Paulusnix
Wed, 11 Jun 2003 18:05:09 -0700
Greetings, all :-
Lotfi Zadeh has asked for decades whether probability theory
handles selected everyday uncertain inferences. Veterans of the UAI
list may remember his "challenge to Bayesians" in August 2000, with
occasional follow-up postings since then.
Recently, a broader inquiry moved on his Berkeley Initiative in
Soft Computing (BISC) list. His questions are no longer put
exclusively to Bayesians, but now to adherents of any "standard"
probability theory based upon bivalent logic.
Most readers are familiar with the kind of inferential task which
bothers Professor Zadeh about probability. Imagine that we are aboard
a ship, out of sight of the land.
When the ship is near the land, we often see birds.
When the ship is far from the land, we see birds less often.
We see birds for the first time since leavng port. From this observation and
the premises, many would conclude that it becomes more credible that we are
near the land, compared to before the sighting.
At least two features place this story outside the scope of some
probability theories.
(1) "The ship is near the land" is vague, or what Professor Zadeh lately
calls "perception-based," rather than categorically and determinably true or
false. Among those who would object are admirers of a Jaynes-style "clarity"
principle, or those whose semantics for probability depends upon unambiguous
betting contracts.
(2) "Often" and "credible" are imprecise descriptions of something
probabilistic.Bayesians typically insist upon precise numerical probability. Some
others relax that, but still require numbers which, say, bound probability
intervals.
While the example is fairly representative of the puzzles that Professor
Zadeh has offered over the decades, this one is not Zadeh's. It is George
Polya's, who derived the expected conclusion by what he took to be valid
probabilistic means.
Polya was prolific about (2), the legitimacy of frankly qualitative
probability. As to issue (1), Richard Threlkeld Cox felt that probability could be
applied to "what song the Sirens sang, or what name Achilles assumed when he
hid himself among women." Closer to the kind of "fuzzy sentence" associated
with Zadeh over the years, Cox discussed
The stranger was a short, fat old man without coat or hat.
as something his notion of 'proposition', and so his notion of probability,
could take in its stride.
Cox and Polya are canonical authors. It is hard to imagine what "standard
probability theory" could mean if taken to exclude them.
That's a start. I hope other readers will also assist our colleague in
his inquiries.
Best regards.
Paul Snow
Cox's prose example can be found on page 5 of his 1946 'Probability,
frequency, and reasonable expectation' (_American Journal of Physics_
volume 14, number 1). The poetic matter is from the introductory quote
of Thomas Browne in Cox's 1978 'Of inference and inquiry' which
appeared in Levine and Tribus (eds.) _The Maximum Entropy Formalism_
(MIT Press, 1979, pages 119-167). Polya's example appears on page 37
of his _Patterns of Plausible Inference_, which is the second volume
of his _Mathematics and Plausible Reasoning_ (Princeton University
Press, 1954).