unique prior μ on the full state space, incorporating implicit probabilities for
unconsidered propositions. To generate a multiple priors model, it is natural
to suppose that there may be more than one such measure. An obvious way
to do this is to look at the measures induced conditional on the possible truth
values for one or more unconsidered propositions.
Considering any p' E P-, there are two induced measures on P'i. namely
and
μ+ (E (p))
μ (E (p)) =
μ (E (p Λ p'Y)
μ (E (p'))
.p E Pi
μ (E (p λ^√)) p■
μ (E ⅛')) 'p E ∙
For a proposition p' that is independent of Pг in the sense that, for all p E Pг
μ (E (p λ p')) = μ (E (p)) μ (E (p')) .
we have μ+ = μ since, for all p E Pг
μ+ (E (p))
μ (E (p λ p')) = μ (E (p)) μ (E (p'))
μ(E (p')) μ(E (p'))
μ(E (p)) μ(E (^p')) = μ(E (p λ +'л = e
μ(E (^p')) μ(E (^p')) μ
In general, however, μ+E (p) = μ-E (p) . and consideration of probability
values for p' in the range [0.1] gives rise to probabilities for p in the range
[μ-E (p) . μ+E (p)] . Thus, we can define a set of priors
M (p') = {Λμ+ + (1 - Aμ- :0 ≤ A ≤ 1} .
The natural interpretation here is that each element of the set of multiple pri-
ors may be derived as a conditional probability measure, given a probability
number for the unconsidered proposition p'. Thus p' has a status interme-
diate between propositions in Pг that are under active consideration, and
unconsidered propositions in the case of restricted Bayesianism. Although
the decision-maker does not explicitly consider p'. the range of multiple priors
corresponds to the probability measure that would arise if p' were a consid-
ered proposition with probability A.
For a more general version of the multiple priors model, let P* be a
set of unconsidered propositions, closed under ^ and Λ. and let Δ be the
unit simplex with dimension equal to K = card (P*) . having typical element
λ=(A1..... Aκ) such that J^fe A⅛ = 1. For each p⅛ E P*. we have, as described
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