Let's consider a set of random variables, x_i, organized in an undirected graph, G=(V, E), as shown in the following diagram: 

Two random variables, a and b, are conditionally independent given the random variable, c if:

Now, consider the graph again; if all generic couples of subsets of variables S_i and S_j are conditionally independent given a separating subset, S_k (so that all connections between variables belonging to S_i to variables belonging to S_j pass through S_k), the graph is called a Markov random field (MRF). 

Given G=(V, E), a subset containing vertices such that every couple is adjacent is called a clique (the set of all cliques is often denoted as cl(G)). For example, consider the graph shown previously; (x₀, x₁) is a clique and if x₀ and x₅ were connected, (x₀, x₁, x₅) would be a clique. A maximal clique is a clique that cannot be expanded by adding new vertices. A particular family of MRF is made up of all those graphs whose joint probability distribution can be factorized as:

In this case, α is the normalizing constant and the product is extended to the set of all maximal cliques. According to the Hammersley–Clifford theorem (for further information, please refer to Proof of Hammersley-Clifford Theorem, Cheung S., University of Kentucky, 2008), if the joint probability density function is strictly positive, the MRF can be factorized and all the ρ_i functions are strictly positive too. Hence p(x), after some straightforward manipulations based on the properties of logarithms, can be rewritten as a Gibbs (or Boltzmann) distribution:

The term E(x) is called energy, as it derives from the first application of such a distribution in statistical physics. 1/Z is now the normalizing constant employing the standard notation. In our scenarios, we always consider graphs containing observed (x_i) and latent variables (h_j). Therefore, it's useful to express the joint probability as:

Whenever it's necessary to marginalize to obtain p(x), we can simply sum over h_j: