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Lecture 1 graphical models

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Duy Tung Pham
Duy Tung Pham

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.

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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
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) 9/17/2012 2
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 9/17/2012 3
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 9/17/2012 4
Representation • Random variables -> Nodes • In Conditional Independences -> Edges (Directed or Undirected) Probability Graphical Models Models 9/17/2012 5
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 9/17/2012 6
Representation->Bayesian Network • Using Directed Acyclic Graph (DAG) • Joint distribution factorizes according to graph 9/17/2012 7
Representation->Bayesian Network 9/17/2012 8
Representation->Bayesian Network • Conditional independence: Or equivalently: Denoted by: 9/17/2012 9
Representation->Bayesian Network • Conditional independence: Example 1 9/17/2012 10
Representation->Bayesian Network • Conditional independence: Example 1 9/17/2012 11
Representation->Bayesian Network • Conditional independence: Example 2 9/17/2012 12
Representation->Bayesian Network • Conditional independence: Example 2 9/17/2012 13
Representation->Bayesian Network • Conditional independence: Example 3 9/17/2012 14
Representation->Bayesian Network • Conditional independence: Example 3 9/17/2012 15
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 . 9/17/2012 16
Representation->Bayesian Network • D-separation: Example 9/17/2012 17
Representation->Bayesian Network • Markov Blanket Factors independent of xi cancel between numerator and denominator. 9/17/2012 18
Example • Mixture of Gaussians 9/17/2012 19
Home Work • LDA model (David M.Blei) 9/17/2012 20
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 9/17/2012 21
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. 9/17/2012 22
Representation->Markov Network • Conditional independence Markov Blanket 9/17/2012 23
Representation->Markov Network • Factorization properties 9/17/2012 24
Representation->Markov Network • Clique Clique Maximal Clique 9/17/2012 25
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 9/17/2012 26
Representation -> Converting Bayesian Networks to Markov Networks 9/17/2012 27
Representation -> Converting Bayesian Networks to Markov Networks • Additional links 9/17/2012 28
Directed vs. Undirected Graphs 9/17/2012 29
Directed vs. Undirected Graphs
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 9/17/2012 31
Naïve Bayes Classifier • Predicting 9/17/2012 32
Naïve Bayes Classifier • How to estimate and ? • Solution: We can separately estimate two parts of the model using Maximum Likelihood 9/17/2012 33
Naïve Bayes Classifier 9/17/2012 34
Naïve Bayes Classifier 9/17/2012 35
Naïve Bayes Classifier 9/17/2012 36
Naïve Bayes Classifier 9/17/2012 37
Naïve Bayes Classifier 9/17/2012 38
Naïve Bayes Classifier 9/17/2012 39
Naïve Bayes Classifier 9/17/2012 40
Naïve Bayes Classifier 9/17/2012 41
Naïve Bayes Classifier 9/17/2012 42
Naïve Bayes Classifier 9/17/2012 43
Ising Model and Image de-noising • Markov Random Field 9/17/2012 44
Ising Model and Image de-noising • An example of Ising Model used in Image processing. Noisy Image Original Image 9/17/2012 45
Ising Model and Image de-noising 9/17/2012 46
Ising Model and Image de-noising • All of our assumptions about images are encoded as follows: 9/17/2012 47
Ising Model and Image de-noising 9/17/2012 48
Ising Model and Image de-noising 9/17/2012 49
Ising Model and Image de-noising 9/17/2012 50
Ising Model and Image de-noising • Results Noisy Image Restored Image (ICM) 9/17/2012 51
Ising Model and Image de-noising • Comparing two optimizing algorithms: Graph cuts vs ICM Restored Image (ICM) Restored Image (Graph cuts) 9/17/2012 52
Ising Model and Image de-noising • Home works – Design and implement an image segmentation algorithms using Ising model. 9/17/2012 53
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 9/17/2012 54
Inference 9/17/2012 55
Learning 9/17/2012 56

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Lecture 1 graphical models