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PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS - - PowerPoint PPT Presentation
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS - - PowerPoint PPT Presentation
Some slides are removed by ATC (See PRML site for the original copy) PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS Bayesian Networks Directed Acyclic Graph (DAG) Bayesian Networks General Factorization Generative Models
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Bayesian Networks
General Factorization
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Generative Models
Causal process for generating images
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Discrete Variables (1)
General joint distribution: parameters Independent joint distribution: parameters
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Discrete Variables (2)
General joint distribution over variables: parameters
- node Markov chain:
- node Markov chain:
parameters
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Conditional Independence
is independent of given Equivalently Equivalently Notation
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Conditional Independence: Example 1
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Conditional Independence: Example 1
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Conditional Independence: Example 2
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Conditional Independence: Example 2
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Conditional Independence: Example 3
Note: this is the opposite of Example 1, with unobserved.
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Conditional Independence: Example 3
Note: this is the opposite of Example 1, with observed.
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“Am I out of fuel?”
= Battery (0=flat, 1=fully charged) = Fuel Tank (0=empty, 1=full) = Fuel Gauge Reading (0=empty, 1=full) and hence
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“Am I out of fuel?”
Probability of an empty tank increased by observing .
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“Am I out of fuel?”
Probability of an empty tank reduced by observing . This referred to as “explaining away”.
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D-separation
, , and are non-intersecting subsets of nodes in a directed graph. A path from to 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 , or to-tail at the node, and the node is in the set , or b) the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, are in the set . If all paths from to are blocked, is said to be d- separated from by . If is d-separated from by , the joint distribution over all variables in the graph satisfies .
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D-separation: Example
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D-separation: I.I.D. Data
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The Markov Blanket
Factors independent of cancel between numerator and denominator.
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Markov Random Fields
Markov Blanket
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Cliques and Maximal Cliques
Clique Maximal Clique
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Joint Distribution
where is the potential over clique and is the normalization coefficient; note: -state variables → terms in . Energies and the Boltzmann distribution
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Illustration: Image De-Noising (1)
Original Image Noisy Image
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Illustration: Image De-Noising (2)
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Illustration: Image De-Noising (3)
Noisy Image Restored Image (ICM)
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Illustration: Image De-Noising (4)
Restored Image (Graph cuts) Restored Image (ICM)
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Converting Directed to Undirected Graphs (1)
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Converting Directed to Undirected Graphs (2)
Additional links
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Directed vs. Undirected Graphs (1)
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Directed vs. Undirected Graphs (2)
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Factor Graphs
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Factor Graphs from Directed Graphs
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