I followed the discussion about Cox axioms with interest. Kathryn B. Laskey
wrote in her response to Kevin S. Van Horn's message:
> "The reason I am uncomfortable with belief functions is that I am not
> convinced that the way beliefs are updated with new evidence (Dempster
> conditioning, or its special case Dempster's Rule) is justifiable."
I don't understand why the justification of Dempster's rule is still a
problem for may proponents of the Bayesian approach. From my point of view,
the important point to realize is that the Bayesian approach is nothing
more than a special case of the Belief Function approach. For people not
being aware of this, note that a conditional probability P(r|q) = x1, for
example, can always be represented as a mass function m_1 of the form
m_1({qr,段r,段毒}) = x1 and m_1({q毒,段r,段毒}) = 1 - x1.
Clearly, the corresponding conditional probability P(r|段) = x2 can be
represented similarly by
m_2({段r,qr,q毒}) = x2 and m_2({段毒,qr,q毒}) = 1 - x2.
Furthermore, prior probabilities P(q) = x0 can be represented easily by
m_0({q}) = x0 and m_0({段}) = 1 - x0.
Combining the mass functions m_0, m_1, and m_2 by Dempster's rule
corresponds then to the multiplication of the potentials P(R|Q) and P(Q).
In this way, every Baysian network can be represented and treated
equivalently as a collection of corresponding belief functions. Therefore,
criticizing Dempster's rule is automatically a criticism of Bayesian
inference.
In my eyes, the main strenght of working with Bayesian networks (compared
to belief functions) is its fast computations. However, this is not
surprising, since restricted special cases are mostly more efficient. The
weakness of the Bayesian approach is that knowledge is forced to be
expressed by conditional and (necessarily) prior probabilities. In this
way, the case of total ignorance, for example, can not be represented
properly. Another important problem is the restriction to directed acyclic
graphs. Regarding at these restrictions makes it hardly understandable, why
research in the domain of uncertain reasoning is dominated by the Bayesian
approach.
To get a better understanding of how probability theory (on which the
Bayesian approach is based) is linked with the belief function approach, I
recommend my technical report about "Probabilistic Argumentation Systems".
This theory integrates quantitative and qualitative approaches to reasoning
under uncertainty. It includes belief function theory as well as Baysian
inference as special cases. The general idea is that the (quantitative)
belief of an event is always obtained through (qualitative) arguments
supporting the event. An important strength of this approach is that
inference can be accelerated considerably by computing only the most
relevant arguments. Note that the inference techniques are mainly based on
logical deduction rather than on Dempster's rule. Interested persons can
downloaded the paper from
http://www2-iiuf.unifr.ch/tcs/publications/article/HKL99.htm.
Best wishes,
Rolf Haenni
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* Dr. Rolf Haenni __/ __/ __/ __/ _______/ *
* Institute of Informatics (IIUF) __/ __/ __/ __/ __/ *
* University of Fribourg, Switzerland __/ __/ __/ __/ _____/ *
* Phone: ++41 26 300 83 31 __/ __/ __/ __/ __/ *
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