Bayesian Belief NetworksCompound Bayesian Decision TheoryBayesian Belief NetworksCausal relationshipsParent-Child RelationshipA simple belief netDetermining a joint probabilityDetermining probabilities in a net with a loopEvidenceExample of Evaluation (Classification)Evaluation StepsMedical Diagnosis Application of Belief NetsCompound Bayesian Decision Theory and ContextSrihari: CSE 5550Bayesian Belief NetworksCompound Bayesian Decision TheorySrihari: CSE 5551Bayesian Belief Networks• In certain situations statistical properties are not directly expressed by a parameter vector but by causal relationships among variablesSrihari: CSE 5552Statistically dependent and independent variablesThree-dimensional distribution which obeys p(x1, x3) = p(x1) p(x3)Thus x1and x3are statistically independent but the other feature pairsare notx1x2x3Srihari: CSE 5553Causal relationships• State of automobile• Temperature of engine• Pressure of brake fluid• Pressure of air in tires• Voltages in the wires• Oil pressure and air pressure are not causally related• Engine temperature and oil temperature areSrihari: CSE 5554Parent-Child RelationshipNode X has variable values (x1,x2,….)Srihari: CSE 5555Bayesian Belief Net or Causal Network or Belief NetNode A has states {a1, a2,…} = a Node B has states {b1, b2,…}= b⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=)|()|()|()|()|()|()|()|(31232221131211acPacPacPacPacPacPacPacPConditional Probability TableRows sum to oneSrihari: CSE 5556A simple belief netSrihari: CSE 5557Determining a joint probabilityP(a3 , b1 , x2 , c3 , d2) = P(a3) P(b1) P(x2|a3,b1) P(c3|x2) P(d2,x2)= 0.25 x 0.6 x 0.4 x 0.5 x 0.4= 0.012 Only X has 2 parents thus only the P(x2|..) has two conditioning variablesSrihari: CSE 5558Determining Probability of variables in a Bayes Belief NetLinear Chain Belief NetTo computeproceed as aboveSrihari: CSE 5559Determining probabilities in a net with a loopComputing the probabilities of variables at H in the networkBelief net with a simple loopDiffers somewhatfrom linear networkbecause of loopSrihari: CSE 55510EvidenceGiven the values of some variables (evidence) determine someconfiguration of other variablesDetermine fish came from North Atlantic, given it is springtimeand fish is a light salmon, or P(b1|a2,x1,c1)Query variableEvidenceb1=North Atlantica2= Springc1=lightx1=salmond = thickness unknownSrihari: CSE 55511Example of Evaluation (Classification)What is the classification when fish is light (c1), caught in South Atlantic (b2)and do not know time of year and thickness?Srihari: CSE 55512Evaluation StepsSimilarly,P(x2|c1,b2) = α 0.066Normalizing,P(x1|c1,b2)=0.63P(x2|c1,b2)=0.37Given the evidenceclassify as a salmonNoteSrihari: CSE 55513Naïve Bayes RuleWhen dependency relationships among the features used bya classfier are unknown, assume features are conditionally independentSrihari: CSE 55514Medical Diagnosis Application of Belief Nets• Uppermost nodes (without parents) • Biological agent such as presence of bacteria or virus• Intermediate nodes• Diseases such as emphysema or flu• Lowermost nodes• Symptoms such as high temperature or coughing• Physician enters measured values in net and finds most likely disease or causeSrihari: CSE 55515Compound Bayesian Decision Theory and