A '''Bayesian network''' is a formofprobabilistic [[graphical model]],alsoknownas'''Bayesianbelief network'''orjust'''belief network'''.
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A '''Bayesian network''' (or a '''beliefnetwork''')is a [[probabilistic graphical model]] thatrepresentsasetof [[variable]]s and their probabilistic independencies. For example, a Bayesian network couldrepresentthe 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:
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ABayesiannetworkcanberepresentedbya [[graph (mathematics)|graph]] (as in [[graph theory]]) with [[probabilities]] attached. Thus, a Bayesian network represents a set of [[variable]]s together with a [[joint probability distribution]]
#TherelianceonBayes'sconditioning as the basis for updating information.
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#The distinction between causal and evidential modes of reasoning, which underscores [[Thomas Bayes]]'s posthumous paper of 1763.<ref>{{Cite journal
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|author = Thomas Bayes
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|year = 1763
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|title = An Essay towards solving a Problem in the Doctrine of Chances. By the late Rev. Mr. Bayes, F.R.S., communicated by Mr. Price, in a letter to John Canton, A.M., F.R.S.
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|journal = Philosophical Transactions of the Royal Society of London
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|volume = 53
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|pages = 370–418
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}}</ref>
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==Definition==
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Formally,Bayesiannetworks are [[directed acyclic graph]]s 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 variable]]s, 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 recognition|speech signals]] or [[peptide sequence|protein sequences]]) are called [[dynamic Bayesian network]]s. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called [[influence diagrams]].
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ABayesiannetwork is a [[directed acyclic graph]] whose
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==Definitionsandconcepts==
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* nodes represent variables,
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* arcs represent statistical dependence relations among the variables and local probability distributions for each variable given values of its parents.
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Nodescanrepresentanykindofvariable, beit a measuredparameter, a [[latentvariable]]or a hypothesis.Theyarenotrestrictedtorepresenting [[randomvariable]]s; this is whatis"[[Bayesian]]"aboutaBayesiannetwork.
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Ifthereisan[[Graph (mathematics)|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''<sub>i</sub>isdenotedbyparents(''X''<sub>i</sub>).A [[directedacyclic graph]] is aBayesian Network relative to a set of variables if the [[joint distribution]] ofthenodevalues can be written as the product of the local distributions of each node and itsparents:
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Ifthere is an arc from node ''A'' to another node ''B'', then variable ''B'' depends directly on variable ''A'', and''A''is called a ''parent'' of ''B''. If for each variable ''X''<sub>i</sub>, ''i''=1 to ''n'',theset of parent variables is denoted by parents(''X''<sub>i</sub>) then the [[joint distribution]] of the variables is product of the local distributions
If ''X''<sub>i</sub> has no parents, its local probability distribution is said to be ''unconditional'', otherwise it is ''conditional''. If the variable representedby a node is ''observed'', then the node is said to be an ''evidence'' node.
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If node <math>X_i</math> has no parents, its local probability distribution is said to be ''unconditional'', otherwise it is ''conditional''. If the valueof a node is ''observed'', then the node is said to be an ''evidence'' node.
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Questionsabout incongruent dependence''among'' variables can be answered by studying the graph alone. It can be shown that ''[[conditional independence]]'' is represented in the graph by the graphical property of [[d-separation|''d''-separation]]: nodes ''X'' and ''Y'' are ''d''-separated in the graph, given specified evidence nodes, if and only if variables ''X'' and ''Y'' are independent given the corresponding evidence variables. The set of all other nodes on which node ''X'' can directly depend is given by ''X'''s [[Markov blanket]].
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===Independenciesand ''d''-separation===
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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 ''X''s [[Markov blanket]].
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OneadvantageofBayesiannetworks is that it is intuitivelyeasierfor a humantounderstanddirectdependencies and localdistributionsthancompletejointdistribution.
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Apathp(allowingpaths that are not directed) is saidtobe d-separated (or blocked) by a setofnodesZif and onlyifone ofthefollowingholds:
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# p contains a ''chain'' i -> m -> j such that the middle node m is in Z,
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# p contains a ''fork'' i <- m -> j such that the middle node m is in Z,
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# 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.
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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.
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Two nodes are (unconditionally) independent if the two nodes have no common ancestors (since this is equivalent to saying all paths between these nodes contain at least one ''collider'', which is equivalent to saying that the two nodes are d-separated by the empty set).
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===Causal Bayesian networks===
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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.
If there are two reasons which could cause grass to be wet, either the sprinkler is on or it's raining, then the situation can be modelled with adjacent Bayesian network. Here,all variables have two possible states T (for true) and F (for false).
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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).
Themodelcan 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 nuisancevariables:
wherethenamesofthevariables have been abbreviated to ''G = Grass wet'', ''S = Sprinkler'',and ''R =Rain''.
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==CausalBayesiannetworks==
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Themodelcananswerquestions 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 variable]]s:
AcausalBayesiannetwork is aBayesiannetwork where the directedarcsofthegraphareinterpretedasrepresenting[[causality|causalrelations]]insome real domain. Thedirectedarcsdonothave to be interpreted as representing causal relations; however in practiceknowledgeaboutcausal relations is very often used as a guide in drawing Bayesian network graphs, thus resulting incausalBayesiannetworks.
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Asin theexamplenumerator is pointedoutexplicitly, the jointprobabilityfunctionisusedtocalculateeachiterationofthesummationfunction. Inthe[[numerator]]marginalizingover<math>\mathit{S}</math>and in the[[denominator]]marginalizingover<math>\mathit{S}</math>and<math>\mathit{R}</math>.
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== Structurelearning ==
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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 <math>\mathrm P(S,R|do(G=T)) = P(S|R)P(R)</math>obtained by removing the factor <math>\mathrm P(G|S,R)</math> from the pre-intervention distribution. As expected, the likelihood of rain is unaffected by the action: <math>\mathrm P(R|do(G=T)) = P(R)</math>.
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In the simplest case, a Bayesian network isspecifiedbyanexpertandisthenusedtoperforminference. Inotherapplicationsthetask of defining the networkistoocomplexforhumans.Inthiscasethenetworkstructureandtheparametersof the local distributions mustbelearnedfromdata.
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Using a Bayesian network cansaveconsiderableamountsofmemory,ifthedependenciesin the joint distributionaresparse. Forexample,anaiveway of storing the conditionalprobabilitiesof10two-valuedvariablesasatablerequiresstoragespacefor<math>2^{10}=1024</math> 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<math>10*2^3=80</math>values.
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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.
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== Inference ==
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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.
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LearningthestructureofaBayesiannetwork(i.e., thegraph)isaveryimportantpartof[[machinelearning]].AssumingthatthedataisgeneratedfromaBayesiannetworkandthatallthevariablesarevisibleineveryiteration,optimization based search method can be usedtofindthestructureofthenetwork.Itrequiresa [[scoring function]] and a [[searchstrategy]].Acommonscoringfunctionis[[posteriorprobability]]of the structuregiventhetrainingdata.Thetimerequirementofan[[exhaustivesearch]]returningbackastructure that maximizes the score is superexponential in the numberof variables. Alocalsearchstrategymakesincrementalchangesaimed at improving the score of the structure. A global search algorithm like [[Markov chain Monte Carlo]] can avoid getting trapped in [[maximaand minima|local minima]].Friedmanet al.{{fact}} talk about using [[mutualinformation]] between variables and findingastructure that maximizes this. They do this by restricting the parent candidate set to ''k'' nodes and exhaustively searching therein.
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Themostcommonexactinferencemethodsare[[variable elimination]], whicheliminates(byintegrationorsummation)thenon-observednon-queryvariablesonebyonebydistributingthesumovertheproduct;[[cliquetreepropagation]],whichcachesthecomputationsothatmanyvariables can be queriedatonetimeandnewevidencecanbepropagatedquickly; and [[recursiveconditioning]],whichallowsforaspace-timetradeoffandmatches the efficiencyofvariableeliminationwhenenoughspaceisused.Allofthesemethodshavecomplexity that is exponential in the network's[[treewidth]]. Themostcommonapproximateinferencealgorithmsarestochastic [[Markov chain Monte Carlo|MCMC]] simulation, [[mini-bucketelimination]] whichgeneralizes [[loopybelief propagation]], and [[variationalBayes|variationalmethods]].
== Parameter learning ==
== Parameter learning ==
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In order to fully specify the Bayesian network and thus fully represent the joint probability distribution, it is necessary to further 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 [[information entropy|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.)
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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 [[normal distribution|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 [[information entropy|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.)
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.
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.
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== Inference ==
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== Structure learning ==
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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.
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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 subsetof variables when other variables (the ''evidence'' variables) are observed. This process of computing the ''posterior'' distribution of variables given evidence is calledprobabilisticinference. The posterior gives auniversal [[sufficientstatistic]] fordetectionapplications,whenone 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 mechanismfor automatically constructing extensions of [[Bayes' theorem]] to more complex problems.
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Learning the structure of a Bayesiannetwork (i.e., the graph) is achallengepursuedwithin [[machinelearning]].Thebasicideagoesback to a recoveryalgorithm
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developed by Rebane and Pearl (1987)<ref>Rebane, G. and Pearl, J., "The Recovery of Causal Poly-trees from Statistical Data," ''Proceedings, 3rd Workshop on Uncertainty in AI,'' (Seattle, WA) pp. 222-228,1987</ref> and rests
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on the distinction between the three possible types of
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adjacent triplets allowed in a directed acyclic graph (DAG):
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<ol>
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<li> <math>X \rightarrow Y \rightarrow Z</math>
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<li> <math>X \leftarrow Y \rightarrow Z</math>
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<li> <math>X \rightarrow Y \leftarrow Z</math>
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</ol>
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Type 1 and type 2 represent the same dependencies (i.e., <math>X</math> and <math>Z</math> are independent given <math>Y</math>) and are, therefore, indistinguishable. Type 3, however, can be uniquely identified, since <math>X</math> and <math>Z</math> 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 <math>X</math> and <math>Z</math> 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.<ref>{{Cite book
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| first = Judea
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| last = Pearl
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| authorlink = Judea Pearl
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| title = Causality: Models, Reasoning, and Inference
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| publisher = [[Cambridge University Press]]
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| year = 2000
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| isbn = 0-521-77362-8
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}}</ref><ref>P. Spirtes and C. Glymour, "An algorithm for fast recovery of sparse causal graphs", ''Social Science Computer Review,'' Vol. 9, pp. 62-72, 1991.
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</ref><ref>P. Spirtes, C. Glymour, and R. Scheines, ''Causation, Prediction, and Search,'' New York: Springer-Verlag, 1993
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</ref><ref>T. Verma and J. Pearl, "Equivalence and Synthesis of Causal Models," ''Proceedings of the Sixth Conference on Uncertainty in Artificial Intelligence,'' (July, Cambridge, MA), pp. 220-227, 1990. Reprinted in P. Bonissone, M. Henrion, L. N. Kanal and J. F. Lemmer (editors), ''Uncertainty in Artificial Intelligence 6,'' Amsterdam: Elsevier Science Publishers, B.V., pp. 225-268, 1991</ref>
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Themostcommonexactinferencemethodsare [[variableelimination]],whicheliminates(byintegrationorsummation)thenon-observednon-queryvariablesonebyonebydistributing the sumovertheproduct; [[cliquetree propagation]],whichcachesthecomputation so that manyvariablescanbequeriedatonetimeandnewevidencecanbepropagatedquickly;and[[recursiveconditioning]],whichallowsforaspace-timetradeoffandmatchestheefficiencyofvariableeliminationwhenenoughspaceisused.Allofthesemethodshavecomplexitythatisexponentialinthenetwork's [[treewidth]].Themostcommonapproximateinferencealgorithmsarestochastic[[Markov_chain_Monte_Carlo|MCMC]]simulation,[[mini-bucketelimination]]whichgeneralizes[[loopybeliefpropagation]], and [[variationalBayes|variationalmethods]].
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Analternativemethodofstructurallearninguses optimization based search. It requires a [[scoringfunction]] anda[[searchstrategy]].Acommonscoringfunctionis[[posteriorprobability]]ofthestructuregiven the trainingdata.The time requirement ofan [[exhaustivesearch]] returningbackastructure that maximizesthescoreis[[superexponential]]inthenumberofvariables.Alocalsearchstrategymakesincrementalchangesaimedatimprovingthescoreofthestructure.Aglobalsearchalgorithmlike[[MarkovchainMonteCarlo]]canavoidgettingtrappedin[[maximaandminima|localminima]].Friedmanetal.{{Fact|date=February2007}}talkabout using [[mutual information]] betweenvariablesandfindingastructurethatmaximizesthis.Theydothisbyrestrictingtheparentcandidate set to ''k'' nodes and exhaustivelysearchingtherein.
== Applications ==
== Applications ==
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Bayesian networks are used for [[mathematical model|modelling]] knowledge in [[bioinformatics]] ([[gene regulatory network]]s, [[protein structure]]), [[medicine]], [[document classification]], [[image processing]], [[data fusion]], [[decision support system]]s,{{Fact|date=October 2007}} [[engineering]]<ref name="davis">Davis (2003)</ref> and [[law]]<ref>Kadane & Schum (1996)</ref><ref name="davis"/>.
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Bayesiannetworksare used for [[mathematical model|modelling]] knowledge in [[gene regulatory network]]s, [[medicine]], [[engineering]], [[text analysis]], [[image processing]], [[data fusion]], and [[decision support system]]s.
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==History==
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Informal variants of such networks were first used by [[legal scholar]] [[John Henry Wigmore]], in the form of [[Wigmore chart]]s, to analyse [[trial (law)|trial]] [[evidence (law)|evidence]] in [[1913]].<ref>Kadane & Schum (1996) 66-76</ref> Another variant, called [[path analysis (statistics)|path diagrams]] was developed by the geneticist [[Sewall Wright]]<ref>Wright, S. (1921) "Correlation and Causation," ''Journal of Agricultural Research'', 20:557-585.</ref> and used in social and
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behavioral sciences (mostly with linear parametric models).
*[http://www.dynamics.unam.edu/DinamicaNoLineal3/bansy3.htm BANSY3] - Freeware. From the Non Linear Dynamics Laboratory. Mathematics Department, Science School, UNAM.
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*SamIam: http://reasoning.cs.ucla.edu/samiam
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*Ace, the Bayesian network compiler: http://reasoning.cs.ucla.edu/ace
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*BN4R: http://bn4r.rubyforge.org/
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*JavaBayes, Bayesian Networks in Java: http://www.pmr.poli.usp.br/ltd/Software/javabayes/
*Promedas (Bayesian medical decision support): http://www.promedas.nl
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*OpenBayes: http://www.openbayes.org
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*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/
* MSBNx: a component-centric toolkit for modeling and inference with Bayesian Network (from [[Microsoft]] Research): http://research.microsoft.com/adapt/MSBNx/
* Finn V. Jensen. Bayesian Networks and Decision Graphs. Springer, 2001.
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<div class="references-small">
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* I. Ben-Gal (2007), [http://www.eng.tau.ac.il/~bengal/BN.pdf Bayesian Networks], in F. Ruggeri, R. Kenett, and F. Faltin (editors), Encyclopedia of Statistics in Quality and Reliability, John Wiley & Sons.
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* Enrique Castillo, José Manuel Gutiérrez, and Ali S. Hadi (1997). ''Expert Systems and Probabilistic Network Models''. New York: [[Springer Science+Business Media|Springer-Verlag]]. ISBN 0-387-94858-9
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*{{cite journal | author=G. A. Davis | title=Bayesian reconstruction of traffic accidents | journal=Law, Probability and Risk | year=2003 | volume=2 | pages=69-89 }}
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* N. E. Fenton, and M. Neil, "Combining evidence in risk analysis using Bayesian Networks". https://www.dcs.qmul.ac.uk/~norman/papers/Combining%20evidence%20in%20risk%20analysis%20using%20BNs.pdf
*{{ cite book | author=J. B. Kadane and D. A. Schum | title=A Probabilistic Analysis of the Sacco and Vanzetti Evidence | location=New York | publisher=Wiley | id=ISBN 0-471-14182-8 | year=1996 }}
<!-- These links don't work! * Eugene Charniak. [http://www.cs.ubc.ca/~murphyk/Bayes/Charniak_91.pdf Bayesian Networks Without Tears]. This is a nice intro for non-specialists.
<!-- These links don't work! * Eugene Charniak. [http://www.cs.ubc.ca/~murphyk/Bayes/Charniak_91.pdf Bayesian Networks Without Tears]. This is a nice intro for non-specialists.
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* Kevin Murphy. An introduction to graphical models. 2001. http://www.ai.mit.edu/~murphyk/Papers/intro_gm.pdf -->
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* Kevin Murphy (2001). An introduction to graphical models. http://www.ai.mit.edu/~murphyk/Papers/intro_gm.pdf -->
* Judea Pearl (1985). "Bayesian Networks:AModelofSelf-ActivatedMemory for Evidential Reasoning". In Proceedings of the7<sup>th</sup>ConferenceoftheCognitiveScienceSociety, UniversityofCalifornia, Irvine, CA,pp. 329-334, August 15-17.
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* NeilM, Fenton N, Tailor M, "UsingBayesianNetworkstomodel Expected and Unexpected Operational Losses",RiskAnalysis: An International Journal, Vol25(4), 963-972, 2005. http://www.dcs.qmul.ac.uk/~norman/papers/oprisk.pdf
* EnriqueCastillo, José Manuel Gutiérrez, and Ali S. Hadi. ''Expert Systems and Probabilistic Network Models''. New York: [[Springer Science+Business Media|Springer-Verlag]], 1997. ISBN 0-387-94858-9
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* {{Citebook
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*FentonNEandNeil M, "Combining evidence in risk analysis using Bayesian Networks". https://www.dcs.qmul.ac.uk/~norman/papers/Combining%20evidence%20in%20risk%20analysis%20using%20BNs.pdf
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|author=JudeaPearl
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* Judea Pearl. Fusion, propagation, and structuring in belief networks. ''Artificial Intelligence'' '''29'''(3):241–288, 1986.
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| authorlink = Judea Pearl
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*JudeaPearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, 1988, ISBN 0-934613-73-7
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|title= Probabilistic Reasoning in Intelligent Systems
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*J.W.Comley and [http://www.csse.monash.edu.au/~dld D.L. Dowe], "[http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2005 Minimum Message Length, MDL and Generalised Bayesian Networks with Asymmetric Languages]", [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10478&mode=tocchapter 11] (pp[http://www.csse.monash.edu.au/~dld/Publications/2005/ComleyDowe2005MMLGeneralizedBayesianNetsAsymmetricLanguages_p265.jpg 265]–[http://www.csse.monash.edu.au/~dld/Publications/2005/ComleyDowe2005MMLGeneralizedBayesianNetsAsymmetricLanguages_p294.jpg294]) in P. Grunwald, M.A. Pitt and I.J. Myung (eds)., [http://mitpress.mit.edu/catalog/item/default.asp?sid=4C100C6F-2255-40FF-A2ED-02FC49FEBE7C&ttype=2&tid=10478 Advances in Minimum Description Length: Theory and Applications], Cambridge, MA: [[MIT Press]], April 2005, ISBN 0-262-07262-9. (This paper puts [[decision tree]]s in internal nodes of Bayes networks using [http://www.csse.monash.edu.au/~dld/MML.html Minimum Message Length] ([[Minimum message length|MML]]). An earlier version is [http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2003 Comley and Dowe (2003)], [http://www.csse.monash.edu.au/~dld/Publications/2003/Comley+Dowe03_HICS2003_GeneralBayesianNetworksAsymmetricLanguages.pdf .pdf].)
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|publisher = [[MorganKaufmann]]
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| year = 1988
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| isbn = 0-934613-73-7
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}}
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* {{Cite book
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| author = Judea Pearl
+
| authorlink = Judea Pearl
+
| title = Causality: Models, Reasoning, and Inference
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| publisher = [[Cambridge University Press]]
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| year = 2000
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| isbn = 0-521-77362-8
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}}
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* Judea Pearl and Stuart Russell. Bayesian Networks, in M. A. Arbib (editor), ''Handbook of Brain Theory and Neural Networks'', pp. 157–160, Cambridge, MA: [[MIT Press]], 2003, ISBN 0-262-01197-2.
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* J. W. Comley and [http://www.csse.monash.edu.au/~dld D. L. Dowe], "[http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2005 Minimum Message Length, MDL and Generalised Bayesian Networks with Asymmetric Languages]", [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10478&mode=toc chapter 11] (pp[http://www.csse.monash.edu.au/~dld/Publications/2005/ComleyDowe2005MMLGeneralizedBayesianNetsAsymmetricLanguages_p265.jpg 265]–[http://www.csse.monash.edu.au/~dld/Publications/2005/ComleyDowe2005MMLGeneralizedBayesianNetsAsymmetricLanguages_p294.jpg 294]) in P. Grunwald, M. A. Pitt and I. J. Myung (editors), [http://mitpress.mit.edu/catalog/item/default.asp?sid=4C100C6F-2255-40FF-A2ED-02FC49FEBE7C&ttype=2&tid=10478 Advances in Minimum Description Length: Theory and Applications], Cambridge, MA: [[MIT Press]], April 2005, ISBN 0-262-07262-9. (This paper puts [[decision tree]]s in internal nodes of Bayes networks using [http://www.csse.monash.edu.au/~dld/MML.html Minimum Message Length] ([[Minimum message length|MML]]). An earlier version is [http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2003 Comley and Dowe (2003)], [http://www.csse.monash.edu.au/~dld/Publications/2003/Comley+Dowe03_HICS2003_GeneralBayesianNetworksAsymmetricLanguages.pdf .pdf].)
* Christian Borgelt and Rudolf Kruse. [http://fuzzy.cs.uni-magdeburg.de/books/gm/ Graphical Models – Methods for Data Analysis and Mining], Chichester, UK: [[Wiley]], 2002, ISBN 0-470-84337-3
* Christian Borgelt and Rudolf Kruse. [http://fuzzy.cs.uni-magdeburg.de/books/gm/ Graphical Models – Methods for Data Analysis and Mining], Chichester, UK: [[Wiley]], 2002, ISBN 0-470-84337-3
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* [http://www.cs.ust.hk/faculty/lzhang/bio.htmlNevin Lianwen Zhang] and [http://www.cs.ubc.ca/spider/poole/ David Poole], [http://www.cs.ubc.ca/spider/poole/papers/canai94.pdf A simple approach to Bayesian network computations], Proceedings of the Tenth Biennial Canadian Artificial Intelligence Conference (AI-94), Banff, May 1994, 171-178. This paper presents variable elimination for belief networks.
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* {{Citebook
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*[http://research.microsoft.com/~heckerman/David Heckerman], [http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSR-TR-95-06ATutorial on Learning with Bayesian Networks]. In Learning in Graphical Models, M. Jordan, ed.. MIT Press, Cambridge, MA, 1999. Also appears as Technical Report MSR-TR-95-06, Microsoft Research, March, 1995. An earlier version appears as Bayesian Networks for Data Mining, Data Mining and Knowledge Discovery, 1:79-119, 1997. This paper is all about parameter and structure learning in Bayesian networks.
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|author = KevinB. Korb
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| coauthors = Ann E. Nicholson
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| url = http://www.csse.monash.edu.au/bai
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| title = Bayesian Artificial Intelligence
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| adress = Boca Raton, FL
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| publisher = [[CRC Press]]
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| year = 2004
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| isbn = 1-58488-387-1
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}}
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* [http://www.cs.ust.hk/faculty/lzhang/bio.html Nevin Lianwen Zhang] and [http://www.cs.ubc.ca/spider/poole/ David Poole], [http://www.cs.ubc.ca/spider/poole/papers/canai94.pdf A simple approach to Bayesian network computations], Proceedings of the Tenth Biennial Canadian Artificial Intelligence Conference (AI-94), Banff, AB, May 1994, 171-178. This paper presents variable elimination for belief networks.
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* [http://research.microsoft.com/~heckerman/ David Heckerman], [http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSR-TR-95-06 A Tutorial on Learning with Bayesian Networks]. In Learning in Graphical Models, M. Jordan, ed. MIT Press, Cambridge, MA, 1999. Also appears as Technical Report MSR-TR-95-06, Microsoft Research, March, 1995. An earlier version appears as Bayesian Networks for Data Mining, Data Mining and Knowledge Discovery, 1:79-119, 1997. The paper is about both parameter and structure learning in Bayesian networks.
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:
The often subjective nature of the input information.
The reliance on Bayes's conditioning as the basis for updating information.
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.
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 Xi is denoted by parents(Xi). 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:
If node 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.
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.
A path p (allowing paths that are not directed) is said to be d-separated (or blocked) by a set of nodes Z if and only if one of the following holds:
p contains a chain i -> m -> j such that the middle node m is in Z,
p contains a fork i <- m -> j such that the middle node m is in Z,
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.
Two nodes are (unconditionally) independent if the two nodes have no common ancestors (since this is equivalent to saying all paths between these nodes contain at least one collider, which is equivalent to saying that the two nodes are d-separated by the empty set).
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.
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).
The joint probability function is:
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:
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 and in the denominator marginalizing over and .
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 obtained by removing the factor from the pre-intervention distribution. As expected, the likelihood of rain is unaffected by the action: .
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 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 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.
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.
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 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.)
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.
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):
Type 1 and type 2 represent the same dependencies (i.e., and are independent given ) and are, therefore, indistinguishable. Type 3, however, can be uniquely identified, since and 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 and 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.[How to reference and link to summary or text] 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.
↑Thomas Bayes (1763). An Essay towards solving a Problem in the Doctrine of Chances. By the late Rev. Mr. Bayes, F.R.S., communicated by Mr. Price, in a letter to John Canton, A.M., F.R.S.. Philosophical Transactions of the Royal Society of London53: 370–418.
↑Rebane, G. and Pearl, J., "The Recovery of Causal Poly-trees from Statistical Data," Proceedings, 3rd Workshop on Uncertainty in AI, (Seattle, WA) pp. 222-228,1987
↑P. Spirtes and C. Glymour, "An algorithm for fast recovery of sparse causal graphs", Social Science Computer Review, Vol. 9, pp. 62-72, 1991.
↑P. Spirtes, C. Glymour, and R. Scheines, Causation, Prediction, and Search, New York: Springer-Verlag, 1993
↑T. Verma and J. Pearl, "Equivalence and Synthesis of Causal Models," Proceedings of the Sixth Conference on Uncertainty in Artificial Intelligence, (July, Cambridge, MA), pp. 220-227, 1990. Reprinted in P. Bonissone, M. Henrion, L. N. Kanal and J. F. Lemmer (editors), Uncertainty in Artificial Intelligence 6, Amsterdam: Elsevier Science Publishers, B.V., pp. 225-268, 1991
↑Wright, S. (1921) "Correlation and Causation," Journal of Agricultural Research, 20:557-585.
I. Ben-Gal (2007), Bayesian Networks, in F. Ruggeri, R. Kenett, and F. Faltin (editors), Encyclopedia of Statistics in Quality and Reliability, John Wiley & Sons.
Enrique Castillo, José Manuel Gutiérrez, and Ali S. Hadi (1997). Expert Systems and Probabilistic Network Models. New York: Springer-Verlag. ISBN 0-387-94858-9
G. A. Davis (2003). Bayesian reconstruction of traffic accidents. Law, Probability and Risk2: 69-89.
Finn V. Jensen (2001). Bayesian Networks and Decision Graphs, Springer.
J. B. Kadane and D. A. Schum (1996). A Probabilistic Analysis of the Sacco and Vanzetti Evidence, New York: Wiley. ISBN 0-471-14182-8.
M. Neil, N. E. Fenton, and M. Tailor (2005), "Using Bayesian Networks to model Expected and Unexpected Operational Losses", Risk Analysis: An International Journal, Vol. 25(4), pp. 963-972. http://www.dcs.qmul.ac.uk/~norman/papers/oprisk.pdf
Judea Pearl (1985). "Bayesian Networks: A Model of Self-Activated Memory for Evidential Reasoning". In Proceedings of the 7th Conference of the Cognitive Science Society, University of California, Irvine, CA, pp. 329-334, August 15-17.
Judea Pearl (1986), "Fusion, propagation, and structuring in belief networks". Artificial Intelligence29(3):241–288.
Judea Pearl and Stuart Russell. Bayesian Networks, in M. A. Arbib (editor), Handbook of Brain Theory and Neural Networks, pp. 157–160, Cambridge, MA: MIT Press, 2003, ISBN 0-262-01197-2.
David Heckerman, A Tutorial on Learning with Bayesian Networks. In Learning in Graphical Models, M. Jordan, ed. MIT Press, Cambridge, MA, 1999. Also appears as Technical Report MSR-TR-95-06, Microsoft Research, March, 1995. An earlier version appears as Bayesian Networks for Data Mining, Data Mining and Knowledge Discovery, 1:79-119, 1997. The paper is about both parameter and structure learning in Bayesian networks.
Edit
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.
Added by PhloxBot
and in the denominator marginalizing over 
.
obtained by removing the factor 
from the pre-intervention distribution. As expected, the likelihood of rain is unaffected by the action: 
.
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 
values.
and 
are independent given 
) and are, therefore, indistinguishable. Type 3, however, can be uniquely identified, since 
