Dear Marco and Kathryn.
Some comments about your AUAI postings:
Marco Zaffalon wrote:
> I am happy with systems that can recognize the limits
> of their knowledge and suspend the judgment when these limits are
> reached,
> in the same way that I prefer to be told "I do not know" when I ask for
> road information rather than being recommended a wrong route. Also good
> human experts know when they should suspend the judgment.
> Having to occasionally suspend the judgment is logical consequence of
> working with probability intervals, or with more general frameworks
> (e.g.,
> lower probabilities and previsions)....
Kathryn Blackmond Laskey wrote:
> Even better is a system that can say: "If forced to give advice, my
> recommendation is X, with justification Y, but my confidence in that
> recommendation and its justification is only Z." If the answer is not
> very trustworthy, I would be well-advised to consult other sources if
> resources permit, but at the very least I have a recommendation and a
> justification to chew over. A system that says nothing but that it
> suspends judgment gives me no help at all.
I totally agree with what Marco and Kathryn say. It is completely in
the spirit of my work and I am happy to see that there is more people
sharing the same thoughts.
However, rather than working with probability intervals, I think it is
more natural to work with probabilities of provability (actually I
prefer to call them "degees of support" or "dsp" for short).
The point is that dsp of X and dsp of NOT-X do not necessarily sum up
to 1 (non-additivity). In the case of total ignorance, for example, we
have dsp(X) = 0 and dsp(NOT-X) = 0. In general, we can define "degree
of ignorance" or "dig" of X as dig(X) = 1-dsp(X)-dsp(NOT-X). The
greater dig(X), the more should one try to postpone the decision and
try to gather more information. On the other hand, if dig(X)=0 or close
to 0, which implies dsp(X)+dsp(NOT-X)=1, one posseses a fully specified
probability measure which allows to make the decision on a solid basis.
It is important to realize that, independently of the amount of
available information, probabilities of provability are always defined,
that is also in cases of incomplete probabilty distributions. One
important example is the case of a Bayesian network with missing priors
or missing conditionals.
Let me illustrate this by a simple example. Suppose we know that:
1) A implies X
2) p(A) = 0.1
What can be said about X?
Well, one can certainly say, that A is a sufficient proof (I call it
"argument") for X, because whenever A is true, X is also true. Since
the probability of the argument A is 0.1, we can say that the
probability of provability of X is dsp(X)=0.1.
On the other hand, our knwoledge does not provide any proof for NOT-X.
This implies dsp(NOT-X)=0 and thus dig(X)=0.9. This tells us to
postpone the decision (if possible) until more information is available.
What would Bayesians do in such a case. They would start by saying
p(X|A)=1 and p(A)=0.1. So what is p(X)?
p(X) = p(X|A)p(A)+p(X|NOT-A)p(NOT-A) = 0.1 + p(X|NOT-A)*0.9.
Correct. But what is p(X|NOT-A)??? Bayesians tend then to assume
p(X|NOT-A)=0.5 and to compute p(X)=0.55. But by doing so, the
information about how ignorant one is gets lost, and there is nothing
left that tells us when to postpone the decision. I think this is
highly unsatisfactory (as Marco and Kathryn already pointed out).
On the other hand, i.e. by switching from probabilities of events to
probabilities of provability of events (note that the former is
included in the latter as a special case), one uses the available
information without making further assumptions. This preserves the
ignorance included in the available informaton. If one is forced to
come up with a decision, I suggest to use Smets' technique of
"pignistic transformations".
For more information about these ideas I refer to my papers about
"probabilistic argumenation" and "ignorance" on my web page
http://haenni.shorturl.com/
Best wishes,
Rolf Haenni
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Dr. Rolf Haenni
University of Konstanz
Center for Junior Research Fellows
D-78457 Konstanz, Germany
Phone:
- Office: (0049) 7531 88 4885
- Home: (0041) 71 670 1049
WWW: http://haenni.shorturl.com
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