Factor graph - Citizendia

In mathematics, a factor graph is an $X, F$ -bipartite graph where $X=\{X_1,X_2,\dots,X_n\}$ is a set of variables and $F=\{f_1,f_2,\dots,f_m\}$ is a set of factors. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In the mathematical field of Graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two A factor $f j$ is a function mapping from a subset of variables $X_j\subseteq X$ to some range (such as the interval between 0 and 1). The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function This graph represents the factorisation

$g(\mathbf{x}) = \prod_{j=1}^m f_j(\mathbf{x_j}),$

where $\mathbf{x}$ is an assignment to all variables in $X$ and $\mathbf{x_j}$ is the assignment of $\mathbf{x}$ to all variables in $X j$ .

When using a factor graph to represent a probability distribution, each factor can be thought of as small distribution over a subset of the variables. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable The joint distribution is made up from the product of the individual distributions. In the study of Probability, given two Random variables X and Y, the joint distribution of X and Y is the distribution Factor graphs can be used to describe large distributions in which many pairs of variables are stochastically independent by explicitly listing only those groups of variables which are stochastically dependent. In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other

Inference over a factor graph can be done using a message passing algorithm such as belief propagation. Bayesian inference is Statistical inference in which evidence or observations are used to update or to newly infer the Probability that a hypothesis may be true In Computer science, message passing is a form of communication used in Parallel computing, Object-oriented programming, and Interprocess communication Belief propagation, also known as the sum-product algorithm, is an iterative Algorithm for computing marginals of functions on a Graphical model This is much more efficient than marginalization over a general distribution (which sums over every possible value of every variable, resulting in an exponential amount of summands), because the message passing approach exploits the locality properties of the factor graph. Conditional probability is the Probability of some event A, given the occurrence of some other event B.

Other probabilistic models such as Markov networks and Bayesian networks can be represented as factor graphs; the latter representation is frequently used when performing inference over such networks using belief propagation. A Markov network, or Markov random field, is a model of the (full Joint probability distribution of a set \mathcal{X} of Random variables A Bayesian network (or a belief network) is a Probabilistic graphical model that represents a set of Variables and their probabilistic independencies Belief propagation, also known as the sum-product algorithm, is an iterative Algorithm for computing marginals of functions on a Graphical model On the other hand, Bayesian networks are more naturally suited for generative models, as they can directly represent the causalities of the model. In Statistics, a generative model is a model for randomly generating observed data typically given some hidden parameters

External links

A tutorial-style dissertation by Volker Koch

References

Frey, Brendan J. Belief propagation, also known as the sum-product algorithm, is an iterative Algorithm for computing marginals of functions on a Graphical model Bayesian inference is Statistical inference in which evidence or observations are used to update or to newly infer the Probability that a hypothesis may be true Conditional probability is the Probability of some event A, given the occurrence of some other event B. A Markov network, or Markov random field, is a model of the (full Joint probability distribution of a set \mathcal{X} of Random variables A Bayesian network (or a belief network) is a Probabilistic graphical model that represents a set of Variables and their probabilistic independencies Frey (2003), “Extending Factor Graphs so as to Unify Directed and Undirected Graphical Models”, in jain, Nitin, UAI'03, Proceedings of the 19th Conference in Uncertainty in Artificial Intelligence, August 7–10, Acapulco, Mexico, Morgan Kaufmann, pp. 257–264
Kschischang, Frank R. ; Brendan J. Frey and Hans-Andrea Loeliger (2001). "Factor Graphs and the Sum-Product Algorithm". IEEE Transactions on Information Theory 47 (2): 498–519. doi:10.1109/18.910572. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.

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