Global semantics
We can calculate the full joint distribution as the product of the local conditional distributions:
n
ź
i“1
Compute: Ppj ^m^a^␣b^␣eq
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P px1, . . . , xnq “
P pxi|parentspXiqq
Global semantics
We can calculate the full joint distribution as the product of the local conditional distributions:
n
ź
i“1
“ P pj |aqP pm|aqP pa|␣b, ␣eqP p␣bqP p␣eq “ 0.9 ̈0.7 ̈0.001 ̈0.999 ̈0.998
« 0.00063
Application of the chain rule.
P px1, . . . , xnq “ Ppj ^m^a^␣b^␣eq
P pxi|parentspXiqq
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Compactness
A CPT for Boolean Xi with k Boolean parents has 2krows for the combinations of parent values
Each row requires one number p for Xi “ true (the number for Xi “ f alse is just 1 ́ p)
If each variable has no more than k parents, the complete network requires Opn ̈ 2kq numbers
Grows linearly with n, vs. Op2nq for the full joint distribution
For burglary net, 1 ` 1 ` 4 ` 2 ` 2 “ 10 numbers (vs.25 ́1“31)
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Local semantics
A node X is conditionally independent of its non-descendants (e.g., the Zi,j s) given its parents (the Uis shown in the gray area).
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Markov blanket
Each node is conditionally independent of all others given its Markov blanket: parents + children + children’s parents
Andrey Markov
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6
Markov blanket
Each node is conditionally independent of all others given its Markov blanket: parents + children + children’s parents
EF ABXGH
CD
Markov blanket of X?
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Markov blanket
Each node is conditionally independent of all others given its Markov blanket: parents + children + children’s parents
EF ABXGH
CD
Markov blanket of X?
Answer: B,C,E,F,G
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Constructing Bayesian networks
Build Bayesian networks like any other form of knowledge representation. First figure out the variables that describe the world.
Then decide how they are connected. Conditional independence.
Then work out the values in the CPTs.
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Ways of compressing further
CPT grows exponentially with number of parents ‚ Usedistributionsthataredefinedcompactly
Deterministic nodes are the simplest case. X “ fpParentspXqq for some function f
‚ Booleanfunctions:
NorthAmerican ô Canadian_US_Mexican
‚ Numericalrelationshipsamongcontinuousvariables
BLevel “ inflow + precipitation – outflow – evaporation
Bt
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