Markov networks

Markov networks (Pearl, 1988), historically also known as Markov random fields, are undirected graphical models. A Markov network does not commit on whether $A$ caused $B$, it is interested only whether there is a dependency or not. A popular application is images where pixels are variables and edges are drawn between neighbouring pixels.

Since inference in Bayesian networks was explained by first transforming it into a Markov network, inference in Markov networks does not require much additional attention. We just start directly from the undirected graph, like in the top right subfigure of Figure 3.1. The joint density can be written directly as in Equation 3.2, but the standard way of writing the joint distribution is different. The joint distribution is proportional to the product of potentials $\psi$ over the cliques of the network, for example:

$$P(E,B,R,A,N_1,N_2) \propto \psi(R,E)\psi(E,B,A)\psi(A,N_1)\psi(A,N_2).$$ (3.12)

Like Bayesian networks, Markov networks can manage continuous values with same simplifying assumptions that can be relaxed by resorting to approximations. Hofmann and Tresp (1998) introduce nonlinear Markov networks. Each continuous valued variable $x_i$ is modelled using all of its neighbours $\mathcal{B}_i$ in the network. The modelled conditional densities $p^M(x_i\mid \mathcal{B}_i)$ can be directly used for Gibbs sampling. The complete likelihood function involves some integrals which cannot be solved in closed form but need to be approximated numerically. Taskar et al. (2002) introduce relational Markov networks (RMN) where the structure of the Markov network is defined by the relational data. Each variable might have a different number of neighbours, but generalisation is possible due to shared clique potentials. Section 6.3 and Publication VI extend these ideas into nonlinear relational Markov networks.