Dear All :-
'The premises provide no information about the desired P' is a tenable
view. Even for those who agree, however, it is untrue that P is undetermined
according to all standard probability theories.
For example, an orthodox Bayesian who accepted that "Robert wears
large-sized shoes" was fit for probabilistic treatment would have a
specific _a priori_ probability for that statement. If the other
statements inspire no opinion change, then the answer for this analyst
is "P equals <my _a priori_ probability>," which is a definite number.
There is no obligation to assent to 'the premises provide no
information about the desired P.' If we understand what is being said
in the shoe problem to imply that
Most members of a population to which Robert belongs wear
large shoes then we might think about direct inference and the fraught
relationships between statements about a population and statements
about a member of that population. In the birds-at-sea example cited
in an earlier posting, Polya makes a typical direct inferential leap,
that what is true of bird sightings in general says something about
this particular occasion.
It is probability theory which tells us that that is a
leap. Whether or not one jumps is up to the believer. Describing "the
logical laws of probability," de Finetti wrote
These laws are the conditions which characterize coherent opinions (that is,
opinions which are admissible in their own right) and which distinguish them
from others that are intrinsically contradictory. The choice of one of these
admissible opinions from among all the others is not objective at all and does
not enter into the logic of the probable...
'More likely than not, Robert wears large shoes, on the available
information' is an admissible conclusion. Many other conclusions are
also admissible.
Professor Zadeh is correct, then, that probability theory does not
dictate an answer in his example. It would be magical if any method
for non-demonstrative deliberation could prescribe the content of
someone's beliefs. Probability offers advice about whether opinions
are consistent, not whether they are right, nor even impeccably
justified.
The question in the BISC root posting was not whether probability theory
decides the issues arising in the example. The question was whether
probability theory addresses them. The correct answer is yes.
Best regards.
Paul Snow
The de Finetti quote appears on page 110 of Henry Kyburg's translation
of "Foresight, its logical laws, its subjective sources" in Kyburg and
Smokler (eds.) _Studies in Subjective Probability_, Wiley, 1964. For
de Finetti's application of his views to direct inference, see pages
115 and following. A discussion of direct inference by Henry himself
starts at the bottom of page 283 in his classic "Bayesian and
non-Bayesian evidential updating" (_Artificial Intelligence_ 31,
1987).
