Graphical models capture structural assumptions about a high dimensional probability distribution, and occur in the context of statistical thermodynamics, telecommunications (decoding) and artificial intelligence. The global probability is assumed to factorise as a product of local functions, conviently represented as nodes in a so-called factor graph. The first part of this talk will introduce the basic concepts of statistical physics and their application to Boltzmann machines. In particular, working at the level of energies (i.e. log-likelihoods) leads to a simpler and unifying picture, where the structural assumptions of the model are translated additively.

The second part of the talk will focus on the algebraic and topological structures involved. The space of parameters will be equipped with a differential structure, whose boundary operator (discrete analog of a divergence) allows to design new marginal estimation algorithms, analogous to diffusion equations. They generalise the traditional belief propagation (BP) algorithm and provide an alternative for contrastive divergence (CD) and Markov chain Monte Carlo (MCMC) methods.