Graphical Model

Creator
Creator
Seonglae ChoSeonglae Cho
Created
Created
2024 Oct 22 23:56
Editor
Edited
Edited
2024 Nov 19 12:0

Probabilistic graphical model (PGM), Structured Probabilistic Model

Joint probability 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
Clique
s, and each factor is a non-negative function over the clique . Note that is called the partition function.
Graphical Model Notion
 
 
 
 
 
 
 
 

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