A Markov Network a.k.a. Markov Random Field is an undirected Dependency Graph which is a minimal I-map of a probability distribution. As a minimal I-map, deleting any edge destroys its Markov property.
Unlike a Bayesian network, a Markov network is undirected and may contain cycles.
The Markov blanket of a variable is the set of other variables which can be conditioned upon to achieve independence with the rest of the graph. The Markov boundary is a minimal Markov blanket.
Also see WP:Markov random field
Construction
Can be constructed for any positive probability distribution which satisfies dependency graph properties.
- Edge deletion
- Start with a full graph
- Check each edge and see if removing it violates the I-mapness of the graph
- Determine whether fixing the values of all other variables in the graph renders these two independent
- Markov boundary
- For each variable, add edges until it has a sufficient Markov boundary
- [http://en.wikipedia.org/wiki/Moral_graph Moralization]
- Starting with a Bayesian Network
- Connect the parents of all convergent nodes
- Fill-In Method
- Starting with a Bayesian Network
- Use the fill-in method to create a Tree Decomposition
Deviant case: $x=y=z=t$ A minimal I-map for this case is any tree because each variable is independent of the rest only conditioned on a third. This is difficult to find with the previous construction methods because it isn't a well-behaved probability distribution.
Inference Using Join Trees
$p(a,b,c,d,e) = \frac{p(a,b,c)p(b,c,d)}{p(b,c)} \frac{p(c,e)}{p©}$
Can be calculated by multiplying the joint of each clique, and dividing by their common variables.
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International