> Does anyone know of any algorithms developed for performing
> multiplication, division, and marginalization of multiple dimensioned
> probability potentials or more general multiple dimensioned structures?
>
Anders,
As we have not witnessed a flow of massively parallel neurocomputers
in the market, despite of the hype back in the 80's, the following may
be of no relevance with respect to your question, but maybe somebody
following this thread may still find this interesting:
In my dissertation I presented a method for mapping a given Bayesian
network to a massively parallel Boltzmann machine architecture so that
the updating process of the Boltzmann machine converges almost surely
to a state which can be mapped back to a MAP state of the Bayesian
network. In a sense, the Boltzmann machine updating process
corresponds to an iterative simulated annealing process on the
Bayesian network structure where all the unclamped variables
are updated simultaneously.
More precisely, the nodes of the neural network can be divided into
two separate clusters, where no two nodes in a cluster are connected
to each other. The first cluster contains as many nodes as there are
variables in the Bayesian network, and the number of nodes in the
second cluster is equal to the number of parameters (entries in the
conditional probability tables) of the Bayesian network. All the nodes
in one cluster can update their state simultaneously. This means that
if suitable (massively parallel) hardware is available, the
computational complexity of one iteration step of the process does not
depend on the size or the structure of the Bayesian network (but the
number of iterations required of course does).
Unfortunately I have not been able to find a way to use this framework
for estimating conditional probabilities of individual unclamped
variables - for technical reasons the idea can only be applied for
finding MAP states.
The thesis, together with a couple of related short conference papers,
can be found through my WWW-page. I am also at the moment revising a
journal paper on this subject for Journal of Applied Intelligence. The
paper will be published most likely in the Spring.
--Petri
Petri Myllymaki, Ph.D., Research Scientist, Department of Computer Science
P.O.Box 26, FIN-00014 University of Helsinki, Finland
Petri.Myllymaki@cs.Helsinki.FI, http://www.cs.Helsinki.FI/~myllymak/
+358 9 708 44441 (fax), +358 9 708 44212 (tel), +358 40 553 1162 (GSM)