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Pattern Recognition and Machine Learning

Pattern Recognition and Machine Learning. Chapter 8: graphical models. Bayesian Networks. Directed Acyclic Graph (DAG). Bayesian Networks. General Factorization. Bayesian Curve Fitting (1) . Polynomial. Bayesian Curve Fitting (2) . Plate. Bayesian Curve Fitting (3) .

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Pattern Recognition and Machine Learning

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  1. Pattern Recognition and Machine Learning Chapter 8: graphical models

  2. Bayesian Networks Directed Acyclic Graph (DAG)

  3. Bayesian Networks General Factorization

  4. Bayesian Curve Fitting (1) Polynomial

  5. Bayesian Curve Fitting (2) Plate

  6. Bayesian Curve Fitting (3) Input variables and explicit hyperparameters

  7. Bayesian Curve Fitting —Learning Condition on data

  8. Bayesian Curve Fitting —Prediction Predictive distribution: where

  9. Generative Models Causal process for generating images

  10. Discrete Variables (1) General joint distribution: K 2 { 1 parameters Independent joint distribution: 2(K{ 1) parameters

  11. Discrete Variables (2) General joint distribution over M variables: KM{ 1 parameters M -node Markov chain: K{ 1 + (M{ 1) K(K{ 1) parameters

  12. Discrete Variables: Bayesian Parameters (1)

  13. Discrete Variables: Bayesian Parameters (2) Shared prior

  14. Parameterized Conditional Distributions If are discrete, K-state variables, in general has O(K M) parameters. The parameterized form requires only M+ 1 parameters

  15. Linear-Gaussian Models Directed Graph Vector-valued Gaussian Nodes • Each node is Gaussian, the mean is a linear function of the parents.

  16. Conditional Independence a is independent of b given c Equivalently Notation

  17. Conditional Independence: Example 1

  18. Conditional Independence: Example 1

  19. Conditional Independence: Example 2

  20. Conditional Independence: Example 2

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

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

  23. “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

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

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

  26. 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 • 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 • 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 .

  27. D-separation: Example

  28. D-separation: I.I.D. Data

  29. Directed Graphs as Distribution Filters

  30. The Markov Blanket Factors independent of xi cancel between numerator and denominator.

  31. Cliques and Maximal Cliques Clique Maximal Clique

  32. Joint Distribution where is the potential over clique C and is the normalization coefficient; note: MK-state variables  KM terms in Z. Energies and the Boltzmann distribution

  33. Illustration: Image De-Noising (1) Original Image Noisy Image

  34. Illustration: Image De-Noising (2)

  35. Illustration: Image De-Noising (3) Noisy Image Restored Image (ICM)

  36. Illustration: Image De-Noising (4) Restored Image (ICM) Restored Image (Graph cuts)

  37. Converting Directed to Undirected Graphs (1)

  38. Converting Directed to Undirected Graphs (2) Additional links

  39. Directed vs. Undirected Graphs (1)

  40. Directed vs. Undirected Graphs (2)

  41. Inference in Graphical Models

  42. Inference on a Chain

  43. Inference on a Chain

  44. Inference on a Chain

  45. Inference on a Chain

  46. Inference on a Chain To compute local marginals: Compute and store all forward messages, . Compute and store all backward messages, . Compute Z at any node xm Computefor all variables required.

  47. Trees Undirected Tree Directed Tree Polytree

  48. Factor Graphs

  49. Factor Graphs from Directed Graphs

  50. Factor Graphs from Undirected Graphs

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