In probability theory and statistics, a graphical model (GM) represents dependencies among random variables by a graph in which each random variable is a node. Probability theory is the mathematical study of probability. ... Statistics is a type of data analysis whose practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... In probability theory, to say that two events are independent intuitively means that knowing whether or not one of them occurs makes it neither more probable nor less probable that the other occurs. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...

In the simplest case, the network structure of the model is a directed acyclic graph (DAG). Then the GM represents a factorization of the joint probability of all random variables. More precisely, if the events are A simple directed acyclic graph In mathematics, a directed acyclic graph, also called a dag or DAG, is a directed graph with no directed cycles; that is, for any vertex v, there is no directed path starting and ending on v. ... The word probability derives from the Latin probare (to prove, or to test). ...

X₁, ..., X_n,

then the joint probability

P(X₁, ..., X_n),

is equal to the product of the conditional probabilities This article defines some terms which characterize probability distributions of two or more variables. ...

P(X_i | parents of X_i) for i = 1,...,n.

In other words, the joint distribution factors into a product of conditional distributions. The graph structure indicates direct dependencies among random variables. Any two nodes that are not in a descendant/ancestor relationship are conditionally independent given the values of their parents. In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory, two events A and B are conditionally independent given a third event C precisely if the occurrence or non-occurrence of A and B are independent events in their conditional probability distribution given C. Two random variables X and Y are conditionally independent given an event C...

This type of graphical model is known as a directed graphical model, Bayesian network, or belief network. A Bayesian network or Bayesian belief network is a directed acyclic graph of nodes representing variables and arcs representing dependence relations among the variables. ...

There are also undirected graphical models, also called Markov networks, in which graph separation encodes conditional independencies (these are also known as graphical Gaussian models, or GGMs). A Markov network is an undirected graph of nodes representing variables and edges representing dependencies amongst these variables. ...

A recent application of graphical models is to describe gene regulatory networks. A gene regulatory network (also called a GRN or genetic regulatory network) is a collection of DNA segments in a cell which interact with each other and with other substances in the cell, thereby governing the rates at which genes in the network are transcribed into mRNA. // Overview Genes can...

See also belief propagation. Belief propagation is an iterative algorithm for computing marginals of functions on a graphical model most commonly used in artificial intelligence and information theory. ...