Generally, probabilistic graphical models use a graph-based representation as the foundation for encoding a distribution over a multi-dimensional space and a graph that is a compact or factorized representation of a set of independences that hold in the specific distribution. Two branches of graphical representations of distributions are commonly used, namely, Bayesian networks and Markov random fields. Both families encompass the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce.[1]
The undirected graph shown may have one of several interpretations; the common feature is that the presence of an edge implies some sort of dependence between the corresponding random variables. From this graph we might deduce that <math>B,C,D</math> are all mutually independent, once <math>A</math> is known, or (equivalently in this case) that
If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are <math>X_1,\ldots,X_n</math> then the joint probability satisfies
where <math>\text{pa}(X_i)</math> is the set of parents of node <math>X_i</math> (nodes with edges directed towards <math>X_i</math>). In other words, the joint distribution factors into a product of conditional distributions. For example, in the directed acyclic graph shown in the Figure this factorization would be
Any two nodes are conditionally independent given the values of their parents. In general, any two sets of nodes are conditionally independent given a third set if a criterion called d-separation holds in the graph. Local independences and global independences are equivalent in Bayesian networks.
An example of a directed graphical model.An example of a directed, cyclic graphical model. Each arrow indicates a dependency. In this example: D depends on A, B, and C; and C depends on B and D; whereas A and B are each independent.
The next figure depicts a graphical model with a cycle. This may be interpreted in terms of each variable 'depending' on the values of its parents in some manner.
The particular graph shown suggests a joint probability density that factors as
Targeted Bayesian network learning (TBNL) Файл:Tbnl corral.jpgTBNL model for "corral dataset"
A factor graph is an undirected bipartite graph connecting variables and factors. Each factor represents a function over the variables it is connected to. This is a helpful representation for understanding and implementing belief propagation.
A chain graph is a graph which may have both directed and undirected edges, but without any directed cycles (i.e. if we start at any vertex and move along the graph respecting the directions of any arrows, we cannot return to the vertex we started from if we have passed an arrow). Both directed acyclic graphs and undirected graphs are special cases of chain graphs, which can therefore provide a way of unifying and generalizing Bayesian and Markov networks.[3]
An ancestral graph is a further extension, having directed, bidirected and undirected edges.[4]