KATHRYN B. LASKEY wrote about her experience with Bayesian networks:
> I get nonsensical answers when I try to build complex probability models,
> too. But when I get nonsensical answers, I go back to my model and look
> for where the answers are coming from, and I can usually understand why I
> got the answer I did. Then I either change the model (add, delete, reverse
> an arc, change some probabilities, add a hidden node...), or else I decide
> the apparently nonsensical answer wasn't so nonsensical at all.
I totally agree with that way of building models. It corresponds to my
experience on modeling with belief functions and probabilistic
argumentation systems.
>I don't know how to do that with belief functions. I understand in the
>formal sense why I'm getting the nonsensical answers I'm getting. I can
>reproduce the calculations. But I can't map the math onto a semantics that
>tells me how to fix my model or my intuition.
When I started working on belief functions, I also had my difficulties to
"map the math onto a semantics". However, my way of understanding belief
functions results from my work on probabilistic argumentation systems. This
theory is based on classical logic and probability theory, so everything is
well founded and well understood. The crucial point is to combine classical
logic and probability theory in an appropriate way. Only at the end, one
realizes that the result of this combination corresponds to the belief
function model. This is my (indirect) way of justifying the use of belief
functions.
>You're going to ask me for examples, and unfortunately I won't be able to
>help you. Most of my interesting examples are buried in the archives of a
>company that no longer exists.
I have already seen a number of "counter-intuitive" examples against the
use of belief functions, so don't worry about the "archives of a company
that no longer exists". Until now, I was always able to find the source of
the counter-intuition in a few minutes. In most cases, the problem is an
incomplete model hiding some implicitly given knowledge.
KEVIN S. VAN HORN asked:
> Rolf Haenni wrote:
> > The weakness of the Bayesian approach is that knowledge is forced to be
> > expressed by conditional and (necessarily) prior probabilities.
> And why is this a weakness?
In my last message I explained how a conditional probability P(R|Q) can be
expressed equivalently as a belief function. The restriction of the
Bayesian approach is due to the fact that in general, many different belief
functions are possible over the product space of two variables RxQ. In
contrast, a conditional probability forces the belief function to be of one
specific form.
Another weakness is the requirement of prior probabilities. For example,
let P(R|Q)=0.9 and P(R|¬Q)=0.8. In such a case, knowing that either Q or ¬Q
is true, belief functions already allow to deduce Bel(R)=0.72 and
Pl(R)=0.98, which makes perfectly sense in my eyes. Why is such a solution
suppressed in the Bayesian approach by requiring a prior probability P(Q)???
> > In this way, the case of total ignorance, for example, can not be
>represented
> > properly.
> Not so. One way of finding a probability distribution that represents total
> ignorance is through the use of group invariance arguments, and this has been
> known for at least 30 years. See the following papers by Jaynes: ...
I don't know these paper, thanks for the references. However, as much as I
know about Bayesian inference, the only way of epxressing total ignorance
about a statement Q is to say P(Q)=0.5. But knowing that P(Q)=0.5 is
clearly not the same as to know nothing about Q. Anyway, note that belief
functions allow to handle such cases properly.
Best wishes to everybody :-)
Rolf Haenni
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