Martedì 6 novembre, ore 12,00 Aula VI
Il Professor A. Philip Dawid, Department of
Statistical Science, University College London
terrà presso il Dipartimento di Economia, Università
Roma Tre, Via Ostiense 139, un seminario dal titolo
MAXIMUM ENTROPY AND ROBUST BAYES DECISION
THEORY
(joint work with Peter Grunwald)
ABSTRACT
It is sometimes reasonable to assume
that the distribution P of a random
variable X belongs to a known family F,
but P is otherwise unknown. For
example, only the mean of X might be
known. In such a case one might wish
to select some single distribution
in F, to act as a representative of F
for the purposes of inference or
decision-making. A celebrated recipe for
doing this is the 'maximum
entropy criterion'. However, its rationale has
often been
unclear. Here we present an alternative justification for
maximum
entropy.
Suppose a decision maker DM has to quote a density q() for X,
and faces a
loss -log q(x) if Nature chooses X = x. DM wants to
minimize the maximum
possible expected loss, as P varies in F (this is a
'robust Bayes'
criterion). Then, under suitable conditions on F, it can
be shown that
DM's optimal q is just the density of the maximum entropy
distribution.
The same logic extends to much more general decision
problems, beyond `log
loss'. With any such problem we can associate a
specific 'generalized
entropy' function; and again, under suitable
conditions, the robust Bayes
act will be simply related to the maximum
generalized entropy
distribution. This property is related to the
existence of a saddle-point
in a suitable game between DM and Nature.
Examples will be given to
illustrate the theory and its
consequences.
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