Abstract
The key purpose of this paper is to present an alternative viewpoint for combining expert opinions based on finite mixture models. Moreover, we consider that the components of the mixture are not necessarily assumed to be from the same parametric family. This approach can enable the agent to make informed decisions about the uncertain quantity of interest in a flexible manner that accounts for multiple sources of heterogeneity involved in the opinions expressed by the experts in terms of the parametric family, the parameters of each component density, and also the mixing weights. Finally, the proposed models are employed for numerically computing quantile-based risk measures in a collective decision-making context.
Highlights
Regarding the implementation of the Expectation Maximization (EM) algorithm for maximum likelihood (ML) estimation in the context of finite mixture models, we follow the standard approach of combining the observed data, which are represented by the random variable X, with the set of unobserved latent random variables w =, where wiz = 1 if the i-th observation belongs to the z-th component, and 0 otherwise, for i = 1, ..., ν and z = 1, ..., n
A decision maker, otherwise called an agent, needs to make a decision about an X random quantity of interest. Since this decision is made under circumstances of uncertainty, the agent seeks for the opinion of an arbitrary number of consultants z = 1, 2, ...n and the combined opinion is seen as a finite mixture model of the type described in 2.3.1 allowing for divergence in expert opinions, both in the class of fz and in components parameters θz
In the context of combining expert opinions for computing quantile-based risk measures, such as Value at Risk (V@R), there is a clear advantage that the suggested finite mixture modelling approach enjoys over the classical approach of calculating quantiles such as the weighted average of individual quantiles coming from the expert judgements; see, Lichtendahl et al (2013)
Summary
“Opinion is the medium between knowledge and ignorance” is an expression that is ascribed to Plato. Our main contribution is that we consider that the component distributions can stem from different parametric families The advantage of this formulation is that it allows the agent to obtain the aggregated opinion of a group of experts, based on a linear opinion pool, and account for the various sources of unobserved heterogeneity in the decision-making process in the following ways: (i) by assuming that the data are drawn from a finite mixture distribution with components representing different opinions about both the distribution family and its parameters regarding the uncertain quantity of interest, and (ii) via the mixing weights that reflect the quality of each opinion.
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