Bayesian framework

Probabilistic modeling of the network

The searched network is a graph x as previously defined.

The field X is assumed to follow a prior probability P(X=x).

Modeling the data

• The map is a graph C with the same structure as X (same number of nodes, same connections between them).
• The image I : SPOT image

These data are assumed to be linked to the field X through the conditional probabilities :

P(C=c|X=x) and P(I=i|X=x)

Application of the Bayesian rule

• C and I are conditionnally independent w.r.t X

  
  

  P(c,i|x)=P(c|x)P(i|x)
  
  

• C and I are known, therefore

  
  

  
  
  

  => The expected $\hat{x}$ is the configuration of x maximizing (MAP criterion) :

  
  

  $$\hat{x} = \arg \max_x (P(c\vert x)P(i\vert x)P(x))$$