Global Structure: Treewidth w O ( n exp( w )) 1 Local - PDF document

Readings: K&F: 4.1, 4.2, 4.3, 4.4, 8.4, 8.5, 8.6 “Recursive Conditioning”, Adnan Darwiche. In Artificial Intelligence Journal, 125:1, pp. 5-41 Context-specific independence Graphical Models – 10708 Carlos Guestrin Carnegie Mellon University October 16 th , 2006 Global Structure: Treewidth w O ( n exp( w )) 1

Local Structure 1: Context specific indepencence �� !�� "�� Local Structure 1: Context specific indepencence ��#�� $�%�$�&�� $��'��&( �%��!��!��)�� !�$�� !�� "�� 2

CSI example: Tree CPD Apply SAT Letter � Represent P(X i | Pa Xi ) using a decision tree � Path to leaf is an assignment to (a subset Job of) Pa Xi � Leaves are distributions over X i given assignment of Pa Xi on path to leaf � Interpretation of leaf : � For specific assignment of Pa Xi on path to this leaf – X i is independent of other parents � Representation can be exponentially smaller than equivalent table Tabular VE with Tree CPDs � If we turn a tree CPD into table � “Sparsity” lost ! � Need inference approach that deals with tree CPD directly ! 3

Local Structure 2: Determinism �� &%��3�� 3�� * �� + -.. -/0 �� !�� -1/ -2/ �� ,��+ �� / 0 �� "�� Determinism and inference � Determinism gives a little �� * �� sparsity in table, but much �� + -.. -/0 bigger impact on inference �� -1/ -2/ ,��+ � Multiplying deterministic factor / 0 �� with other factor introduces many new zeros � Operations related to theorem proving, e.g., unit resolution 4

Today’s Models … � Often characterized by: � Richness in local structure (determinism, CSI) � Massiveness in size (10,000’s variables) � High connectivity (treewidth) � Enabled by: � High level modeling tools: relational, first order � Advances in machine learning � New application areas (synthesis): � Bioinformatics (e.g. linkage analysis) � Sensor networks � Exploiting local structure a must! Exact inference in large models is possible… � BN from a relational model 5

Recursive Conditioning � Treewidth complexity (worst case) � Better than treewidth complexity with local structure � Provides a framework for time-space tradeoffs � Only quick intuition today, details in readings �� %�� " �� !�� "�� -��$�� 6

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Global Structure: Treewidth w O ( n exp( w )) 1 Local - PDF document

Readings: K&F: 4.1, 4.2, 4.3, 4.4, 8.4, 8.5, 8.6 Recursive Conditioning, Adnan Darwiche. In Artificial Intelligence Journal, 125:1, pp. 5-41 Context-specific independence Graphical Models 10708 Carlos Guestrin Carnegie Mellon

Global Structure: Treewidth w ( exp( )) O n w 1 Local Structure 1: Context specific

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