The name is absent

if we receive information on x we can update each of the conditional priors
on y in a consistent fashion.

As an example suppose that x is gross domestic product y is national
income, z is depreciation and w is international transfers. Given a sequence
of observations on x, the decision-maker may act as a restricted Bayesian
with respect to x and employ multiple priors with respect to y, each of which
corresponds implicitly to an alternative hypothesis about z.

Observe in addition that there may exist unconsidered propositions p
that are uninformative wrt any of w,x,y and z. Trivially, observation of the
truth value of such propositions does not imply any change in the restricted
Bayesian posterior, nor in the induced prior.

9 Concluding comments

In formulating more general representations of choice under uncertainty, it
is highly desirable to show that, under appropriate conditions, existing rep-
resentations can be derived as special cases. These conditions are often re-
strictive. Nevertheless, it is often the case that they may be satisfied exactly
or as a reasonable approximation.

The Bayesian approach to decision theory is powerful and appealing.
However, the assumption, necessary for the model to be applied, that the
decision-maker possesses an exhaustive description of all possible states of
the world, with an associated probability distribution, is obviously unrealis-
tic. In this paper, we have derived necessary and sufficient conditions for the
consistency of Bayesian updating when applied to a restricted set of propo-
sitions, a subset of an exhaustive propositional description of the world.

The necessary conditions are highly restrictive, suggesting that, in many
cases, a multiple priors model may be more realistic. For this case also,
conditions have been derived under which each element of a set of multiple
priors may be updated consistently.

In general, neither of these sets of conditions may be satisfied. Typi-
cally, learning is not simply a matter of updating priors but involves new
discoveries, imaginative conjectures, abandonment of previously maintained
hypotheses and so on. A central task for decision theory, partially addressed
by Grant and Quiggin (2004) is to develop models of these processes.

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