Probabilistic Graphical Model
Probabilistic Graphic Model (Probabilistic Graphic Model) can well dig out potential content.
The nodes in the probability graph are divided into hidden nodes and observation nodes, and the edges are divided into directed edges and undirected edges. From the perspective of probability theory, nodes correspond to random variables, and edges correspond to the dependence or correlation of random variables, in which directed edges represent unidirectional dependence, and undirected edges represent mutual dependence.
Probabilistic graph models are divided into two categories: Bayesian Network and Markov Network. Bayesian is represented by a directed graph structure, and Markov network is represented by an undirected graph network structure.
Probabilistic graph models include naive Bayes models, maximum entropy models, hidden Markov models, conditional random fields, topic models, and so on.
Bayesian joint probability distribution
Bayesian network on the left, Markov network on the right
Bayesian network and Markov network
It can be seen from the figure that, given A, B and C are conditionally independent, based on the definition of conditional conditional probability.
In the same way, given the conditions of B and C, A and D are conditionally independent, and we can get
The above two formulas can be joint probability
Markov joint probability distribution
In the Markov network, the definition of the joint probability distribution is as follows: 
where C is the set of the largest clique in the graph, and 
is the normalization factor to ensure that P(x) is the correctly defined probability, which is the same as the clique Q Corresponding potential function, the potential function is non-negative, and it should obtain a larger value on a variable with a larger probability, such as the exponential function, 
which is a subset 
of all nodes in the graph. If this subset, any two points If there are edges connected between them, all the nodes of this self form a clique. If any other nodes are added to this subset, they cannot form a clique. We call this subset a largest clique. Bayesian network on the left, Markov network on the right
Bayesian network and Markov network
Obviously, only (A,B), (A,C), (B,D), (C,D) constitute a clique, and it is the largest clique. The joint probability density can be expressed as
If you use the above exponential function as the potential function, then
Get
