bani K.; Dey, Dipak K.
Journal of the American Statistical Association , Vol. 90, No. 430
Expert opinion is often sought with regard to unknowns in a decision-making setting to improve the quality of the decision-making process. We presume that this opinion pertains to a continuous univariate unknown, denoted by [Theta]. Examples of such unknowns are quite diverse: a survival time for a particular patient, the annual precipitation for a given region, the LD50 for a proposed drug, and the dollar return on a national advertising promotion. Opinion may be collected from several experts but in each case it is assumed probabilistic in nature.
In this context, most work assumes that an individual expert fully supplies a subjective probability measure for [Theta]. By now, the literature on elicitation of a probability measure is substantial. (See Kahneman, Slovic, and Tversky 1982 for a readable review and Kadane, Dickey, Winkler, Smith, and Peters 1980 for implementation suggestions.) A naive approach is to insist that the individual probability measures for [Theta] are members of a standard parametric family, to which the expert need only supply its parameters. This seems too restrictive and likely inappropriate. Rather, we assume that each expert expresses belief regarding the unknown [Theta] in a partial way. That is, either probabilities are provided for a small collection of disjoint exhaustive intervals in the domain of [Theta], or a small set of quantiles for the distribution of [Theta] are provided. We anticipate such incomplete description to be more reliable than, say, partial specification of the first few moments of the distribution of [Theta] (Genest and Schervish 1985), which seem less intuitive than probabilities or quantiles.
Note that we have traded one modeling problem for another. We now need to develop appropriate probabilistic models for the "data," which are these partial specifications. This problem is the subject of our work; we address it primarily at the individual level (though see Sec. 5). Our work builds on that of Lindley (1983, 1985) and West (1988).
We adopt the so-called "supra Bayesian" approach as our mechanism for combining expert opinions. The literature on normative approaches for the formation of aggregate opinion is substantial; see, for instance, the survey articles of French (1985), Genest and Zidek (1986), and Chatterjee and Chatterjee (1987). Much of the work in this area presumes an axiomatic specification of a set of properties that the aggregation mechanism must obey and then deduces the class of pooling functions meeting these properties. We are drawn to the supra Bayesian stance of adopting Bayes's rule as the pooling operator synthesizing opinion through the posterior distribution. Often there is an implicit external decision-maker who has his or her prior opinions, who gathers the experts' opinions and who can calibrate the relative quality of each expert's opinion. We seek to help this decision-maker to determine a joint density for this collection of opinions after which the Bayesian paradigm would enable the desired pooling. The supra Bayesian approach has its roots in work of Winkler (1968); the name was coined by Keeney and Raiffa (1976). Genest and Zidek (1986) provided insightful additional discussion. Some group decision-making situations would not naturally identify an overseeing decision-maker and so would not fit this structure.
For data which is opinion in the form of partial specification, Lindley (1985) and West (1988) clearly articulated the key issues in providing suitable density functions. In the case of specification of probabilities of disjoint exhaustive sets, Lindley essentially took a multivariate normal distribution for the logits of these probabilities. If the supra Bayesian specifies a prior for [Theta] over these sets, then Lindley observed that the posterior or combined opinion updates these prior probabilities in a linear fashion on the log scale. In the case of specification of quantiles, West developed a density emanating from a Dirichlet process that implicitly determines the distribution of these quantiles. …