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Learning. Probabilistic Graphical Models. Parameter Estimation. Max Likelihood for BNs. MLE for Bayesian Networks. Parameters: Data instances: <x[m],y[m]>. X. Y. MLE for Bayesian Networks. X. Parameters:. X. Y|X. Y. Data d. MLE for Bayesian Networks.
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Learning Probabilistic Graphical Models Parameter Estimation Max Likelihoodfor BNs
MLE for Bayesian Networks • Parameters: • Data instances: <x[m],y[m]> X Y
MLE for Bayesian Networks X • Parameters: X Y|X Y Data d
MLE for Bayesian Networks • Likelihood for Bayesian network if Xi|Uiare disjoint, then MLE can be computed by maximizing each local likelihood separately
Shared Parameters S’|S S(0) S(1) S(2) S(3)
Shared Parameters S(0) S(1) S(2) S(3) O(1) O(2) O(3) S’|S O’|S’
Summary • For BN with disjoint sets of parameters in CPDs, likelihood decomposes as product of local likelihood functions, one per variable • For table CPDs, local likelihood further decomposes as product of likelihood for multinomials, one for each parent combination • For networks with shared CPDs, sufficient statistics accumulate over all uses of CPD