Start by going back an (re)reading:
\refrence {Shafer, G. [1982]} ``Belief Functions and Parametric
Models.'' {\it Journal of the Royal Statistical Society, Series B,\/}
{\bf 44}, 322-352.
This is the primary reference for belief function conditioning an
independence. It points out the fundamental problem:
Namely, Bel(B) and Bel(A|b) for all b in B is not sufficient to
uniquely determine Bel(A,B). [Most of the nonsense I've read about
belief functions independence starts with missing this. They then
usually goes on to confuse Smets' conditional embedding rule for
producing *a* joint distribution compatable with a set of margins with
Dempster's product-intersection rule for combination which it uses.]
I've conjectured that it is sufficient if you condition on all subsets
b of B, then you won't get any surprizes, but I have different
feelings about it on different days.
The second fundamental problem is characterizing indepedence.
Dempster's last word on the subject was that two marginal belief
functions are indepenent if you combine them using the
product-intersection rule and you get the proper joint distribution.
This is a little bit circular, but it is essentially the same as the
definition of independent probabilities.
Walley [1991] defines two characteristics: (1) there must be full
support (e.g., no logicial impliciation of one variable given the
other.) (2) the "random mechansims" used to generate the random
messages of the two marginal belief functions must be independent. He
rightly criticizes that last condition as being somewhat abstract.
[On the other hand, this is remarkably close to the "global" and
"local" independence assumption which are often made when making prior
distributions for Bayesian graphical model parameters. I know
precisely what kind of trouble these can cause with belief function
models, which makes me skeptical every time they are assumed as a
matter of course.]
I think most of these arguments are in my book.
\refrence {Almond, R.G. [1995]} {\it Graphical Belief Modeling\/}
Chapman and Hall.
Another important reference is:
\refrence {Kong, C.T.A. [1988]}``A Belief Function Generalization of Gibbs
Ensembles,'' Joint Technical Report,~S-122 Harvard University and
No.~239 University of Chicago, Departments of Statistics.
Augistine Kong here characterizes weak and strong depedence and gives
conditions for each. The critical result from this paper is that
Gibbs-Markov equivalence does not generally hold for belief function
models. In particular, factorization implies conditional indepedence,
but independence does not necessarily imply factorization. (This is
also true for probabilities, but in that case the additional condition
is merely that the distribution is positive everywhere.) I thought
that this paper was published in the Annals of Statistics, but the
technical report reference is the only one I can find. (I may be that
the revision was never completed.)
Shafer, Shenoy and colleagues put out a number of papers on graphical
belief function models in the late 80s early 90s, included the Shenoy
and Shafer general axiomization of the fusion and propagation
algorithm including both probabilities and belief functions in a
single framework. Most of those references can be found in my book,
along with a fairly extensive treatment on building graphical belief
function models.
--Russell Almond
Educational Testing Service
Research Statistics Group, 15-T
Princeton, NJ 08541
Phone: 609-734-1557 FAX: 609-734-5420
Email: --almond@acm.org, --ralmond@ets.org
http://www.stat.washington.edu/bayes/almond/almond.html
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