I'm working on applying singly-connected bayesian networks to tracking
people in computer vision, and I've come up against a bit of a problem. I
was wondering if anybody has some advice.
I have an observed node X with N parent nodes U_1, U_2, ... U_N. The
conditional probabilities p(X | U) are described by a noisy OR relation:
p(X|U) = F(U) (1 - prod_i (1 - delta(X, U)));
delta(X,U) = 1 if X=U, 0 otherwise
X and the U's are variables in 1,2,...d denoting the position of the
object. Therefore the noisy OR rule gives a possibility for the presence of
X if at least one of the U's is present there also. The relation ignores
and F(U) is a normalising constant to make the thing integrate over x to
This is a generalisation of the noisy OR relation in Pearl's book "Prob.
reasoning in intelligent systems", but the difference is the variables here
are d-dimensional whereas in the book they are boolean (2 - dimensional).
The problem I'm experiencing is normalisation of p(X|U) such that
integral_x p(X|U) dx = 1. In general it depends on the values of U_i. To
further complicate things, I'm actually doing this with continuous domain
variables and I need the analytic integral. This works if p(X|U) can be
decomposed into the U_i contributions, but the normalising factor is still a
joint function of all the U_i's.
Does anybody have any experience with this, or seen any related work?
Thanks for your help,
Dr. Jamie Sherrah
Safehouse International Ltd
Suite 1, Level 6, Como Centre
650 Chapel Street
South Yarra, Victoria, 3141
Telephone: +61 3 9827 5411
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