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AsimpleBayesiannetwork.Raininfluenceswhetherthesprinklerisactivated,andbothrainandthesprinklerinfluencewhetherthegrassiswet.

BayesiannetworkFromWikipedia,thefreeencyclopedia

ABayesiannetwork,Bayesnetwork,beliefnetwork,Bayes(ian)modelorprobabilisticdirectedacyclicgraphicalmodelisaprobabilisticgraphicalmodel(atypeofstatisticalmodel)thatrepresentsasetofrandomvariablesandtheirconditionaldependenciesviaadirectedacyclicgraph(DAG).Forexample,aBayesiannetworkcouldrepresenttheprobabilisticrelationshipsbetweendiseasesandsymptoms.Givensymptoms,thenetworkcanbeusedtocomputetheprobabilitiesofthepresenceofvariousdiseases.

Formally,BayesiannetworksareDAGswhosenodesrepresentrandomvariablesintheBayesiansense:theymaybeobservablequantities,latentvariables,unknownparametersorhypotheses.Edgesrepresentconditionaldependenciesnodesthatarenotconnectedrepresentvariablesthatareconditionallyindependentofeachother.Eachnodeisassociatedwithaprobabilityfunctionthattakes,asinput,aparticularsetofvaluesforthenode'sparentvariables,andgives(asoutput)theprobability(orprobabilitydistribution,ifapplicable)ofthevariablerepresentedbythenode.Forexample,if parentnodesrepresent Booleanvariablesthentheprobabilityfunctioncouldberepresentedbyatableof entries,oneentryforeachofthe possiblecombinationsofitsparentsbeingtrueorfalse.Similarideasmaybeappliedtoundirected,andpossiblycyclic,graphssucharecalledMarkovnetworks.

EfficientalgorithmsexistthatperforminferenceandlearninginBayesiannetworks.Bayesiannetworksthatmodelsequencesofvariables(e.g.speechsignalsorproteinsequences)arecalleddynamicBayesiannetworks.GeneralizationsofBayesiannetworksthatcanrepresentandsolvedecisionproblemsunderuncertaintyarecalledinfluencediagrams.

Contents

1Example2Inferenceandlearning

2.1Inferringunobservedvariables2.2Parameterlearning2.3Structurelearning

3Statisticalintroduction3.1Introductoryexamples3.2Restrictionsonpriors

4Definitionsandconcepts4.1Factorizationdefinition4.2LocalMarkovproperty4.3DevelopingBayesiannetworks4.4Markovblanket

4.4.1dseparation4.5Hierarchicalmodels4.6Causalnetworks

5Applications

5.1Software
http://en.wikipedia.org/wiki/Probability_functionhttp://en.wikipedia.org/wiki/Peptide_sequencehttp://en.wikipedia.org/wiki/Conditional_independencehttp://en.wikipedia.org/wiki/Machine_learninghttp://en.wikipedia.org/wiki/Dynamic_Bayesian_networkhttp://en.wikipedia.org/wiki/Latent_variablehttp://en.wikipedia.org/wiki/Boolean_data_typehttp://en.wikipedia.org/wiki/Directed_acyclic_graphhttp://en.wikipedia.org/wiki/Inferencehttp://en.wikipedia.org/wiki/Random_variableshttp://en.wikipedia.org/wiki/Graphical_modelhttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/File:SimpleBayesNetNodes.svghttp://en.wikipedia.org/wiki/Markov_networkhttp://en.wikipedia.org/wiki/Influence_diagramshttp://en.wikipedia.org/wiki/Conditional_independencehttp://en.wikipedia.org/wiki/Glossary_of_graph_theory#Directed_acyclic_graphshttp://en.wikipedia.org/wiki/Speech_recognitionhttp://en.wikipedia.org/wiki/Bayesian_probability



AsimpleBayesiannetworkwithconditionalprobabilitytables

5.1Software6History7Seealso8Notes9References10Furtherreading11Externallinks

Example

Supposethattherearetwoeventswhichcouldcausegrasstobewet:eitherthesprinklerisonorit'sraining.Also,supposethattherainhasadirecteffectontheuseofthesprinkler(namelythatwhenitrains,thesprinklerisusuallynotturnedon).ThenthesituationcanbemodeledwithaBayesiannetwork(shown).Allthreevariableshavetwopossiblevalues,T(fortrue)andF(forfalse).

Thejointprobabilityfunctionis:

wherethenamesofthevariableshavebeenabbreviatedtoG=Grasswet(yes/no),S=Sprinklerturnedon(yes/no),andR=Raining(yes/no).

Themodelcananswerquestionslike"Whatistheprobabilitythatitisraining,giventhegrassiswet?"byusingtheconditionalprobabilityformulaandsummingoverallnuisancevariables:

Usingtheexpansionforthejointprobabilityfunction andtheconditionalprobabilitiesfromtheconditionalprobabilitytables(CPTs)statedinthediagram,onecanevaluateeachterminthesumsinthenumeratoranddenominator.Forexample,

Thenthenumericalresults(subscriptedbytheassociatedvariablevalues)are
http://en.wikipedia.org/wiki/Conditional_probability_tablehttp://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/wiki/Conditional_probability_tablehttp://en.wikipedia.org/wiki/File:SimpleBayesNet.svghttp://en.wikipedia.org/wiki/Nuisance_variablehttp://en.wikipedia.org/wiki/Joint_probability_distribution



If,ontheotherhand,wewishtoansweraninterventionalquestion:"Whatisthelikelihoodthatitwouldrain,giventhatwewetthegrass?"theanswerwouldbegovernedbythepostinterventionjointdistributionfunction obtainedbyremovingthefactor

fromthepreinterventiondistribution.Asexpected,thelikelihoodofrainisunaffectedbytheaction: .

If,moreover,wewishtopredicttheimpactofturningthesprinkleron,wehave

withtheterm removed,showingthattheactionhasaneffectonthegrassbutnotontherain.

Thesepredictionsmaynotbefeasiblewhensomeofthevariablesareunobserved,asinmostpolicyevaluationproblems.Theeffectoftheaction canstillbepredicted,however,wheneveracriterioncalled"backdoor"issatisfied.[1][2]Itstatesthat,ifasetZofnodescanbeobservedthatdseparates[3](orblocks)allbackdoorpathsfromXtoYthen

.AbackdoorpathisonethatendswithanarrowintoX.Setsthatsatisfythebackdoorcriterionarecalled"sufficient"or"admissible."Forexample,thesetZ=RisadmissibleforpredictingtheeffectofS=TonG,becauseRdseparatethe(only)backdoorpathSRG.However,ifSisnotobserved,thereisnoothersetthatdseparatesthispathandtheeffectofturningthesprinkleron(S=T)onthegrass(G)cannotbepredictedfrompassiveobservations.WethensaythatP(G|do(S=T))isnot"identified."Thisreflectsthefactthat,lackinginterventionaldata,wecannotdetermineiftheobserveddependencebetweenSandGisduetoacausalconnectionorisspurious(apparentdependencearisingfromacommoncause,R).(seeSimpson'sparadox)

TodeterminewhetheracausalrelationisidentifiedfromanarbitraryBayesiannetworkwithunobservedvariables,onecanusethethreerulesof"docalculus"[1][4]andtestwhetheralldotermscanberemovedfromtheexpressionofthatrelation,thusconfirmingthatthedesiredquantityisestimablefromfrequencydata.[5]

UsingaBayesiannetworkcansaveconsiderableamountsofmemory,ifthedependenciesinthejointdistributionaresparse.Forexample,anaivewayofstoringtheconditionalprobabilitiesof10twovaluedvariablesasatablerequiresstoragespacefor values.Ifthelocaldistributionsofnovariabledependsonmorethan3parentvariables,theBayesiannetworkrepresentationonlyneedstostoreatmost values.

OneadvantageofBayesiannetworksisthatitisintuitivelyeasierforahumantounderstand(asparsesetof)directdependenciesandlocaldistributionsthancompletejointdistributions.

Inferenceandlearning

TherearethreemaininferencetasksforBayesiannetworks.
http://en.wikipedia.org/wiki/Simpson%27s_paradox



Inferringunobservedvariables

BecauseaBayesiannetworkisacompletemodelforthevariablesandtheirrelationships,itcanbeusedtoanswerprobabilisticqueriesaboutthem.Forexample,thenetworkcanbeusedtofindoutupdatedknowledgeofthestateofasubsetofvariableswhenothervariables(theevidencevariables)areobserved.Thisprocessofcomputingtheposteriordistributionofvariablesgivenevidenceiscalledprobabilisticinference.Theposteriorgivesauniversalsufficientstatisticfordetectionapplications,whenonewantstochoosevaluesforthevariablesubsetwhichminimizesomeexpectedlossfunction,forinstancetheprobabilityofdecisionerror.ABayesiannetworkcanthusbeconsideredamechanismforautomaticallyapplyingBayes'theoremtocomplexproblems.

Themostcommonexactinferencemethodsare:variableelimination,whicheliminates(byintegrationorsummation)thenonobservednonqueryvariablesonebyonebydistributingthesumovertheproductcliquetreepropagation,whichcachesthecomputationsothatmanyvariablescanbequeriedatonetimeandnewevidencecanbepropagatedquicklyandrecursiveconditioningandAND/ORsearch,whichallowforaspacetimetradeoffandmatchtheefficiencyofvariableeliminationwhenenoughspaceisused.Allofthesemethodshavecomplexitythatisexponentialinthenetwork'streewidth.Themostcommonapproximateinferencealgorithmsareimportancesampling,stochasticMCMCsimulation,minibucketelimination,loopybeliefpropagation,generalizedbeliefpropagation,andvariationalmethods.

Parameterlearning

InordertofullyspecifytheBayesiannetworkandthusfullyrepresentthejointprobabilitydistribution,itisnecessarytospecifyforeachnodeXtheprobabilitydistributionforXconditionaluponX'sparents.ThedistributionofXconditionaluponitsparentsmayhaveanyform.ItiscommontoworkwithdiscreteorGaussiandistributionssincethatsimplifiescalculations.Sometimesonlyconstraintsonadistributionareknownonecanthenusetheprincipleofmaximumentropytodetermineasingledistribution,theonewiththegreatestentropygiventheconstraints.(Analogously,inthespecificcontextofadynamicBayesiannetwork,onecommonlyspecifiestheconditionaldistributionforthehiddenstate'stemporalevolutiontomaximizetheentropyrateoftheimpliedstochasticprocess.)

Oftentheseconditionaldistributionsincludeparameterswhichareunknownandmustbeestimatedfromdata,sometimesusingthemaximumlikelihoodapproach.Directmaximizationofthelikelihood(oroftheposteriorprobability)isoftencomplexwhenthereareunobservedvariables.Aclassicalapproachtothisproblemistheexpectationmaximizationalgorithmwhichalternatescomputingexpectedvaluesoftheunobservedvariablesconditionalonobserveddata,withmaximizingthecompletelikelihood(orposterior)assumingthatpreviouslycomputedexpectedvaluesarecorrect.Undermildregularityconditionsthisprocessconvergesonmaximumlikelihood(ormaximumposterior)valuesforparameters.

AmorefullyBayesianapproachtoparametersistotreatparametersasadditionalunobservedvariablesandtocomputeafullposteriordistributionoverallnodesconditionaluponobserveddata,thentointegrateouttheparameters.Thisapproachcanbeexpensiveandleadtolargedimensionmodels,soinpracticeclassicalparametersettingapproachesaremorecommon.

Structurelearning

Inthesimplestcase,aBayesiannetworkisspecifiedbyanexpertandisthenusedtoperforminference.Inotherapplicationsthetaskofdefiningthenetworkistoocomplexforhumans.Inthiscasethenetworkstructureandtheparametersofthelocaldistributionsmustbelearnedfromdata.
http://en.wikipedia.org/w/index.php?title=Recursive_conditioning&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Clique_tree_propagation&action=edit&redlink=1http://en.wikipedia.org/wiki/Loopy_belief_propagationhttp://en.wikipedia.org/wiki/Joint_probability_distributionhttp://en.wikipedia.org/wiki/Treewidthhttp://en.wikipedia.org/wiki/Markov_chain_Monte_Carlohttp://en.wikipedia.org/wiki/Approximate_inferencehttp://en.wikipedia.org/wiki/Posterior_probabilityhttp://en.wikipedia.org/wiki/Space-time_tradeoffhttp://en.wikipedia.org/wiki/Dynamic_Bayesian_networkhttp://en.wikipedia.org/wiki/Expectation-maximization_algorithmhttp://en.wikipedia.org/w/index.php?title=AND/OR_search&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Mini-bucket_elimination&action=edit&redlink=1http://en.wikipedia.org/wiki/Maximum_likelihoodhttp://en.wikipedia.org/wiki/Sufficient_statistichttp://en.wikipedia.org/wiki/Bayes%27_theoremhttp://en.wikipedia.org/wiki/Principle_of_maximum_entropyhttp://en.wikipedia.org/wiki/Entropy_ratehttp://en.wikipedia.org/wiki/Variational_Bayeshttp://en.wikipedia.org/wiki/Generalized_belief_propagationhttp://en.wikipedia.org/wiki/Importance_samplinghttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Information_entropyhttp://en.wikipedia.org/wiki/Variable_elimination



AutomaticallylearningthegraphstructureofaBayesiannetworkisachallengepursuedwithinmachinelearning.ThebasicideagoesbacktoarecoveryalgorithmdevelopedbyRebaneandPearl(1987)[6]andrestsonthedistinctionbetweenthethreepossibletypesofadjacenttripletsallowedinadirectedacyclicgraph(DAG):

1.2.3.

Type1andtype2representthesamedependencies( and areindependentgiven )andare,therefore,indistinguishable.Type3,however,canbeuniquelyidentified,since and aremarginallyindependentandallotherpairsaredependent.Thus,whiletheskeletons(thegraphsstrippedofarrows)ofthesethreetripletsareidentical,thedirectionalityofthearrowsispartiallyidentifiable.Thesamedistinctionapplieswhen and havecommonparents,exceptthatonemustfirstconditiononthoseparents.Algorithmshavebeendevelopedtosystematicallydeterminetheskeletonoftheunderlyinggraphand,then,orientallarrowswhosedirectionalityisdictatedbytheconditionalindependenciesobserved.[1][7][8][9]

Analternativemethodofstructurallearningusesoptimizationbasedsearch.Itrequiresascoringfunctionandasearchstrategy.Acommonscoringfunctionisposteriorprobabilityofthestructuregiventhetrainingdata.Thetimerequirementofanexhaustivesearchreturningastructurethatmaximizesthescoreissuperexponentialinthenumberofvariables.Alocalsearchstrategymakesincrementalchangesaimedatimprovingthescoreofthestructure.AglobalsearchalgorithmlikeMarkovchainMonteCarlocanavoidgettingtrappedinlocalminima.Friedmanetal.[10][11]discussusingmutualinformationbetweenvariablesandfindingastructurethatmaximizesthis.Theydothisbyrestrictingtheparentcandidatesettoknodesandexhaustivelysearchingtherein.

Anothermethodconsistsoffocusingonthesubclassofdecomposablemodels,forwhichtheMLEhaveaclosedform.Itisthenpossibletodiscoveraconsistentstructureforhundredsofvariables.[12]

ABayesiannetworkcanbeaugmentedwithnodesandedgesusingrulebasedmachinelearningtechniques.Inductivelogicprogrammingcanbeusedtominerulesandcreatenewnodes.[13]Statisticalrelationallearning(SRL)approachesuseascoringfunctionbasedontheBayesnetworkstructuretoguidethestructuralsearchandaugmentthenetwork.[14]AcommonSRLscoringfunctionistheareaundertheROCcurve.

Statisticalintroduction

Givendata andparameter ,asimpleBayesiananalysisstartswithapriorprobability(prior)andlikelihood tocomputeaposteriorprobability .

Oftentheprioron dependsinturnonotherparameters thatarenotmentionedinthelikelihood.So,theprior mustbereplacedbyalikelihood ,andaprior onthenewlyintroducedparameters isrequired,resultinginaposteriorprobability

ThisisthesimplestexampleofahierarchicalBayesmodel.
http://en.wikipedia.org/wiki/Statistical_relational_learninghttp://en.wikipedia.org/wiki/Posterior_probabilityhttp://en.wikipedia.org/wiki/Likelihood_functionhttp://en.wikipedia.org/wiki/Scoring_functionhttp://en.wikipedia.org/wiki/Exhaustive_searchhttp://en.wikipedia.org/wiki/Tetrationhttp://en.wikipedia.org/wiki/Prior_probabilityhttp://en.wikipedia.org/wiki/Inductive_logic_programminghttp://en.wikipedia.org/wiki/Mutual_informationhttp://en.wikipedia.org/wiki/ROC_curvehttp://en.wikipedia.org/w/index.php?title=Search_strategy&action=edit&redlink=1http://en.wikipedia.org/wiki/Maximum_likelihood_estimatehttp://en.wikipedia.org/wiki/Markov_chain_Monte_Carlohttp://en.wikipedia.org/wiki/Scoring_functionhttp://en.wikipedia.org/wiki/Bayesian_statisticshttp://en.wikipedia.org/wiki/Machine_learninghttp://en.wikipedia.org/wiki/Maxima_and_minimahttp://en.wikipedia.org/wiki/Machine_learninghttp://en.wikipedia.org/wiki/Posterior_probability



Theprocessmayberepeatedforexample,theparameters maydependinturnonadditionalparameters ,whichwillrequiretheirownprior.Eventuallytheprocessmustterminate,withpriorsthatdonotdependonanyotherunmentionedparameters.

Introductoryexamples

Supposewehavemeasuredthequantities eachwithnormallydistributederrorsofknownstandarddeviation ,

Supposeweareinterestedinestimatingthe .Anapproachwouldbetoestimatethe usingamaximumlikelihoodapproachsincetheobservationsareindependent,thelikelihoodfactorizesandthemaximumlikelihoodestimateissimply

However,ifthequantitiesarerelated,sothatforexamplewemaythinkthattheindividual havethemselvesbeendrawnfromanunderlyingdistribution,thenthisrelationshipdestroystheindependenceandsuggestsamorecomplexmodel,e.g.,

withimproperpriors flat, flat .When ,thisisanidentifiedmodel(i.e.thereexistsauniquesolutionforthemodel'sparameters),andtheposteriordistributionsoftheindividualwilltendtomove,orshrinkawayfromthemaximumlikelihoodestimatestowardstheircommonmean.ThisshrinkageisatypicalbehaviorinhierarchicalBayesmodels.

Restrictionsonpriors

Somecareisneededwhenchoosingpriorsinahierarchicalmodel,particularlyonscalevariablesathigherlevelsofthehierarchysuchasthevariable intheexample.TheusualpriorssuchastheJeffreysprioroftendonotwork,becausetheposteriordistributionwillbeimproper(notnormalizable),andestimatesmadebyminimizingtheexpectedlosswillbeinadmissible.

Definitionsandconcepts

ThereareseveralequivalentdefinitionsofaBayesiannetwork.Forallthefollowing,letG=(V,E)beadirectedacyclicgraph(orDAG),andletX=(Xv)vVbeasetofrandomvariablesindexedbyV.

Factorizationdefinition

XisaBayesiannetworkwithrespecttoGifitsjointprobabilitydensityfunction(withrespecttoaproductmeasure)canbewrittenasaproductoftheindividualdensityfunctions,conditionalontheirparentvariables:[15]
http://en.wikipedia.org/wiki/Directed_acyclic_graphhttp://en.wikipedia.org/wiki/Probability_density_functionhttp://en.wikipedia.org/w/index.php?title=Identified_model&action=edit&redlink=1http://en.wikipedia.org/wiki/Maximum_likelihoodhttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Loss_function#Expected_losshttp://en.wikipedia.org/wiki/Shrinkage_estimatorhttp://en.wikipedia.org/wiki/Admissible_decision_rulehttp://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Jeffreys_priorhttp://en.wikipedia.org/wiki/Improper_priorhttp://en.wikipedia.org/wiki/Product_measure



wherepa(v)isthesetofparentsofv(i.e.thoseverticespointingdirectlytovviaasingleedge).

Foranysetofrandomvariables,theprobabilityofanymemberofajointdistributioncanbecalculatedfromconditionalprobabilitiesusingthechainrule(givenatopologicalorderingofX)asfollows:[15]

Comparethiswiththedefinitionabove,whichcanbewrittenas:

foreach whichisaparentof

Thedifferencebetweenthetwoexpressionsistheconditionalindependenceofthevariablesfromanyoftheirnondescendants,giventhevaluesoftheirparentvariables.

LocalMarkovproperty

XisaBayesiannetworkwithrespecttoGifitsatisfiesthelocalMarkovproperty:eachvariableisconditionallyindependentofitsnondescendantsgivenitsparentvariables:[16]

wherede(v)isthesetofdescendantsandV\de(v)isthesetofnondescendantsofv.

Thiscanalsobeexpressedintermssimilartothefirstdefinition,as

foreach whichisnotadescendantofforeach whichisaparentof

Notethatthesetofparentsisasubsetofthesetofnondescendantsbecausethegraphisacyclic.

DevelopingBayesiannetworks

TodevelopaBayesiannetwork,weoftenfirstdevelopaDAGGsuchthatwebelieveXsatisfiesthelocalMarkovpropertywithrespecttoG.SometimesthisisdonebycreatingacausalDAG.WethenascertaintheconditionalprobabilitydistributionsofeachvariablegivenitsparentsinG.Inmanycases,inparticularinthecasewherethevariablesarediscrete,ifwedefinethejointdistributionofXtobetheproductoftheseconditionaldistributions,thenXisaBayesiannetworkwithrespecttoG.[17]

Markovblanket

TheMarkovblanketofanodeisthesetofnodesconsistingofitsparents,itschildren,andanyotherparentsofitschildren.ThissetrendersitindependentoftherestofthenetworkthejointdistributionofthevariablesintheMarkovblanketofanodeissufficientknowledgeforcalculatingthedistributionofthenode.XisaBayesiannetworkwithrespecttoGifeverynodeisconditionallyindependentofallothernodesinthenetwork,givenitsMarkovblanket.[16]
http://en.wikipedia.org/wiki/Conditional_independencehttp://en.wikipedia.org/wiki/Conditional_independencehttp://en.wikipedia.org/wiki/Chain_rule_(probability)http://en.wikipedia.org/wiki/Markov_blankethttp://en.wikipedia.org/wiki/Joint_distributionhttp://en.wikipedia.org/wiki/Cycle_(graph_theory)http://en.wikipedia.org/wiki/Topological_orderinghttp://en.wikipedia.org/wiki/Markov_blanket



dseparation

Thisdefinitioncanbemademoregeneralbydefiningthe"d"separationoftwonodes,wheredstandsfordirectional.[18][19]LetPbeatrail(thatis,acollectionofedgeswhichislikeapath,buteachofwhoseedgesmayhaveanydirection)fromnodeutov.ThenPissaidtobedseparatedbyasetofnodesZifandonlyif(atleast)oneofthefollowingholds:

1. Pcontainsachain,umv,suchthatthemiddlenodemisinZ,2. Pcontainsafork,umv,suchthatthemiddlenodemisinZ,or3. Pcontainsaninvertedfork(orcollider),umv,suchthatthemiddlenodemisnotinZand

nodescendantofmisinZ.

ThusuandvaresaidtobedseparatedbyZifalltrailsbetweenthemaredseparated.Ifuandvarenotdseparated,theyarecalleddconnected.

XisaBayesiannetworkwithrespecttoGif,foranytwonodesu,v:

whereZisasetwhichdseparatesuandv.(TheMarkovblanketistheminimalsetofnodeswhichdseparatesnodevfromallothernodes.)

Hierarchicalmodels

ThetermhierarchicalmodelissometimesconsideredaparticulartypeofBayesiannetwork,buthasnoformaldefinition.Sometimesthetermisreservedformodelswiththreeormorelevelsofrandomvariablesothertimes,itisreservedformodelswithlatentvariables.Ingeneral,however,anymoderatelycomplexBayesiannetworkisusuallytermed"hierarchical".

Causalnetworks

AlthoughBayesiannetworksareoftenusedtorepresentcausalrelationships,thisneednotbethecase:adirectededgefromutovdoesnotrequirethatXviscausallydependentonXu.ThisisdemonstratedbythefactthatBayesiannetworksonthegraphs:

areequivalent:thatistheyimposeexactlythesameconditionalindependencerequirements.

AcausalnetworkisaBayesiannetworkwithanexplicitrequirementthattherelationshipsbecausal.TheadditionalsemanticsofthecausalnetworksspecifythatifanodeXisactivelycausedtobeinagivenstatex(anactionwrittenasdo(X=x)),thentheprobabilitydensityfunctionchangestotheoneofthenetworkobtainedbycuttingthelinksfromtheparentsofXtoX,andsettingXtothecausedvaluex.[1]Usingthesesemantics,onecanpredicttheimpactofexternalinterventionsfromdataobtainedpriortointervention.

Applications

Bayesiannetworksareusedformodellingbeliefsincomputationalbiologyandbioinformatics(generegulatorynetworks,proteinstructure,geneexpressionanalysis,[20]learningepistasisfromGWASdatasets[21])medicine,[22]biomonitoring,[23]documentclassification,informationretrieval,[24]semantic
http://en.wikipedia.org/wiki/Markov_blankethttp://en.wikipedia.org/wiki/Protein_structurehttp://en.wikipedia.org/wiki/Medicinehttp://en.wikipedia.org/wiki/Information_retrievalhttp://en.wikipedia.org/wiki/Computational_biologyhttp://en.wikipedia.org/wiki/Bioinformaticshttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Gene_regulatory_networkhttp://en.wikipedia.org/wiki/Causalityhttp://en.wikipedia.org/wiki/Latent_variablehttp://en.wikipedia.org/wiki/Biomonitoringhttp://en.wikipedia.org/wiki/Document_classificationhttp://en.wikipedia.org/wiki/Gene_expressionhttp://en.wikipedia.org/wiki/Semantic_search



search,[25]imageprocessing,datafusion,decisionsupportsystems,[26]engineering,sportsbetting,[27][28]

gaming,law,[29][30][31]studydesign[32]andriskanalysis.[33][34][35]TherearetextsapplyingBayesiannetworkstobioinformatics[36]andfinancialandmarketinginformatics.[37]

Software

WinBUGSOpenBUGS(website(http://www.openbugs.net/w/FrontPage)),further(opensource)developmentofWinBUGS.OpenMarkov(http://www.openmarkov.org/),opensourcesoftwareandAPIimplementedinJavaGraphicalModelsToolkit(http://melodi.ee.washington.edu/gmtk)(GMTK)GMTKisanopensource,publiclyavailabletoolkitforrapidlyprototypingstatisticalmodelsusingdynamicgraphicalmodels(DGMs)anddynamicBayesiannetworks(DBNs).GMTKcanbeusedforapplicationsandresearchinspeechandlanguageprocessing,bioinformatics,activityrecognition,andanytimeseriesapplication.JustanotherGibbssampler(JAGS)(website(http://wwwfis.iarc.fr/~martyn/software/jags/))Stan(software)(website(http://mcstan.org/))StanisanopensourcepackageforobtainingBayesianinferenceusingtheNoUTurnsampler,avariantofHamiltonianMonteCarlo.It'ssomewhatlikeBUGS,butwithadifferentlanguageforexpressingmodelsandadifferentsamplerforsamplingfromtheirposteriors.RStanistheRinterfacetoStan.PyMC(http://pymcdevs.github.io/pymc/)PyMCisapythonmodulethatimplementsBayesianstatisticalmodelsandfittingalgorithms,includingMarkovchainMonteCarlo.Itsflexibilityandextensibilitymakeitapplicabletoalargesuiteofproblems.Alongwithcoresamplingfunctionality,PyMCincludesmethodsforsummarizingoutput,plotting,goodnessoffitandconvergencediagnostics.GeNIe&Smile(website(http://genie.sis.pitt.edu/))SMILEisaC++libraryforBNandID,andGeNIeisaGUIforitSamIam(website(http://reasoning.cs.ucla.edu/samiam/)),aJavabasedsystemwithGUIandJavaAPIBayesServer(http://www.BayesServer.com/)UserInterfaceandAPIforBayesiannetworks,includessupportfortimeseriesandsequencesBeliefandDecisionNetworksonAIspace(http://www.aispace.org/bayes/index.shtml)BayesiaLab(http://library.bayesia.com/display/HOME/The+BayesiaLab+Library/)byBayesiaHugin(http://www.hugin.com/)Netica(http://www.norsys.com/netica.html)byNorsysdVelox(http://www.aparasw.com/index.php/en)byAparaSoftwareSystemModeler(http://www.inatas.com)byInatasABUnBBayes(http://sourceforge.net/projects/unbbayes/)byGIAUnB(IntelligenceArtificialGroupUniversityofBrasilia)

History

Theterm"Bayesiannetworks"wascoinedbyJudeaPearlin1985toemphasizethreeaspects:[38]

1. Theoftensubjectivenatureoftheinputinformation.2. TherelianceonBayes'conditioningasthebasisforupdatinginformation.3. Thedistinctionbetweencausalandevidentialmodesofreasoning,whichunderscoresThomas

Bayes'posthumouslypublishedpaperof1763.[39]
http://www.norsys.com/netica.htmlhttp://en.wikipedia.org/wiki/Decision_support_systemhttp://www.openbugs.net/w/FrontPagehttp://reasoning.cs.ucla.edu/samiam/http://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Lawhttp://pymc-devs.github.io/pymc/http://en.wikipedia.org/wiki/Thomas_Bayeshttp://en.wikipedia.org/wiki/Stan_(software)http://www.inatas.com/http://www.hugin.com/http://mc-stan.org/http://www.openmarkov.org/http://www.bayesserver.com/http://en.wikipedia.org/wiki/Data_fusionhttp://www.aispace.org/bayes/index.shtmlhttp://en.wikipedia.org/wiki/Judea_Pearlhttp://genie.sis.pitt.edu/http://www.aparasw.com/index.php/enhttp://en.wikipedia.org/wiki/OpenBUGShttp://melodi.ee.washington.edu/gmtkhttp://sourceforge.net/projects/unbbayes/http://library.bayesia.com/display/HOME/The+BayesiaLab+Library/http://en.wikipedia.org/wiki/Just_another_Gibbs_samplerhttp://en.wikipedia.org/wiki/Risk_analysishttp://en.wikipedia.org/wiki/WinBUGShttp://en.wikipedia.org/wiki/Image_processinghttp://www-fis.iarc.fr/~martyn/software/jags/http://en.wikipedia.org/wiki/Semantic_search



Inthelate1980sJudeaPearl'stextProbabilisticReasoninginIntelligentSystems[40]andRichardE.Neapolitan'stextProbabilisticReasoninginExpertSystems[41]summarizedthepropertiesofBayesiannetworksandestablishedBayesiannetworksasafieldofstudy.

InformalvariantsofsuchnetworkswerefirstusedbylegalscholarJohnHenryWigmore,intheformofWigmorecharts,toanalysetrialevidencein1913.[30]:6676Anothervariant,calledpathdiagrams,wasdevelopedbythegeneticistSewallWright[42]andusedinsocialandbehavioralsciences(mostlywithlinearparametricmodels).

Seealso

Notes

ArtificialintelligenceBayes'theoremBayesianinferenceBayesianprobabilityBayesianprogrammingBeliefpropagationCausalloopdiagramChowLiutreeComputationalintelligenceComputationalphylogeneticsDeepbeliefnetworkDempsterShafertheoryaGeneralizationofBayes'theoremDynamicBayesiannetworkExpectationmaximizationalgorithmFactorgraphGraphicalmodelHierarchicaltemporalmemoryInfluencediagramJudeaPearlKalmanfilterMachinelearningMemorypredictionframeworkMixturedistributionMixturemodelNaiveBayesclassifierPathanalysisPolytreeSensorfusionSequencealignmentSpeechrecognitionStructuralequationmodelingSubjectivelogicVariableorderBayesiannetworkWigmorechartWorldview

1. Pearl,Judea(2000).Causality:Models,Reasoning,andInference.CambridgeUniversityPress.ISBN0521773628.OCLC42291253(https://www.worldcat.org/oclc/42291253).
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BenGal,Irad(2007)."BayesianNetworks".InRuggeri,FabrizioKennett,RonS.Faltin,FrederickW.EncyclopediaofStatisticsinQualityandReliability(http://www.eng.tau.ac.il/~bengal/BN.pdf)(PDF).EncyclopediaofStatisticsinQualityandReliability.JohnWiley&Sons.doi:10.1002/9780470061572.eqr089(https://dx.doi.org/10.1002%2F9780470061572.eqr089).ISBN9780470018613.BertschMcGrayne,Sharon.TheTheoryThatWouldnotDie.Yale.Borgelt,ChristianKruse,Rudolf(March2002).GraphicalModels:MethodsforDataAnalysisandMining(http://fuzzy.cs.unimagdeburg.de/books/gm/).Chichester,UK:Wiley.ISBN0470843373.Borsuk,MarkEdward(2008)."Ecologicalinformatics:Bayesiannetworks".InJrgensen,SvenErik,Fath,Brian.EncyclopediaofEcology.Elsevier.ISBN9780444520333.Cardenas,I.etal.(April2015)."ModelingtheInfluenceofUnknownFactorsinRiskAnalysisusingBayesianNetworks"(http://www.researchgate.net/publication/274456154_Modeling_the_Influence_of_Unknown_Factors_in_Risk_Analysis_using_Bayesian_Networks)(PDF).Underreviewbyarefereedjournal.Castillo,EnriqueGutirrez,JosManuelHadi,AliS.(1997)."LearningBayesianNetworks".ExpertSystemsandProbabilisticNetworkModels.Monographsincomputerscience.NewYork:SpringerVerlag.pp.481528.ISBN0387948589.Comley,JoshuaW.Dowe,DavidL.(http://www.csse.monash.edu.au/~dld)(October2003)."MinimumMessageLengthandGeneralizedBayesianNetswithAsymmetricLanguages"(http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2005).WrittenatVictoria,Australia.InGrnwald,PeterD.Myung,InJaePitt,MarkA.AdvancesinMinimumDescriptionLength:TheoryandApplications.Neuralinformationprocessingseries.Cambridge,Massachusetts:BradfordBooks(MITPress)(publishedApril2005).pp.265294.ISBN0262072629.(ThispaperputsdecisiontreesininternalnodesofBayesnetworksusingMinimumMessageLength(http://www.csse.monash.edu.au/~dld/MML.html)(MML).AnearlierversionisComleyandDowe(2003)(http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2003),.pdf(http://www.csse.monash.edu.au/~dld/Publications/2003/Comley+Dowe03_HICS2003_GeneralBayesianNetworksAsymmetricLanguages.pdf).)Darwiche,Adnan(2009).ModelingandReasoningwithBayesianNetworks(http://www.cambridge.org/9780521884389).CambridgeUniversityPress.ISBN9780521884389.
http://www.cambridge.org/9780521884389http://en.wikipedia.org/wiki/Special:BookSources/978-0-444-52033-3http://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Mediahttp://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2005http://en.wikipedia.org/wiki/Digital_object_identifierhttp://www.csse.monash.edu.au/~dld/MML.htmlhttp://en.wikipedia.org/wiki/Philosophical_Transactions_of_the_Royal_Societyhttp://en.wikipedia.org/wiki/Special:BookSources/978-0521884389http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Decision_tree_learninghttp://ftp.cs.ucla.edu/tech-report/198_-reports/850017.pdfhttp://en.wikipedia.org/wiki/Special:BookSources/978-0-470-01861-3http://en.wikipedia.org/wiki/Yalehttp://www.amazon.com/Probabilistic-Reasoning-Expert-Systems-Algorithms/dp/1477452540/ref=sr_1_3?s=books&ie=UTF8&qid=1389578837&sr=1-3&keywords=probabilistic+reasoning+in+expert+systemshttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/books?id=AvNID7LyMusChttp://en.wikipedia.org/wiki/Special:BookSources/1558604790http://en.wikipedia.org/wiki/Special:BookSources/0-470-84337-3http://en.wikipedia.org/wiki/Judea_Pearlhttp://en.wikipedia.org/wiki/MIT_Presshttp://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2003http://www.researchgate.net/publication/274456154_Modeling_the_Influence_of_Unknown_Factors_in_Risk_Analysis_using_Bayesian_Networkshttp://en.wikipedia.org/wiki/Special:BookSources/978-0-471-61840-9http://en.wikipedia.org/wiki/Sven_Erik_J%C3%B8rgensenhttp://en.wikipedia.org/wiki/John_Wiley_%26_Sonshttp://dx.doi.org/10.1002%2F9780470061572.eqr089http://www.eng.tau.ac.il/~bengal/BN.pdfhttp://www.csse.monash.edu.au/~dld/Publications/2003/Comley+Dowe03_HICS2003_GeneralBayesianNetworksAsymmetricLanguages.pdfhttp://en.wikipedia.org/wiki/Victoria_(Australia)http://www.csse.monash.edu.au/~dldhttp://en.wikipedia.org/wiki/Chichesterhttp://en.wikipedia.org/wiki/Minimum_message_lengthhttp://fuzzy.cs.uni-magdeburg.de/books/gm/http://en.wikipedia.org/wiki/John_Wiley_%26_Sonshttp://en.wikipedia.org/wiki/Thomas_Bayeshttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://www.ssc.wisc.edu/soc/class/soc952/Wright/Wright_Correlation%20and%20Causation.pdfhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/An_Essay_towards_solving_a_Problem_in_the_Doctrine_of_Chanceshttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Sewall_Wrighthttp://en.wikipedia.org/wiki/Cambridge_University_Presshttp://en.wikipedia.org/wiki/Special:BookSources/0-262-07262-9http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0-387-94858-9http://dx.doi.org/10.1098%2Frstl.1763.0053http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Cambridge,_Massachusetts



Dowe,DavidL.(2010).MML,hybridBayesiannetworkgraphicalmodels,statisticalconsistency,invarianceanduniqueness(http://www.csse.monash.edu.au/~dld/Publications/2010/Dowe2010_MML_HandbookPhilSci_Vol7_HandbookPhilStat_MML+hybridBayesianNetworkGraphicalModels+StatisticalConsistency+InvarianceAndUniqueness_pp901982.pdf),inHandbookofPhilosophyofScience(Volume7:HandbookofPhilosophyofStatistics),Elsevier,ISBN9780444518620(http://japan.elsevier.com/products/books/HPS.pdf),pp901982(http://www.csse.monash.edu.au/~dld/Publications/2010/Dowe2010_MML_HandbookPhilSci_Vol7_HandbookPhilStat_MML+hybridBayesianNetworkGraphicalModels+StatisticalConsistency+InvarianceAndUniqueness_pp901982.pdf).Fenton,NormanNeil,MartinE.(November2007).ManagingRiskintheModernWorld:ApplicationsofBayesianNetworks(http://www.agenarisk.com/resources/apps_bayesian_networks.pdf)AKnowledgeTransferReportfromtheLondonMathematicalSocietyandtheKnowledgeTransferNetworkforIndustrialMathematics.London(England):LondonMathematicalSociety.Fenton,NormanNeil,MartinE.(July23,2004)."CombiningevidenceinriskanalysisusingBayesianNetworks"(https://www.dcs.qmul.ac.uk/~norman/papers/Combining%20evidence%20in%20risk%20analysis%20using%20BNs.pdf)(PDF).SafetyCriticalSystemsClubNewsletter13(4)(NewcastleuponTyne,England).pp.813.AndrewGelmanJohnBCarlinHalSSternDonaldBRubin(2003)."PartII:FundamentalsofBayesianDataAnalysis:Ch.5Hierarchicalmodels"(http://books.google.com/books?id=TNYhnkXQSjAC&pg=PA120).BayesianDataAnalysis(http://books.google.com.au/books?id=TNYhnkXQSjAC).CRCPress.pp.120.ISBN9781584883883.Heckerman,David(March1,1995)."TutorialonLearningwithBayesianNetworks"(http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSRTR9506).InJordan,MichaelIrwin.LearninginGraphicalModels.AdaptiveComputationandMachineLearning.Cambridge,Massachusetts:MITPress(published1998).pp.301354.ISBN0262600323..

AlsoappearsasHeckerman,David(March1997)."BayesianNetworksforDataMining".DataMiningandKnowledgeDiscovery(Netherlands:SpringerNetherlands)1(1):79119.doi:10.1023/A:1009730122752(https://dx.doi.org/10.1023%2FA%3A1009730122752).ISSN13845810(https://www.worldcat.org/issn/13845810).AnearlierversionappearsasTechnicalReportMSRTR9506(http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSRTR9506),MicrosoftResearch,March1,1995.ThepaperisaboutbothparameterandstructurelearninginBayesiannetworks.

Jensen,FinnVNielsen,ThomasD.(June6,2007).BayesianNetworksandDecisionGraphs.InformationScienceandStatisticsseries(2nded.).NewYork:SpringerVerlag.ISBN9780387682815.Karimi,KamranHamilton,HowardJ.(2000)."Findingtemporalrelations:Causalbayesiannetworksvs.C4.5"(http://www.kamrankarimi.com/pubs/khISMIS2000.pdf)(PDF).TwelfthInternationalSymposiumonMethodologiesforIntelligentSystems.Korb,KevinB.Nicholson,AnnE.(December2010).BayesianArtificialIntelligence.CRCComputerScience&DataAnalysis(2nded.).Chapman&Hall(CRCPress).doi:10.1007/s1004400402145(https://dx.doi.org/10.1007%2Fs1004400402145).ISBN1584883871.Lunn,D.Thomas,ABest,Netal.(2009)."TheBUGSproject:Evolution,critiqueandfuturedirections".StatisticsinMedicine28(25):30493067.doi:10.1002/sim.3680(https://dx.doi.org/10.1002%2Fsim.3680).PMID19630097(https://www.ncbi.nlm.nih.gov/pubmed/19630097).|first2=missing|last2=inAuthorslist(help)Neil,MartinFenton,NormanE.Tailor,Manesh(August2005).Greenberg,MichaelR.,ed."UsingBayesianNetworkstoModelExpectedandUnexpectedOperationalLosses"(http://www.dcs.qmul.ac.uk/~norman/papers/oprisk.pdf)(PDF).RiskAnalysis:anInternationalJournal(JohnWiley&Sons)25(4):963972.doi:10.1111/j.15396924.2005.00641.x(https://dx.doi.org/10.1111%2Fj.15396924.2005.00641.x).PMID16268944(https://www.ncbi.nlm.nih.gov/pubmed/16268944).Pearl,Judea(September1986)."Fusion,propagation,andstructuringinbeliefnetworks".ArtificialIntelligence(Elsevier)29(3):241288.doi:10.1016/00043702(86)90072X(https://dx.doi.org/10.1016%2F00043702%2886%2990072X).ISSN00043702(https://www.worldcat.org/issn/00043702).
http://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttp://en.wikipedia.org/wiki/John_Wiley_%26_Sonshttp://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSR-TR-95-06http://en.wikipedia.org/wiki/Help:CS1_errors#first_missing_lasthttps://www.dcs.qmul.ac.uk/~norman/papers/Combining%20evidence%20in%20risk%20analysis%20using%20BNs.pdfhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/CRC_Presshttp://en.wikipedia.org/wiki/Special:BookSources/0-262-60032-3http://www.ncbi.nlm.nih.gov/pubmed/16268944http://en.wikipedia.org/wiki/Digital_object_identifierhttp://www.worldcat.org/issn/0004-3702http://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Mediahttp://en.wikipedia.org/wiki/Chapman_%26_Hallhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://dx.doi.org/10.1111%2Fj.1539-6924.2005.00641.xhttp://books.google.com/books?id=TNYhnkXQSjAC&pg=PA120http://en.wikipedia.org/wiki/Cambridge,_Massachusettshttp://en.wikipedia.org/wiki/Elsevierhttp://www.csse.monash.edu.au/~dld/Publications/2010/Dowe2010_MML_HandbookPhilSci_Vol7_HandbookPhilStat_MML+hybridBayesianNetworkGraphicalModels+StatisticalConsistency+InvarianceAndUniqueness_pp901-982.pdfhttp://en.wikipedia.org/wiki/MIT_Presshttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Artificial_Intelligence_(journal)http://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1023%2FA%3A1009730122752http://www.ncbi.nlm.nih.gov/pubmed/19630097http://en.wikipedia.org/wiki/International_Standard_Serial_Numberhttp://en.wikipedia.org/wiki/PubMed_Identifierhttp://en.wikipedia.org/wiki/Special:BookSources/1-58488-387-1http://en.wikipedia.org/wiki/London_Mathematical_Societyhttp://japan.elsevier.com/products/books/HPS.pdfhttp://en.wikipedia.org/wiki/Data_Mining_and_Knowledge_Discoveryhttp://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSR-TR-95-06http://en.wikipedia.org/wiki/Digital_object_identifierhttp://www.dcs.qmul.ac.uk/~norman/papers/oprisk.pdfhttp://www.agenarisk.com/resources/apps_bayesian_networks.pdfhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://www.csse.monash.edu.au/~dld/Publications/2010/Dowe2010_MML_HandbookPhilSci_Vol7_HandbookPhilStat_MML+hybridBayesianNetworkGraphicalModels+StatisticalConsistency+InvarianceAndUniqueness_pp901-982.pdfhttp://en.wikipedia.org/wiki/Judea_Pearlhttp://www.worldcat.org/issn/1384-5810http://www.kamran-karimi.com/pubs/khISMIS2000.pdfhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Netherlandshttp://dx.doi.org/10.1007%2Fs10044-004-0214-5http://en.wikipedia.org/wiki/PubMed_Identifierhttp://en.wikipedia.org/wiki/Society_for_Risk_Analysishttp://dx.doi.org/10.1002%2Fsim.3680http://books.google.com.au/books?id=TNYhnkXQSjAChttp://en.wikipedia.org/wiki/Special:BookSources/978-0-387-68281-5http://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Mediahttp://en.wikipedia.org/wiki/Londonhttp://dx.doi.org/10.1016%2F0004-3702%2886%2990072-Xhttp://en.wikipedia.org/wiki/New_Yorkhttp://en.wikipedia.org/wiki/Newcastle_upon_Tynehttp://en.wikipedia.org/wiki/Special:BookSources/978-1-58488-388-3



Furtherreading

ComputationalIntelligence:AMethodologicalIntroductionbyKruse,Borgelt,Klawonn,Moewes,Steinbrecher,Held,2013,Springer,ISBN9781447150121GraphicalModelsRepresentationsforLearning,ReasoningandDataMining,2ndEdition,byBorgelt,Steinbrecher,Kruse,2009,J.Wiley&Sons,ISBN9780470749562

Externallinks

AtutorialonlearningwithBayesianNetworks(http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSRTR9506)AnIntroductiontoBayesianNetworksandtheirContemporaryApplications(http://www.niedermayer.ca/papers/bayesian/bayes.html)OnlineTutorialonBayesiannetsandprobability(http://www.dcs.qmw.ac.uk/%7Enorman/BBNs/BBNs.htm)WebApptocreateBayesiannetsandrunitwithaMonteCarlomethod(http://princesofserendib.com/)ContinuousTimeBayesianNetworks(http://robotics.stanford.edu/~nodelman/papers/ctbn.pdf)BayesianNetworks:ExplanationandAnalogy(http://wiki.syncleus.com/index.php/DANN:Bayesian_Network)AlivetutorialonlearningBayesiannetworks(http://videolectures.net/kdd07_neapolitan_lbn/)AhierarchicalBayesModelforhandlingsampleheterogeneityinclassificationproblems(http://www.biomedcentral.com/14712105/7/514/abstract),providesaclassificationmodeltakingintoconsiderationtheuncertaintyassociatedwithmeasuringreplicatesamples.HierarchicalNaiveBayesModelforhandlingsampleuncertainty(http://www.labmedinfo.org/download/lmi339.pdf),showshowtoperformclassificationandlearningwithcontinuousanddiscretevariableswithreplicatedmeasurements.

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Pearl,Judea(1988).ProbabilisticReasoninginIntelligentSystems:NetworksofPlausibleInference.RepresentationandReasoningSeries(2ndprintinged.).SanFrancisco,California:MorganKaufmann.ISBN0934613737.Pearl,JudeaRussell,Stuart(November2002)."BayesianNetworks".InArbib,MichaelA.HandbookofBrainTheoryandNeuralNetworks.Cambridge,Massachusetts:BradfordBooks(MITPress).pp.157160.ISBN0262011972.Russell,StuartJ.Norvig,Peter(2003),ArtificialIntelligence:AModernApproach(http://aima.cs.berkeley.edu/)(2nded.),UpperSaddleRiver,NewJersey:PrenticeHall,ISBN0137903952.Zhang,NevinLianwen(http://www.cs.ust.hk/faculty/lzhang/bio.html)Poole,David(http://www.cs.ubc.ca/spider/poole/)(May1994)."AsimpleapproachtoBayesiannetworkcomputations".ProceedingsoftheTenthBiennialCanadianArtificialIntelligenceConference(AI94).(Banff,Alberta):171178.Thispaperpresentsvariableeliminationforbeliefnetworks.
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