Bernal S D_2010.pdf - University of Plymouth

4.4. FEEDFORWARD PROCESSING
of this computation is often outside the typical numeric boundaries in simulation environments
(for Matlab these boundaries range from iO"-*^' to 10'-^'^'^). Iw this reason ii is necessary to
malte several approximations during the belief propagalion calculation:
• Given a node X with child nodes C],-- ,CM, the numlier of input A messages is re­
duced such (hat A(;c) ^ fl A(V(J;). where {jmax] C I..M. represents ihe indices of the
I'^max ^, W messages with highest variance, and Mma, < M. The maximum number of
input messages, M„„t, is calculated as a function of the number of states of the messages.
Kx, Matlab's maximum real value, R„,a, — 10'^^^, and Ihe minimium value allowed in
probabilily distributions, Vm,n, as follows:
M^--—^ (4.10)
Thus, the likelihood function of each node is obtained by multiplying only the Mmax
input A messages with higher variance, where !/„„, is set to ensure that the result of
the computation never reaches Matlab's numeric upperbound. Probability distributions
with higher variance are chosen as they are likely to carry more infonnation. In the
majority of cases Mmnx > ^' so the resulting computation is equivalent to the original
belief propagation formulation.
To check how well this sampling procedure managed lo approximate Ihe exact likelihood
functions the method was tested siatislically. Using randomly generated A messages
from a normal distribution, the difference between the exact likelihood and the approx­
imated likelihood distribution obtained after sampling was measured, for different val­
ues of M^io^. The difference was measured using Ihe Ku 11 back-Lei bier (K-L) divergence
which calculates the cross-correlation between an approximate distribution and the true
distribution. This method cannol be considered a distance measure, as it is not symmet­
ric, but has been used extensively to measure the goodness of fil between two discrete
probabilily distributions (Trislon and Kiebel 2009, Winn and Bishop 2005, Hinton el al.
2006).
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'mm