Hi,
The current discussion on ignorance, beliefs, etc,
started (I think) with a question: if one begins
with an imprecise prior, does it get more precise
as data is collected? There seems to be two ways
to interpret the question --- it is an "imprecise question":)
a) One starts with beliefs that are non-informative
with respect to the truth of possible options.
b) One starts with beliefs that are incomplete, and
translates them into probabilities that
are not point-valued; maybe intervals or sets of
probabilities.
Option (a) suggests that one is beginning with a flat prior.
There have been answers to (a); namely, standard results
in Bayesian theory prove that data tends to reduce
the importance of the prior --- even a flat prior will
be "forgotten" as more and more data arrives.
I would like to add an answer to (b); I'm not sure this
has actually been answered. The answer has to assume a
method to process the observations, and most of the
literature assumes various forms of the Bayes rule.
The answer to alternative (b) is: In general
imprecise probabilities will not collapse to a precise
probability (except in special cases, for example if the
observation logically implies something). The reason is
that Bayes rule applied to a set of measures does produce
a set of measures; it does not magically reduces the
set of measures to a single point. This kind of reduction
(from an interval to a point) is possible with Dempster rule.
In fact, the "size" of the imprecise probabilities (for
example, the length of an interval between maximal and
minimal probability) may increase if an observation conflicts
with previous observations. There is a nice example of
this in Peter Walley's book (the example is at page 225):
Statistical Reasoning with Imprecise Probabilities
Peter Walley
Monographs on Statistics and Applied Probability, Chapman and Hall, 1991.
The book also contains lengthy discussions of differences
between concepts of "ignorance", "knowledge", "beliefs", etc.
This is an old discussion in AI, and much older in statistics;
there are several references to work by Keynes, Popper, etc.
Speaking of Walley's work, the current discussion on dice
and buttons is quite in line with a paper by him called "Learning
from a Bag of Marbles", read for the Journal of the Royal
Statistical Society in 1996. The paper is exactly about
the statistical estimation of the probability of drawing
a marble from a bag when the constitution of the bag is unknown.
The debate over imprecise probability is quite important
and goes over many subjects like economics and psychology.
People who are interested in theories and applications that cover
many views and positions may find it useful to look at
the electronic proceedings of the First Int. Symp. on
Imprecise Probabilities and Their Applications, at
http://ensmain.rug.ac.be/~isipta99/index.html
There are many papers there that look at exactly the questions
that are being discussed in this thread.
Fabio Cozman
PS: On this thread, there have been references to many
thought experiments, including the probability of God.
In fact, guessing the probability of God was a common
exercise three centuries ago, when philosophers were
trying to find ways to prove the existence of God.
There is an amusing passage in Laplace's book
where he ridicules some efforts by Liebnitz and Daniel
Bernoulli. Liebnitz used strange "Bayesian" arguments
to justify that if someone did not know the value of
a convergent series, the value should be 1/2 (!).
And then others jumped to prove the existence of
God and other things. The passage is at page 169 of
Dover's edition of "A Philosophical Essay on
Probabilities", by the Marquis de Laplace.
Laplace did not take such subjective approaches to
probability as a problem, only said that they showed "to
what extent the prejudices of infancy can mislead
the greatest man".
