Probabilistic Opinion Pooling

Abstract

Suppose several individuals (e.g., experts on a panel) each assign probabilities to some events. How can these individual probability assignments be aggregated into a single collective probability assignment? This chapter is a review of several proposed solutions to this problem, focusing on three salient proposals: linear pooling (the weighted or unweighted linear averaging of probabilities), geometric pooling (the weighted or unweighted geometric averaging of probabilities), and multiplicative pooling (where probabilities are multiplied rather than averaged). Axiomatic characterizations of each class of pooling functions are presented (most characterizations are classic results, but one is new), with the argument that linear pooling can be justified “procedurally” but not “epistemically”, while the other two pooling methods can be justified “epistemically”. The choice between them, in turn, depends on whether the individuals' probability assignments are based on shared information or on private information. In conclusion a number of other pooling methods are mentioned.

Probabilistic Opinion Pooling with Imprecise Probabilities

“…The question of how the probabilistic opinions of different individuals should be aggregated to form a group opinion is controversial. But one assumption seems to be pretty much common ground: for a group of Bayesians, the representation of group opinion should itself be a unique probability distribution (Madansky, 1964;Lehrer and Wagner, 1981;McConway, 1981;Bordley, 1982;Genest and Zidek, 1986;Mongin, 1995;Clemen and Winkler, 1999;Dietrich and List, 2014;Herzberg, 2014). We argue that this assumption is not always in order.…”

“…It is important to consider, for each property, the extent to which it is normatively compelling for a particular interpretation and use of pooling functions. Surveys of the material presented here include Simon French's (1985), Genest and Zidek's (1986), and Dietrich and List's (2014). McConway (1981) and Lerher and Wagner (1981) introduce a convenient property of pooling functions called the strong setwise function property and strong label neutrality by the respective authors.…”

“…For every profile pp 1 , ..., p n q in the domain of F and every likelihood function λ such that pp λ , ..., p λ n q remains in the domain of F , F pp λ 1 , ..., p λ n q " F λ pp 1 , ..., p n q. A similar axiom requires that a single individual conditionalizing on λ before pooling is the same as conditionalizing the pool on λ (Dietrich and List, 2014).…”

Reasons, Coherence, and Group Rationality

“…I will not attempt a survey of impossibility theorems in the theory of probability aggregation (see Genest and Zidek 1986;Russell et al 2015;Dietrich and List 2016). Instead, I will simply look at the most obvious and popular probability aggregation rule, note one condition it violates, and then turn to a reasons-based approach to group credences.…”

Learning from others: conditioning versus averaging

“…There is a large statistics literature, and a modest philosophical one, on propositionindependent linear averaging and its properties, in which the weights are taken to be constant across propositions, see Genest and Zidek (1986) and Dietrich and List (2016) for good surveys. 2 Much of this literature is focused on the problem of how a group should form a consensual opinion on some issue, a question that is not directly related to that of how individuals should improve their own opinions by taking into account the information provided by the opinions of others.…”