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A Bayesian network (or a belief network) is a probabilistic graphical model that represents a set of variables and their probabilistic independencies. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases. The term "Bayesian networks" was coined by Pearl (1985) to emphasize three aspects: Image File history File links This is a lossless scalable vector image. ... 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, and the edges between the nodes represent conditional dependencies. ... In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ...

1. The often subjective nature of the input information
2. The reliance on Bayes's conditioning as the basis for updating information
3. The distinction between causal and evidential modes of reasoning, which underscores Thomas Bayes's posthumous paper of 1763.^[1]

Formally, Bayesian networks are directed acyclic graphs whose nodes represent variables, and whose arcs encode conditional independencies between the variables. Nodes can represent any kind of variable, be it a measured parameter, a latent variable or a hypothesis. They are not restricted to representing random variables, which represents another "Bayesian" aspect of a Bayesian network. Efficient algorithms exist that perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (such as for example speech signals or protein sequences) are called dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams. Thomas Bayes (c. ... A simple directed acyclic graph In computer science and 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 nonempty directed path that starts and ends on v. ... Latent variables, as opposed to observable variables, are those variables that cannot be directly observed but are rather inferred from other variables that can be observed and directly measured. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... Bayesian refers to probability and statistics -- either methods associated with the Reverend Thomas Bayes (ca. ... Inference is the act or process of deriving a conclusion based solely on what one already knows. ... Speech recognition (in many contexts also known as automatic speech recognition, computer speech recognition or erroneously as voice recognition) is the process of converting a speech signal to a sequence of words in the form of digital data, by means of an algorithm implemented as a computer program. ... Peptide sequence or amino acid sequence is the order in which amino acid residues, connected by peptide bonds, lie in the chain. ... A Bayesian network or Bayesian belief network is a directed acyclic graph of nodes representing variables and arcs representing dependence relations among the variables. ... In decision analysis, an influence diagram (ID) (also called a relevance diagram or decision network) is a graphical and mathematical representation of probabilistic inference and decision problems. ...

Definitions and concepts

If there is an arc from node A to another node B, A is called a parent of B, and B is a child of A. The set of parent nodes of a node X_i is denoted by parents(X_i). A directed acyclic graph is a Bayesian Network relative to a set of variables if the joint distribution of the node values can be written as the product of the local distributions of each node and its parents: This article presents the essential definitions. ...

If node X_i has no parents, its local probability distribution is said to be unconditional, otherwise it is conditional. If the value of a node is observed, then the node is said to be an evidence node.

Independencies and d-separation

The graph encodes independencies between variables. Conditional independence can be determined by the graphical property of d-separation. If two sets of nodes X and Y are d-separated in the graph by a third set Z, then the corresponding variable sets X and Y are independent given the variables in Z. The minimal set of nodes which d-separates node X from all other nodes is given by Xs Markov blanket. 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. In other words, Two random variables X and Y are conditionally independent given... In machine learning, the Markov blanket for a node in a Bayes net is the set of nodes composed by the s parents, its children, and its childrens parents. ...

A path p is said to be d-separated (or blocked) by a set of nodes Z if and only if all paths (regardless of direction) have one of the following properties:

1. p contains a chain i -> m -> j such that the middle node m is in Z,
2. p contains a fork i <- m -> j such that the middle node m is in Z,
3. p contains an inverted fork (or collider) i -> m <- j such that the middle node m is not in Z and no descendant of m is in Z.

A set Z is said to d-separate x from y in a directed acyclic graph G if all paths from x to y in G are d-separated by Z. The 'd' in d-separation stands for 'directional', since the behavior of a three node link on a path depends on the direction of the arrows in the link. A simple directed acyclic graph In computer science and 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 nonempty directed path that starts and ends on v. ...

Two nodes are (unconditionally) independent if the two nodes have no common ancestors - which is equivalent to saying that all paths between those nodes contain at least one collider and thus d-separate the nodes.

Causal Bayesian networks

A Bayesian network is a carrier of the conditional independencies of a set of variables, not of their causal connections. However, causal relations can be modelled by the closely related causal Bayesian network. The additional semantics of the causal Bayesian networks specify that if a node X is actively caused to be in a given state x (an operation written as do(x)), then the probability density function changes to the one of the network obtained by cutting the links from X's parents to X, and setting X to the caused value x (Pearl, 2000). Using this semantics, one can predict the impact of external interventions from data obtained prior to intervention.

Example

A simple Bayesian network.

Suppose that there are two reasons which could cause grass to be wet: either the sprinkler is on or it's raining. Also, suppose that the rain has a direct effect on the use of the sprinkler (namely that when it rains, the sprinkler is usually not turned on.) Then the situation can be modelled with the adjacent Bayesian network. All three variables have two possible values T (for true) and F (for false). Image File history File links SimpleBayesNet. ... Image File history File links SimpleBayesNet. ...

The joint probability function is:

P(G,S,R) = P(G | S,R)P(S | R)P(R)

where the names of the variables have been abbreviated to G = Grass wet, S = Sprinkler, and R = Rain.

The model can answer questions like "What is the likelihood that it is raining, given the grass is wet?" by using the conditional probability formula and summing over all nuisance variables: This article defines some terms which characterize probability distributions of two or more variables. ... In probability theory, a nuisance variable is a random variable which is fundamental to the probabilistic model, but which is of no particular interest in itself. ...

As in the example numerator is pointed out explicitly, the joint probability function is used to calculate each iteration of the summation function. In the numerator marginalizing over S and in the denominator marginalizing over S and R. In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ... A denominator is a name. ...

If, on the other hand, we wish to answer an interventional question: "What is the likelihood that it would rain, given that we wet the grass?" the answer would be governed by the post-intervention joint distribution function P(S,R | do(G = T)) = P(S | R)P(R) obtained by removing the factor P(G | S,R) from the pre-intervention distribution. As expected, the likelihood of rain is unaffected by the action: P(R | do(G = T)) = P(R).

Using a Bayesian network can save considerable amounts of memory, if the dependencies in the joint distribution are sparse. For example, a naive way of storing the conditional probabilities of 10 two-valued variables as a table requires storage space for 2¹⁰ = 1024 values. If the local distributions of no variable depends on more than 3 parent variables, the Bayesian network representation only needs to store at most 10 * 2³ = 80 values.

One advantage of Bayesian networks is that it is intuitively easier for a human to understand (a sparse set of) direct dependencies and local distributions than complete joint distribution.

Inference

Because a Bayesian network is a complete model for the variables and their relationships, it can be used to answer probabilistic queries about them. For example, the network can be used to find out updated knowledge of the state of a subset of variables when other variables (the evidence variables) are observed. This process of computing the posterior distribution of variables given evidence is called probabilistic inference. The posterior gives a universal sufficient statistic for detection applications, when one wants to choose values for the variable subset which minimize some expected loss function, for instance the probability of decision error. A Bayesian network can thus be considered a mechanism for automatically applying Bayes' theorem to complex problems. In statistics, one often considers a family of probability distributions for a random variable X (and X is often a vector whose components are scalar-valued random variables, frequently independent) parameterized by a scalar- or vector-valued parameter, which let us call &#952;. A quantity T(X) that depends on... Bayes theorem (also known as Bayes rule or Bayes law) is a result in probability theory, which relates the conditional and marginal probability distributions of random variables. ...

The most common exact inference methods are variable elimination, which eliminates (by integration or summation) the non-observed non-query variables one by one by distributing the sum over the product; clique tree propagation, which caches the computation so that many variables can be queried at one time and new evidence can be propagated quickly; and recursive conditioning, which allows for a space-time tradeoff and matches the efficiency of variable elimination when enough space is used. All of these methods have complexity that is exponential in the network's treewidth. The most common approximate inference algorithms are stochastic MCMC simulation, mini-bucket elimination which generalizes loopy belief propagation, and variational methods. In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to speed up solving certain problems on the original graph. ... Markov chain Monte Carlo (MCMC) methods (which include random walk Monte Carlo methods) are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its stationary distribution. ... Belief propagation, also known as the sum-product algorithm, is an iterative algorithm for computing marginals of functions on a graphical model most commonly used in artificial intelligence and information theory. ... In Bayesian network theory, the joint conditional posterior $P(h|v)$ is often intractable but the joint posterior $P(h,v)$ is tractable. ...

Parameter learning

In order to fully specify the Bayesian network and thus fully represent the joint probability distribution, it is necessary to specify for each node X the probability distribution for X conditional upon X's parents. The distribution of X conditional upon its parents may have any form. It is common to work with discrete or Gaussian Distributions since that simplifies calculations. Sometimes only constraints on a distribution are known; one can then use the principle of maximum entropy to determine a single distribution, the one with the greatest entropy given the constraints. (Analogously, in the specific context of a dynamic Bayesian network, one commonly specifies the conditional distribution for the hidden state's temporal evolution to maximize the entropy rate of the implied stochastic process.) The normal distribution, also called Gaussian distribution (named after Carl Friedrich Gauss, a German mathematician, although Gauss was not the first to work with it), is an extremely important probability distribution in many fields. ... The principle of maximum entropy is a method for analyzing the available information in order to determine a unique epistemic probability distribution. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... A Bayesian network or Bayesian belief network is a directed acyclic graph of nodes representing variables and arcs representing dependence relations among the variables. ... The entropy rate of a stochastic process is, informally, the time density of the average information in a stochastic process. ...

Often these conditional distributions include parameters which are unknown and must be estimated from data, sometimes using the maximum likelihood approach. Direct maximization of the likelihood (or of the posterior probability) is often complex when there are unobserved variables. A classical approach to this problem is the expectation-maximization algorithm which alternates computing expected values of the unobserved variables conditional on observed data, with maximizing the complete likelihood (or posterior) assuming that previously computed expected values are correct. Under mild regularity conditions this process converges on maximum likelihood (or maximum posterior) values for parameters. A more fully Bayesian approach to parameters is to treat parameters as additional unobserved variables and to compute a full posterior distribution over all nodes conditional upon observed data, then to integrate out the parameters. This approach can be expensive and lead to large dimension models, so in practise classical parameter-setting approaches are more common. Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ... The posterior probability can be calculated by Bayes theorem from the prior probability and the likelihood function. ... In statistical computing, an expectation-maximization (EM) algorithm is an algorithm for finding maximum likelihood estimates of parameters in probabilistic models, where the model depends on unobserved latent variables. ...

Structure learning

In the simplest case, a Bayesian network is specified by an expert and is then used to perform inference. In other applications the task of defining the network is too complex for humans. In this case the network structure and the parameters of the local distributions must be learned from data.

Learning the structure of a Bayesian network (i.e., the graph) is a challenge pursued within machine learning. The basic idea goes back to a recovery algorithm developed by Rebane and Pearl (1987)^[2] and rests on the distinction between the three possible types of adjacent triplets allowed in a directed acyclic graph (DAG): As a broad subfield of artificial intelligence, machine learning is concerned with the design and development of algorithms and techniques that allow computers to learn. At a general level, there are two types of learning: inductive, and deductive. ...

Type 1 and type 2 represent the same dependencies (i.e., X and Z are independent given Y) and are, therefore, indistinguishable. Type 3, however, can be uniquely identified, since X and Z are marginally independent and all other pairs are dependent. Thus, while the skeletons (the graphs stripped of arrows) of these three triplets are identical, the directionality of the arrows is partially identifiable. The same distinction applies when X and Z have common parents, except that one must first condition on those parents. Algorithms have been developed to systematically determine the skeleton of the underlying graph and, then, orient all arrows whose directionality is dictated by the conditional independencies observed.^[3][4][5][6]

An alternative method of structural learning uses optimization based search. It requires a scoring function and a search strategy. A common scoring function is posterior probability of the structure given the training data. The time requirement of an exhaustive search returning back a structure that maximizes the score is superexponential in the number of variables. A local search strategy makes incremental changes aimed at improving the score of the structure. A global search algorithm like Markov chain Monte Carlo can avoid getting trapped in local minima. Friedman et al.^{[citation needed]} talk about using mutual information between variables and finding a structure that maximizes this. They do this by restricting the parent candidate set to k nodes and exhaustively searching therein. The posterior probability can be calculated by Bayes theorem from the prior probability and the likelihood function. ... In computer science, brute-force search is a trivial but very general problem-solving technique, that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problems statement. ... Markov chain Monte Carlo (MCMC) methods (which include random walk Monte Carlo methods) are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its stationary distribution. ... Local and global maxima and minima for cos(3Ï€x)/x, 0. ... In probability theory and, in particular, information theory, the mutual information, or transinformation, of two random variables is a quantity that measures the mutual dependence of the two variables. ...

Applications

Bayesian networks are used for modelling knowledge in bioinformatics (gene regulatory networks, protein structure), medicine, document classification, image processing, data fusion, decision support systems,^{[citation needed]} engineering^[7] and law^[8][7]. A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... Map of the human X chromosome (from the NCBI website). ... 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 (indirectly through their RNA and protein expression products) and with other substances in the cell, thereby governing the rates at which genes in the... Proteins are an important class of biological macromolecules present in all biological organisms, made up of such elements as carbon, hydrogen, nitrogen, oxygen, and sulfur. ... For the chemical substances known as medicines, see medication. ... Document classification/categorization is a problem in information science. ... UPIICSA IPN - Binary image Image processing is any form of information processing for which the input is an image, such as photographs or frames of video; the output is not necessarily an image, but can be for instance a set of features of the image. ... Sensor fusion is the combining of sensory data or data derived from sensory data from disparate sources such that the resulting information is in some sense better than would be possible when these sources were used individually. ... Decision support systems are a class of computer-based information systems including knowledge based systems that support decision making activities. ... Engineering is the discipline of acquiring and applying knowledge of design, analysis, and/or construction of works for practical purposes. ... For other uses, see Law (disambiguation). ...

History

Informal variants of such networks were first used by legal scholar John Henry Wigmore, in the form of Wigmore charts, to analyse trial evidence in 1913.^[9] Another variant, called path diagrams was developed by the geneticist Sewall Wright^[10] and used in social and behavioral sciences (mostly with linear parametric models). A jurist is a professional who studies, develops, applies or otherwise deals with the law. ... John Henry Wigmore (1863 – 1943) was an American jurist and expert in the law of evidence. ... A Wigmore chart is a graphical method for the analysis of legal evidence in trials, developed by John Henry Wigmore. ... In legal parlance, a trial is an event in which parties to a dispute present information (in the form of evidence) in a formal setting, usually a court, before a judge, jury, or other designated finder of fact, in order to achieve a resolution to their dispute. ... The law of evidence governs the use of testimony (e. ... Year 1913 (MCMXIII) was a common year starting on Wednesday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Tuesday of the 13-day-slower Julian calendar). ... In statistics, path analysis is a type of multiple regression analysis. ... Sewall Green Wright ForMemRS (December 21, 1889 – March 3, 1988) was an American geneticist known for his influential work on evolutionary theory. ...

See also

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. ... Bayes theorem (also known as Bayes rule or Bayes law) is a result in probability theory, which relates the conditional and marginal probability distributions of random variables. ... Bayesian inference is statistical inference in which probabilities are interpreted not as frequencies or proportions or the like, but rather as degrees of belief. ... A second-order dependency tree representing the product below. ... 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. ... An influence diagram (ID) (also called a decision network) is a compact graphical and mathematical representation of a decision situation. ... As a broad subfield of artificial intelligence, machine learning is concerned with the design and development of algorithms and techniques that allow computers to learn. At a general level, there are two types of learning: inductive, and deductive. ... A polytree is a type of Bayesian network, in which the graph has at most one undirected path. ... Variable-order Bayesian network (VOBN) models provide an important extension of both the Bayesian network models and the variable-order Markov models. ... Structural equation modeling (SEM) is a statistical technique for building and testing statistical models, which are sometimes called causal models. ... A world view (or worldview) is a term calqued from the German word Weltanschauung ( ) Welt is the German word for world, and Anschauung is the German word for view or outlook. It is a concept fundamental to German philosophy and epistemology and refers to a wide world perception. ...

External links

• Association for Uncertainty in Artificial Intelligence: http://www.auai.org/
• Intro to Bayesian networks: http://www.niedermayer.ca/papers/bayesian/bayes.html
• On-line Tutorial on Bayesian nets and probability: http://www.dcs.qmw.ac.uk/%7Enorman/BBNs/BBNs.htm

Software

Free and open source software

• Kevin Murphy's Bayesian Network Toolbox for MatLab: http://bnt.sourceforge.net/
• GeNIe & SMILE: http://genie.sis.pitt.edu
• OpenBayes: http://www.openbayes.org
• RISO: http://sourceforge.net/projects/riso/ (distributed belief networks)
• BANSY3 - Freeware. From the Non Linear Dynamics Laboratory. Mathematics Department, Science School, UNAM.
• SamIam: http://reasoning.cs.ucla.edu/samiam
• Ace, the Bayesian network compiler: http://reasoning.cs.ucla.edu/ace
• BN4R: http://bn4r.rubyforge.org/

Commercial software

• AgenaRisk Bayesian network tool: http://www.agenarisk.com
• BayesBuilder: http://www.snn.ru.nl/nijmegen/index.php3?page=31
• Bayesian network application library: http://www.norsys.com/netlibrary/index.htm
• Bayesia: http://www.bayesia.com
• Hugin: http://www.hugin.com
• Netica: http://www.norsys.com
• BNet: http://www.cra.com/bnet
• Dezide: http://www.dezide.com
• Promedas (Bayesian medical decision support): http://www.promedas.nl
• ProBayes: http://www.probayes.com
• MSBNx: a component-centric toolkit for modeling and inference with Bayesian Network (from Microsoft Research): http://research.microsoft.com/adapt/MSBNx/
• Bayes Net Toolbox for Matlab: http://bnt.sourceforge.net/
• dVelox: http://www.aparasw.com/en/index.php?option=com_content&task=view&id=65&Itemid=102
• Causeway: http://www.inet.saic.com/

Microsoft Corporation, (NASDAQ: MSFT, HKSE: 4338) is a multinational computer technology corporation with global annual revenue of US$44. ...

References