prises, unconsidered possibilities and so on. A number of recent studies, no-
tably those of Modica & Rustichini (1999), Dekel, Lipman and Rusticchini
(2001), Halpern (2001), Li (2003) and Heifetz, Meier & Schipper (2004) have
attempted to model these problems.
In this paper, we employ the propositional approach developed by Grant
and Quiggin (2004) and consider the properties of Bayesian updating in the
presence of unconsidered propositions. The key idea is to take the prior distri-
bution over a set of propositions under consideration (which is an exogenous
given in the standard Bayesian model) as the conditional distribution de-
rived from a probability distribution over a complete state space, which is
not accessible to the decisionmaker. We then consider updating in the light of
new information. Consistency requires that the usual Bayesian posterior over
the considered propositions be the same as the prior that would be derived
from the updated probability distribution over the complete space. We show
that consistency is equivalent to an independence property, which justifies
treating the considered propositions in isolation.
Next we consider the multiple priors model. In our approach, multi-
ple priors are the conditional distributions derived from different implicit
assumptions about unconsidered propositions. For consistent updating of
multiple priors, we require not only that the independence property should
hold for each prior, but that observations on considered propositions should
not be informative with respect to those priors.
These are stringent conditions. However, we show by example, that in
many of the standard situations in which Bayesian reasoning is applied, these
conditions are reasonable, at least in the sense that they are implicit in
standard approaches to such problems.
2 Setup
We consider a single individual decision-maker, denoted i, making choices
over time, in a situation of uncertainty regarding the state of nature, and an
incomplete description of the set of states of nature. To describe the individ-
ual’s representation of the world, it is necessary to embed this representation
in a more complete description.
2.1 Representation
Let the set of states of the world be Ω. We focus on the representation
Ω = 2n