The core idea is that, for Gibbs random fields and in particular for Ising models, when comparing several neighbourhood structures, the computation of the posterior probabilities of the models under competition can be operated by
likelihood-free simulation techniques (ABC). The turning point for this
resolution is that, due to the specific structure of Gibbs random field
distributions, there exists a sufficient statistic across models which allows
for an exact (rather than Approximate) simulation from the posterior
probabilities of the models. Obviously, when the structures grow more complex,
it becomes necessary to introduce a true ABC step with a tolerance threshold in
order to avoid running the algorithm for too long. Our toy example shows that
the accuracy of the approximation of the Bayes factor can be greatly improved by
resorting to the original ABC approach, since it allows for the inclusion of
many more simulations. In a biophysical application to the choice of a folding
structure for two proteins, we also demonstrate that we can implement the ABC
solution on realistic datasets and, in the examples processed there, that the
Bayes factors allow for a ranking more standard methods (FROST, TM-score) do
not.