Bayesian network

The Bayesian network is a network probabilistic, graphical model that describes dependencies.

Application

Bayesian networks have applications in machine learning tasks that deal with dependent features.

An example is Part-of-speech, where words are grammatically classified in a string. This involves a complex network of dependencies between object, subject, verbs, nouns, etc. that can be modeled and optimized with a Bayesian network.

Properties

DAG

Bayesian networks are directed, acyclic graphs.

Each edge identifies a causal relation, usually temporal: something that happened in the future cannot cause something to happen in the past. As such, the graph is acyclic.

Each node is a conditional probability: Given that the parents happen, what is the probability of the node event happening.

Joint probability

Bayesian networks are primarily used to calculate the joint probability of an event given its dependencies. This can be done with the following formula

$P(x_{1},x_{2},\ldots ,x_{n})=\prod _{i=1}^{n}P(x_{i}|Parents(x_{i}))$

By relaxing some assumptions, Naiive Bayes reduces the computational complexity.