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A theory of Bayesian groups

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Franz Dietrich

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It is standard practice in economics and the social sciences more broadly to rank alternatives from a social perspective: to ascribe preferences or betterness judgments to a group or society as a whole. Less common in social science is to ascribe beliefs – subjective probability assignments – to groups. Yet everyday language routinely invokes group beliefs, not just group-member beliefs: the UK government is said to believe that the Brexit is long-term beneficial, the IPCC is said to believe in certain patterns of climate change, and so on. If group beliefs are to be “rational”, they should change “rationally” in the light of new information. The gold standard for belief revision is, of course, Bayesian revision: the new beliefs should equal the old beliefs conditionalized on the information (Bayes’ rule).

In this very recent theoretical paper, Franz Dietrich proposes to apply Bayesian revision to group beliefs rather than individual beliefs, and to explore the formal constraints which this imposes on group beliefs. Such “group Bayesianism” faces three challenges. First, group beliefs are not free-floating, but a function of the beliefs of the group members. There is an extensive literature on how to pool beliefs (probability measures) of individuals into group beliefs (a group probability measure). Yet most standard proposals – including that of taking the arithmetic average of the individual beliefs – heavily violate group Bayesianism: the resulting group beliefs will not be revised in a Bayesian way.
Second, what does it mean that “the group” learns information? The author proposes to distinguish between public information (learnt by all members), private information (learnt by only one member), and partially spread information (learnt by some but not all members). This raises the question of the type(s) of information for which one should require Bayesian revision of group beliefs. The third challenge for group Bayesianism pertains to the fact that some information might not be representable by any event in the domain (algebra) on which beliefs (probabilities) are defined. The group might learn that the radio forecasts rainy weather, but it might hold beliefs only relative to ‘weather events’, not ‘weather-forecast events’. In such a case ordinary Bayesian revision is not even defined. Yet a generalized form of Bayesian revision can still be applied, as explained in the paper. This question is here: should one require Bayesian revision of group beliefs even for non-representable information?
Combining the second and third challenges, Franz Dietrich obtains different types of information: public representable information, private representable information, public non-representable information, and so on. Each such type of information gives rise to a specific form of group Bayesianism, requiring Bayesian conditionalization on information of this type. The paper’s mathematical contribution is to determine the exact class of belief pooling rules which guarantee group Bayesianism of any given type. This is done in six theorems, one for each of the six types group Bayesianism considered. As it turns out, each type of group Bayesianism permits only certain weighted geometric pooling rules. Such rules define the group’s probability of a “world” as a weighted geometric average of the individuals’ probabilities of that world (with a subsequent normalization). The different versions of group Bayesianism impose different constraints on the weights given to the individuals in the geometric average: for instance the weights might have to be non-zero, or they might have to sum to one, and so on. The idea of group Bayesianism has already been around in the literature, for instance in the form of the axiom of ‘external Bayesianity’ often imposed in probabilistic opinion pooling theory. One of F. Dietrich’s theorems—that for public non-representable information—indeed fills an important gap in the theory of externally Bayesian opinion pooling.

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Original title : A theory of Bayesian groups
Published in : Forthcoming
Available at : http://franzdietrich.net/Papers/Dietrich-BayesianGroups.pdf
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