[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?

    Thanks for your help,

    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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