This document provides an overview of probabilistic graphical models. It discusses two types of probabilistic graphical models - Bayesian networks and Markov networks. Bayesian networks use directed graphs to represent conditional independence relationships between random variables. Markov networks use undirected graphs for the same purpose. The document outlines topics like representation, examples including naive Bayes classifiers and the Ising model, and inference and learning algorithms for probabilistic graphical models.

1. Machine Learning
Lecture 1: Probabilistic Graphical Models
Phạm Duy Tùng
Email: duytung88@gmail.com
9/9/2012
9/17/2012 Some slides copied from Pattern Recognition and Machine Learning (Bishop 2006) 1

2. Introduction
• Probabilistic graphical models
• A visual presentation of probability distributions, using
diagrams, called PGMs
• Two types of PGM
• Bayesian Network (Directed Graphical Model)
• Markov Network (Undirected Graphical Model)
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3. Introduction
• Offering several useful properties:
• They provide a simply way to visualize the
structure of probabilistic models and motivate
new models.
• Insights into the properties of the models,
including the conditional independence
properties, can be obtain by inspection of the
graph.
• Graph based algorithms for calculation and
computation
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4. Topics
• Introduction
• Representation
• Bayesian Networks (Directed Graphical Model)
• Markov Networks (Undirected Graphical Model)
• Converting Bayesian Networks to Markov Networks
• Directed vs. Undirected Graphs
• Some examples
• Naïve Bayes Classifier
• Ising Model
• Inference
• Learning
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5. Representation
• Random variables -> Nodes
• In Conditional Independences -> Edges
(Directed or Undirected)
Probability Graphical
Models Models
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6. Topics
• Introduction
• Representation
• Bayesian Networks (Directed Graphical Model)
• Markov Networks (Undirected Graphical Model)
• Converting Bayesian Networks to Markov Networks
• Directed vs. Undirected Graphs
• Some examples
• Naïve Bayes Classifier
• Ising Model
• Inference
• Learning
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7. Representation->Bayesian Network
• Using Directed Acyclic Graph (DAG)
• Joint distribution factorizes according to graph
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8. Representation->Bayesian Network
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9. Representation->Bayesian Network
• Conditional independence:
Or equivalently:
Denoted by:
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10. Representation->Bayesian Network
• Conditional independence: Example 1
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11. Representation->Bayesian Network
• Conditional independence: Example 1
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12. Representation->Bayesian Network
• Conditional independence: Example 2
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13. Representation->Bayesian Network
• Conditional independence: Example 2
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14. Representation->Bayesian Network
• Conditional independence: Example 3
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15. Representation->Bayesian Network
• Conditional independence: Example 3
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16. Representation->Bayesian Network
• 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 .
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17. Representation->Bayesian Network
• D-separation: Example
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18. Representation->Bayesian Network
• Markov Blanket
Factors independent of xi cancel
between numerator and denominator.
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19. Example
• Mixture of Gaussians
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20. Home Work
• LDA model (David M.Blei)
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21. Topics
• Introduction
• Representation
• Bayesian Networks (Directed Graphical Model)
• Markov Networks (Undirected Graphical Model)
• Converting Bayesian Networks to Markov Networks
• Directed vs. Undirected Graphs
• Some examples
• Naïve Bayes Classifier
• Ising Model
• Inference
• Learning
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22. Representation->Markov Network
• Many phenomenon in real life, we can not
determine exactly the directionality to the
interaction between random variables.
• We use Markov Network to modeling these
phenomenon instead of Bayesian Network.
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23. Representation->Markov Network
• Conditional independence
Markov Blanket
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24. Representation->Markov Network
• Factorization properties
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25. Representation->Markov Network
• Clique
Clique
Maximal Clique
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26. Representation->Markov Network
• where is the potential over clique C and
is the normalization coefficient; note: M K-state variables
KM terms in Z.
• Energies and the Boltzmann distribution
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27. Representation -> Converting Bayesian
Networks to Markov Networks
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28. Representation -> Converting Bayesian
Networks to Markov Networks
• Additional links
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29. Directed vs. Undirected Graphs
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30. Directed vs. Undirected Graphs

31. Topics
• Introduction
• Representation
• Bayesian Networks (Directed Graphical Model)
• Markov Networks (Undirected Graphical Model)
• Converting Bayesian Networks to Markov Networks
• Directed vs. Undirected Graphs
• Some examples
• Naïve Bayes Classifier
• Ising Model
• Inference
• Learning
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32. Naïve Bayes Classifier
• Predicting
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33. Naïve Bayes Classifier
• How to estimate and ?
• Solution: We can separately estimate two
parts of the model using Maximum Likelihood
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34. Naïve Bayes Classifier
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35. Naïve Bayes Classifier
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36. Naïve Bayes Classifier
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37. Naïve Bayes Classifier
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38. Naïve Bayes Classifier
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39. Naïve Bayes Classifier
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40. Naïve Bayes Classifier
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41. Naïve Bayes Classifier
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42. Naïve Bayes Classifier
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43. Naïve Bayes Classifier
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44. Ising Model and Image de-noising
• Markov Random Field
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45. Ising Model and Image de-noising
• An example of Ising Model used in Image
processing.
Noisy Image
Original Image
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46. Ising Model and Image de-noising
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47. Ising Model and Image de-noising
• All of our assumptions about images are
encoded as follows:
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48. Ising Model and Image de-noising
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49. Ising Model and Image de-noising
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50. Ising Model and Image de-noising
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51. Ising Model and Image de-noising
• Results
Noisy Image Restored Image (ICM)
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52. Ising Model and Image de-noising
• Comparing two optimizing algorithms: Graph
cuts vs ICM
Restored Image (ICM) Restored Image (Graph cuts)
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53. Ising Model and Image de-noising
• Home works
– Design and implement an image segmentation
algorithms using Ising model.
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54. Topics
• Introduction
• Representation
• Bayesian Networks (Directed Graphical Model)
• Markov Networks (Undirected Graphical Model)
• Converting Bayesian Networks to Markov Networks
• Directed vs. Undirected Graphs
• Some examples
• Naïve Bayes Classifier
• Ising Model
• Inference
• Learning
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55. Inference
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56. Learning
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