Seonglae ChoProbabilistic graphical model (PGM), Structured Probabilistic Model
Joint Distribution that uses a graph structure to encode conditional independence assumptions. The nodes are random variables, if they have an edge between them they are directly dependent, if they do not they are Conditionally Independent.
In a directed graph, if a node receives an arrow, it goes into the conditional part, and if it sends an arrow, it goes into the marginal part. Multiplication is done as many times as the number of nodes. When , and are predecessors of , while is not a parent of (only B is).
From factorized probability to Directed Acyclic Graph
Independent random variables are nodes without any previous input. Random variables which are conditional on others will have an arrow from those conditional variables to themselves.
where denotes the number of nodes in the graph.
From DAG to joint probability distribution
All of the variables have a joint probability, the nodes which do not have any arrows pointing to them are independent, the nodes which have arrows pointing to them are dependent of the nodes where the arrows originate.
Un-directed graphs
When the graph is un-directed, the probabilistic distribution is called a Markov Random Field. Un-directed graph is more general and indicates less information than directed graphs.
where denotes the set of Cliques, and each factor is a non-negative function over the clique . Note that is called the partition function.
Graphical Model Notion
