Consider a Markov Random Field and the associated Gibbs distribution written in the following way :
The estimation of the partition function is performed from samples of the model. As we can not sample the model for each value of parameters, we introduce important sampling :
The partition function can thus be estimated for each parameter value A up to a multiplicative constant :
Define the log-likelihood in the following way :
From samples with parameter value B, we can estimate the log-likelihood and its partial derivatives given by :
To get accurate estimation in practice, we have to consider distributions with enough overlapping. So, A and B must be close enough.
We propose the following algorithm :
SYNOPSIS OF THE DEMO 