Nonparametric elicitation for heavy-tailed prior distributions

Abstract

In the context of statistical analysis, elicitation is the process of translating someone's beliefs about some uncertain quantities into a probability distribution. The person's judgements about the quantities are usually fitted to some member of a convenient parametric family. This approach does not allow for the possibility that any number of distributions could fit the same judgements.

In this paper, elicitation of an expert's beliefs is treated as any other inference problem: the facilitator of the elicitation exercise has prior beliefs about the form of the expert's density function, the facilitator elicits judgements about the density function, and the facilitator's beliefs about the expert's density function are updated in the light of these judgements. This paper investigates prior beliefs about an expert's density function and shows how many different types of judgement can be handled by this method.

This elicitation method begins with the belief that the expert's density will roughly have the shape of a $t$~density. This belief is then updated through a Gaussian process model using judgements from the expert. The method gives a framework for quantifying the facilitator's uncertainty about a density given judgements about the mean and percentiles of the expert's distribution. A property of Gaussian processes can be manipulated to include judgements about the derivatives of the density, which allows the facilitator to incorporate mode judgements and judgements on the sign of the density at any given point. The benefit of including the second type of judgement is that substantial computational time can be saved.