PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS.

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Presentation transcript:

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS

Bayesian Networks Directed Acyclic Graph (DAG)

Bayesian Networks General Factorization

Conditional Independence a is independent of b given c Equivalently Notation

Conditional Independence: Example 1

Conditional Independence: Example 2

Conditional Independence: Example 3 Note: this is the opposite of Example 1, with c unobserved.

Conditional Independence: Example 3 Note: this is the opposite of Example 1, with c observed.

“Am I out of fuel?” B =Battery (0=flat, 1=fully charged) F =Fuel Tank (0=empty, 1=full) G =Fuel Gauge Reading (0=empty, 1=full) and hence

“Am I out of fuel?” Probability of an empty tank increased by observing G = 0.

“Am I out of fuel?” Probability of an empty tank reduced by observing B = 0. This referred to as “explaining away”.

D-separation A, B, and C are non-intersecting subsets of nodes in a directed graph. A path from A to B is blocked if it contains a node such that either a)the arrows on the path meet either head-to-tail or tail- to-tail at the node, and the node is in the set C, or b)the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, are in the set C. If all paths from A to B are blocked, A is said to be d- separated from B by C. If A is d-separated from B by C, the joint distribution over all variables in the graph satisfies.

D-separation: Example

Inference in Graphical Models