Beliefs about beliefs, a theory for stochastic assessment of subjective probabilities

Parameterized families of subjective probability distributions can be used to great advantage to model beliefs of experts, especially when such models include dependence on concomitant variables. In one such model, probabilities of simple events can be expressed in loglinear form. In another, a generalization of the multivariate t distribution has concomitant variables entering linearly through the location vector. Interactive interview methods for assessing this second model and matrix extensions thereof were given in recent joint work of the author with A. P. Dawid, J. B. Kadane and others. In any such verbal assessment method, elicited quantiles must be fitted by subjective probability models. The fitting requires the use of a further probability method for errors of elicitation. This paper gives new theory relating the form of the distribution of elicited probabilities and elicited quantiles to the form of the subjective probability distribution. The first and the second order moment structures are developed to permit generalized least squares fit