Download Graphical Model www.AssignmentPoint.com A Graphical Model or

Graphical Model
www.AssignmentPoint.com
www.AssignmentPoint.com
A Graphical Model or probabilistic graphical model (PGM) is a probabilistic model for
which a graph expresses the conditional dependence structure between random variables.
They are commonly used in probability theory, statistics—particularly Bayesian statistics—
and machine learning.
Types of graphical models
Generally, probabilistic graphical models use a graph-based representation as the foundation
for encoding a complete distribution over a multi-dimensional space and a graph that is a
compact or factorized representation of a set of independences that hold in the specific
distribution. Two branches of graphical representations of distributions are commonly used,
namely, Bayesian networks and Markov networks. Both families encompass the properties of
factorization and independences, but they differ in the set of independences they can encode
and the factorization of the distribution that they induce.
Bayesian network
If the network structure of the model is a directed acyclic graph, the model represents a
factorization of the joint probability of all random variables. More precisely, if the events are
then the joint probability satisfies
where
is the set of parents of node Xi. In other words, the joint distribution factors into a
product of conditional distributions. For example, the graphical model in the Figure shown
above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the
random variables A, B, C, D with a joint probability density that factors as
Any two nodes are conditionally independent given the values of their parents. In general,
any two sets of nodes are conditionally independent given a third set if a criterion called d-
www.AssignmentPoint.com
separation holds in the graph. Local independences and global independences are equivalent
in Bayesian networks.
This type of graphical model is known as a directed graphical model, Bayesian network, or
belief network. Classic machine learning models like hidden Markov models, neural
networks and newer models such as variable-order Markov models can be considered special
cases of Bayesian networks.
Markov random field
A Markov random field, also known as a Markov network, is a model over an undirected
graph. A graphical model with many repeated subunits can be represented with plate
notation.
Other types
A factor graph is an undirected bipartite graph connecting variables and factors. Each factor
represents a function over the variables it is connected to. This is a helpful representation for
understanding and implementing belief propagation.
A clique tree or junction tree is a tree of cliques, used in the junction tree algorithm.
A chain graph is a graph which may have both directed and undirected edges, but without any
directed cycles (i.e. if we start at any vertex and move along the graph respecting the
directions of any arrows, we cannot return to the vertex we started from if we have passed an
arrow). Both directed acyclic graphs and undirected graphs are special cases of chain graphs,
which can therefore provide a way of unifying and generalizing Bayesian and Markov
networks.

An ancestral graph is a further extension, having directed, bidirected and undirected
edges.

A conditional random field is a discriminative model specified over an undirected
graph.
www.AssignmentPoint.com

A restricted Boltzmann machine is a bipartite generative model specified over an
undirected graph.
Applications
The framework of the models, which provides algorithms for discovering and analyzing
structure in complex distributions to describe them succinctly and extract the unstructured
information, allows them to be constructed and utilized effectively. Applications of graphical
models include information extraction, speech recognition, computer vision, decoding of
low-density parity-check codes, modeling of gene regulatory networks, gene finding and
diagnosis of diseases, and graphical models for protein structure.
www.AssignmentPoint.com