# [UAI] noisy OR for non-boolean variables

From: Jamie Sherrah (jamie.sherrah@safehouse.com.au)
Date: Wed Jan 16 2002 - 18:47:14 PST

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Hi,

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
noise,
and F(U) is a normalising constant to make the thing integrate over x to
unity.

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?

Jamie

- --
Dr. Jamie Sherrah
Safehouse International Ltd
Suite 1, Level 6, Como Centre
650 Chapel Street
South Yarra, Victoria, 3141
Australia
Telephone: +61 3 9827 5411
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Web: http://safehouse.com.au
Email: jamies@safehouse.com.au
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