The name is absent

where a = E [z] , ε = z — a. Under plausible conditions, the decisionmaker
may be able to make the implicit assumption E [z] = 0 without any detailed
knowledge of 7 and z. For example, suppose that x is gross domestic product
and y is gross national product for some unspecified country. The differ-
ence between x and y is determined by international income flows, which
necessarily sum to zero for the world as a whole. So, a decisionmaker could
reasonably use the model

y = x + ε

while having no knowledge of factors determining ε. More generally, even
when the value of E [z] is unknown and unconsidered, a decisionmaker might
have reasonable knowledge about β = ∂y∕∂x, so that Bayesian updating
applied to x may be useful even when y is the variable of interest.

6.1.1 An observation on decision theory

Allthough we have not formally considered applications to decision theory,
it is easy to see that for appropriately linear choice problems, restricted
Bayesianism applied to x yields optimal decisions (on the assumption that
no information is available about y or z). Assume for simplicity that the
unconditional expectation of z is equal to zero, and consider the problem of
choosing θ to maximise E [v] where

^v (^θ,y} = ^θy ^— c (^θ) ^,

and c is a convex cost function. We have

E [v] = θE [y] — c (θ)
= θβE [x] — c (θ).

Hence, the optimal policy, given an estimate of E [x] , is to choose θ such
that

c' (θ) = βE [x] ,

and the best estimate of E [x] may be obtained by restricted Bayesian up-
dating of the unconditional prior distribution μ₁.

In this context, then, it seems reasonable to suggest that an optimal
policy could be achieved solely by considering hypotheses about β and V.

7 Multiple priors

Thus far, we have considered cases where the prior distribution μ_t on the
restricted domain generated by the considered propositions is induced by a

10

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