bayesian belief networks compound bayesian decision theory
TRANSCRIPT
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Bayesian Belief NetworksCompound Bayesian Decision
Theory
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Bayesian Belief Networks
• In certain situations statistical properties are not directly expressed by a parameter vector but by causal relationships among variables
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Statistically dependent and independent variables
Three-dimensional distribution which obeys p(x1, x3) = p(x1) p(x3)Thus x1 and x3 are statistically independent but the other feature pairsare not
x1
x2
x3
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Causal 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 are
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Parent-Child Relationship
Node X has variable values (x1,x2,….)
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Bayesian Belief Net or Causal Network or Belief Net
Node A has states {a1, a2,…} = a Node B has states {b1, b2,…}= b
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Conditional Probability Table
Rows sum to one
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A simple belief net
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Determining a joint probability
P(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 variables
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Determining Probability of variables in a Bayes Belief Net
Linear Chain Belief Net
To compute
proceed as above
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Determining probabilities in a net with a loop
Computing the probabilities of variables at H in the network
Belief net with a simple loop
Differs somewhatfrom linear networkbecause of loop
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Evidence
Given 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 variable
Evidence
b1=North Atlantica2 = Spring
c1=lightx1=salmon
d = thickness unknown
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Example 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?
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Evaluation StepsSimilarly,P(x2|c1,b2) = α 0.066
Normalizing,P(x1|c1,b2)=0.63P(x2|c1,b2)=0.37
Given the evidenceclassify as a salmonNote
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Naïve Bayes Rule
When dependency relationships among the features used bya classfier are unknown, assume features are conditionally independent
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Medical 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 cause
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Compound Bayesian Decision Theory and Context