Bernal S D_2010.pdf - University of Plymouth

3..1. DEHSmOS AND MATHEM/mCAL FORMULATION
the inference problem defined by the network (Kschischang el al. 21)01).
In this thesis however, I have used Pearl's original belief propagation algorithm applied to
Bayesian networks (Pearl 1988). The rationale behind this choice is that, although factor graphs
can capture the same phenomena, Hayesian networks provide a more intuitive and explicit ac­
count of the causal relations between the variables, T believe this is crucial when modelling
hierarchical object recognition in Ihe visual system from the the generative model perspective.
3.3.3.2 Combination of evidence and belief calculation
The aim of behef propagation is to calculate the marginal probability of u variable X given
some evidence e. The inllucnce of evidence can propagate to ntxie X either Ihrough its parent
or ihrough its child nodes, thus evidence can be divided into two subsets such that.
c-e^UCx (3.13)
e^nex-0 (3.14)
where C;t represents the evidence oftovf node X, and 6;^ represents ihc evidence below node X.
This is shown in Figure 3.4. Similarly, in this section we will use the symbol e,;;^ to designate
the evidence above the link (I, -> X: while e[^r^ refers to the evidence below the link Uj —> X.
The probability of a node X given some evidence e, i.e. P(xle), is usually referred to as the
posterior probability, P'iX), or the belief. Bel{X). Taking into account all the above, as well a-s
Bayes rule (Equation (3.8)), we can write,
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