Markov Networks

Markov Random Field

In a Undirected Graphical Model (undirected graph), meaning their connection between nodes are bidirectional, and can be interpreted as functions $\varphi$.

For example in this UGM: We can write the joint probability between all the variables in the graph can be factorized as following:

$$P(A,B,C,D,E)\propto \varphi (A,B)\varphi (B,C)\varphi (B,D)\varphi (C,E)\varphi (D,E)$$

and so we can compute the marginal probability of a single variable as:

$$P(X)=\frac{1}{Z}=\prod _{c\in \text{cliques}(G)}\varphi c(x_c)$$

Where a clique is just a subset of the nodes in the graph where each couple of nodes is adjacent (the subgraph is complete).

Markov Random Fields are a class of Undirected Graphical Models, which obey the Markov property:

• Any two subsets $S$ and $T$ of the graph are conditionally independent given a separating subset (subset for which there doesn’t exist a path between $S$ and $T$ that doesn’t travel trough it) From this property other properties (corollaries) can be derived:
• Any two non-adjacent variables are conditionally independent given all other variables.
• Any variable is conditionally independent of the other variables given its neighbours (this is another form of the Markov blanket).

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tags:#probability-theory -#graph-theory references:

• Undirected Graphical Models - YouTube