Bayesian Network

definition

Bayesian network is one of the most effective theoretical models in the field of uncertain knowledge expression and reasoning. Bayesian network consists of nodes representing variables and directed edges connecting these nodes.

Nodes represent random variables, and directed edges between nodes represent the relationships between nodes. Conditional probability is used to express the strength of the relationship, and prior probability is used to express information for nodes without parent nodes.

Mathematical definition

Let G = (I,E) represent a directed acyclic graph (DAG), where I represents the set of all nodes in the graph, and E represents the set of directed connecting segments, and let X = (Xi)i∈I be the random variable represented by a node i in the directed acyclic graph. If the joint probability distribution of node X can be expressed as:

Then X is called a Bayesian network relative to a directed acyclic graph G, where pa(i)Represents the "cause" of node i.

For any random variable, its joint distribution can be obtained by multiplying their local conditional probability distributions:

According to the above formula, we can write the joint probability distribution of a Bayesian network as:

The difference between the above two expressions lies in the conditional probability part. In the Bayesian network, if the "dependent" variable is known, some nodes will be conditionally independent of their "dependent" variable. Only nodes related to the "dependent" variable will have conditional probability.

If the number of dependencies in the joint distribution is small, using the Bayesian function method can save a considerable amount of memory capacity. For example, if we want to store 10 variables whose values are all 0 or 1 in a conditional probability table, an intuitive idea is that we have to calculate a total of 2^10=1024 values; however, if none of the 10 variables has more than three dependent variables, then the conditional probability table of the Bayesian network only needs to calculate 10* 2^3=80 A value is enough.

feature

• The Bayesian network itself is an indeterminate causal association model;
• Bayesian networks have powerful capabilities for handling uncertain problems;
• Bayesian networks can effectively express and fuse multi-source information.