Graphical Models. Pradeep Ravikumar Department of Computer Science The University of Texas at Austin

Size: px
Start display at page:

Download "Graphical Models. Pradeep Ravikumar Department of Computer Science The University of Texas at Austin"

Transcription

1 Graphical Models Pradeep Ravikumar Department of Computer Science The University of Texas at Austin

2 Useful References Graphical models, exponential families, and variational inference. M. J. Wainwright and M. I. Jordan. Foundations and Trends in Machine Learning, Vol. 1, Numbers 1--2, pp , December (Available for free online) Graphical Models with R. Soren Hojsgaard, David Edwards, Steffen Lauritzen Probabilistic Graphical Models: Principles and Techniques. D. Koller and N. Friedman.

3 Applications of Graphical Models Computer vision and graphics Natural language processing Information retrieval Robotic control Computational biology Medical diagnosis Finance and economics Error-control codes Decision making under uncertainty...

4 Speech Recognition

5

6 Bioinformatics

7

8

9

10 Computer Vision

11

12

13 Abstraction Layers and Interfaces Internet Protocol Layers Computer Architecture Layers

14 Abstraction Layers and Interfaces Internet Protocol Layers Computer Architecture Layers Interface layer for machine learning?

15 Abstraction Layers and Interfaces Internet Protocol Layers Computer Architecture Layers Interface layer for machine learning? Graphical Models!

16 Graphical Models: The Why Compact way of representing distributions among random variables

17 Graphical Models: The Why Compact way of representing distributions among random variables Visually appealing way (accessible to domain experts) of representing distributions among random variables

18 Graphical Models: The Why Compact way of representing distributions among random variables Visually appealing way (accessible to domain experts) of representing distributions among random variables They represent distributions among random variables probability theory

19 Graphical Models: The Why Compact way of representing distributions among random variables Visually appealing way (accessible to domain experts) of representing distributions among random variables They represent distributions among random variables probability theory They use graphs to do so graph theory

20 Graphical Models: The Why Compact way of representing distributions among random variables Visually appealing way (accessible to domain experts) of representing distributions among random variables They represent distributions among random variables probability theory They use graphs to do so graph theory A marriage of graph and probability theory

21 Graphical Models: Representation

22 Joint Distributions Consider a set of random variables {X 1,X 2,...,X n } Let x i be a realization of random variable X i ; x i may be real, discrete, vector in vector space; for now assume variables are discrete. Probability Mass Function p(x 1,...,x n ):=P (X 1 = x 1,...,X n = x n ). Shorthand: X =(X 1,...,X n ), x =(x 1,...,x n ), so that p(x) =P (X = x).

23

24

25

26

27 Directed Graphs Let V = {1, 2,...,n}. Given a set of non-negative functions {f i (x i,x i ): i 2 V }, each of which sum to one as a function of first argument, we define a joint probability distribution: p(x 1,...,x n ):= Exercise: Calculate p(x i x i ). ny f i (x i,x i ). i=1

28 Directed Graphs Let V = {1, 2,...,n}. Given a set of non-negative functions {f i (x i,x i ): i 2 V }, each of which sum to one as a function of first argument, we define a joint probability distribution: p(x 1,...,x n ):= Exercise: Calculate p(x i x i ). ny f i (x i,x i ). i=1 Can be shown that p(x i x i )=f i (x i,x i ).

29 Directed Graphs Let V = {1, 2,...,n}. Given a set of non-negative functions {f i (x i,x i ): i 2 V }, each of which sum to one as a function of first argument, we define a joint probability distribution: p(x 1,...,x n ):= Exercise: Calculate p(x i x i ). ny f i (x i,x i ). i=1 Can be shown that p(x i x i )=f i (x i,x i ).

30

31

32

33 Representational Economy Directed graphical model distributions thus can be represented or stored much more cheaply than a general distribution What s the catch? The graphical model distribution satisfy certain conditional independence assumptions

34

35

36

37

38

39

40

41

42

43 Graphical Models and Conditional Independences A Graphical Model is a family of probability distributions. Given a graph, specify any set of conditional distributions of nodes given parents.

44 Graphical Models and Conditional Independences A Graphical Model is a family of probability distributions. Given a graph, specify any set of conditional distributions of nodes given parents. Any such distribution satisfies conditional independences

45 Graphical Models and Conditional Independences A Graphical Model is a family of probability distributions. Given a graph, specify any set of conditional distributions of nodes given parents. Any such distribution satisfies conditional independences We will now provide an algorithm (Bayes-Ball, d-separation) to list all conditional independence statements satisfied by the given family of graphical model distributions

46 Three Canonical Graphs

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66 Is there a graph-theoretic way to write down all conditional independence statements given graph?

67

68 Blocking Rules I X Y Z X Y Z

69

70

71

72

73

74

75

76

77

78

79

80 Undirected Graphical Models

81

82

83 Undirected Graphical Models In the directed graphical model case, we started with a factorized form before discussing conditional independence assumptions because the former was more natural (conditional probabilities vs Bayes-Ball algorithm) In the undirected graphical model case, the conditional independence assertions are much more natural and intuitive Just depend on classical graph separation (contrast with blocking rules) X_A is cond. independent of X_C given X_B, if nodes in X_A are separated from nodes in X_C given nodes in X_B Remove nodes in X_B: no path from nodes in X_A to those in X_B

84

85 Undirected Graphical Models: Parameterization

86 Undirected Graphical Models: Parameterization Given an undirected graph, we can define the set of conditional independence assertions using graph separation as above

87 Undirected Graphical Models: Parameterization Given an undirected graph, we can define the set of conditional independence assertions using graph separation as above the set of distributions that satisfy all these assertions is the undirected graphical model family given the graph

88 Undirected Graphical Models: Parameterization Given an undirected graph, we can define the set of conditional independence assertions using graph separation as above the set of distributions that satisfy all these assertions is the undirected graphical model family given the graph We would like to obtain a local parameterization of an undirected graphical model

89 Undirected Graphical Models: Parameterization Given an undirected graph, we can define the set of conditional independence assertions using graph separation as above the set of distributions that satisfy all these assertions is the undirected graphical model family given the graph We would like to obtain a local parameterization of an undirected graphical model Why? For compact representation of joint distribution!

90 Undirected Graphical Models: Parameterization Given an undirected graph, we can define the set of conditional independence assertions using graph separation as above the set of distributions that satisfy all these assertions is the undirected graphical model family given the graph We would like to obtain a local parameterization of an undirected graphical model Why? For compact representation of joint distribution! For directed graphs, the parameterization was based on local conditional probabilities of nodes given their parents

91 Undirected Graphical Models: Parameterization Given an undirected graph, we can define the set of conditional independence assertions using graph separation as above the set of distributions that satisfy all these assertions is the undirected graphical model family given the graph We would like to obtain a local parameterization of an undirected graphical model Why? For compact representation of joint distribution! For directed graphs, the parameterization was based on local conditional probabilities of nodes given their parents local had the interpretation of a set consisting of a node and its parents

92 Undirected Graphical Models: Parameterization We would like to obtain a local parameterization of an undirected graphical model Conditional probabilities of nodes given neighbors? Hard to ensure consistency of such conditional probabilities across nodes, so won t be able to choose such functions independently for each node

93 Undirected Graphical Models: Parameterization We would like to obtain a local parameterization of an undirected graphical model Conditional probabilities of nodes given neighbors? Hard to ensure consistency of such conditional probabilities across nodes, so won t be able to choose such functions independently for each node Their product need not be a valid joint distribution

94 Undirected Graphical Models: Parameterization How do we use conditional probabilities in the undirected graphical model case (where there are no parents)? Conditional probabilities of nodes given neighbors? Hard to ensure consistency of such conditional probabilities across nodes, so won t be able to choose such functions independently for each node Their product need not be a valid joint distribution Better to not use conditional probabilities, but to use product of some other local functions that can be chosen independently

95 Conditional Independence and Factorization If X A?? X C X B,then P (X A X B,X C )=P (X A X C ) P (X A,X B,X C )=P (X A X C ) P (X B,X C ) The joint distribution P (X A,X B,X C )=f(x A,X B ) g(x B,X C ), for some functions f and g. Conditional Independence entails factorization!

96 Conditional Independence and Factorization If X A?? X C X B,then P (X A X B,X C )=P (X A X C ) P (X A,X B,X C )=P (X A X C ) P (X B,X C ) The joint distribution P (X A,X B,X C )=f(x A,X B ) g(x B,X C ), for some functions f and g. Conditional Independence entails factorization! If there is no edge between X i and X j, then the joint distribution cannot have a factor such as (X i,x j,x k ). Factors of the joint distribution can depend only on variables within a fully connected subgraph, also called a clique. Without loss of generality, assume factors only over maximum cliques

97 Conditional Independence and Factorization If X A?? X C X B,then P (X A X B,X C )=P (X A X C ) P (X A,X B,X C )=P (X A X C ) P (X B,X C ) The joint distribution P (X A,X B,X C )=f(x A,X B ) g(x B,X C ), for some functions f and g. Conditional Independence entails factorization! If there is no edge between X i and X j, then the joint distribution cannot have a factor such as (X i,x j,x k ). Factors of the joint distribution can depend only on variables within a fully connected subgraph, also called a clique. Without loss of generality, assume factors only over maximum cliques

98 Conditional Independence and Factorization If X A?? X C X B,then P (X A X B,X C )=P (X A X C ) P (X A,X B,X C )=P (X A X C ) P (X B,X C ) The joint distribution P (X A,X B,X C )=f(x A,X B ) g(x B,X C ), for some functions f and g. Conditional Independence entails factorization! If there is no edge between X i and X j, then the joint distribution cannot have a factor such as (X i,x j,x k ). Factors of the joint distribution can depend only on variables within a fully connected subgraph, also called a clique. Without loss of generality, assume factors only over maximum cliques

99

100 Undirected Graphical Models: Factorization x x x x X 2 X 4 X 1 X 6 0 x 1 1 x X 3 0 x 3 1 x X 5 0 x 5 1 x x 2

101 What do potential functions mean? We just discussed how local conditional probabilities are not satisfactory as potential functions Why not replace potential functions C (x C ) with marginal probabilities p(x C )?

102

103 What do potential functions mean? Potential functions are neither conditional nor marginal probabilities, and do not have a local probabilistic interpretation But do often have a pre-probabilistic interpretation as agreement, constraint or energy Potential function favors certain configurations by assigning larger value

104

105 Dual Characterization of Undirected Graphical Models Characterization I: Given a graph G, all distributions expressed as product of potential functions over maximal cliques Characterization II: Given a graph G, all distributions that satisfy all conditional independence statements specified by graph separation Hammersley Clifford Theorem: Characterizations are equivalent! As in the directed case, a profound link between probability theory and graph theory

106 Beyond Directed/Undirected Graphical Models

107

108

109

110

111

112

113

114 Factorization Factorization is a richer concept than conditional independence There exist families of distributions defined in terms of factorizations that cannot be captured solely within (un)directed graphical model formalism. From point of view of graphical model inference algorithms, not a problem: would work for any distribution within any sub-family defined on graph. But by missing out on algebraic structure, might not have the most e - cient inference algorithm possible

115 Factorization Factorization is a richer concept than conditional independence There exist families of distributions defined in terms of factorizations that cannot be captured solely within (un)directed graphical model formalism. From point of view of graphical model inference algorithms, not a problem: would work for any distribution within any sub-family defined on graph. But by missing out on algebraic structure, might not have the most e - cient inference algorithm possible We will soon see factor graphs, a graphical representation that makes such reduced parameterizations explicit, and in particular could capture the factorized family example earlier Factor graphs provide nothing new in terms of exploiting conditional independences, but provide a way to exploit algebraic relationships.

116

117 Factor Graphs Given a set of variables {x 1,...,x n }, let C denote a set of subsets of {1,...,n}. Example: C = {{1, 3}, {3, 4}, {2, 4, 5}, {1, 3}}. Denote: C = {C s : s 2 F}. To each index s 2 F, associate factor f s (x Cs ), function on variables indexed by C s. Example above: with index set F = {a, b, c, d}, factors are f a (x 1,x 3 ),f b (x 3,x 4 ),f 3 (x 2,x 4,x 5 ),f d (x 1,x 3 ) Note that C is an arbitrary collection of subsets of indices; need not correspond to cliques of some graph (note that we have not even specified any graph at this point!)

118 Factor Graphs Such factorized forms occur in many areas of mathematics, and has applications outside probability theory Our interest is in application to probability distributions of course Can see that both undirected and directed graphical models can also be written as a product of factors We have yet not defined the factor graph a graphical representation of the factored distribution above

119 Factor Graph Bipartite graph G(V, F, E), where vertices V index variables, the vertices F index the factors. The edges E: each factor node s 2 F linked to all variable nodes in C s ; these are the only edges.

120

121

122 Graphical Models: Inference

123 Probabilistic Inference Let E and F be disjoint subsets of nodes of a graphical model; with XE and XF the disjoint subsets of variables corresponding to E, F We want to calculate P(xF xe) for arbitrary subsets E and F This is the general probabilistic inference problem We will focus initially on a single query node F, and directed graphical models

124

125

126 Simplest Case X Y

127 Simplest Case X X Y Y P(Y X) is specified directly by the conditional probability table

128

129

130

131

132

133

134 Inference Algorithms Exact Algorithms: these solve the inference problem exactly Caveat: Time taken is in general exponential in tree-width of graph (which is typically not constant, and scales poorly with the number of nodes) Variable Elimination, Sum-Product (for tree-structured graphs) Approximate algorithms: these solve the inference problem approximately While time taken is linear/quadratic in number of nodes, even to obtain methods with approximation guarantees can be shown to be NP-hard Sum-Product (for general graphs)

135 Sum Product Message-Passing Protocol: A node can send a message to a neighboring node when (and only when) it has received messages from all of its remaining neighbors

136 Sum Product Message-Passing Protocol: A node can send a message to a neighboring node when (and only when) it has received messages from all of its remaining neighbors Two ways to implement this protocol: Parallel From leaves to root, and from root to leaves (as in previous diagram, this will provide messages along both directions for any edge)

137

138 Sum Product: Parallel (a)

139 Sum Product: Parallel (a) (b)

140

141 Sum Product: Parallel (a) (b) (c) (d)

142

143 Graphical Models: Learning

144 We will focus on pairwise graphical models p(x;,g)= 1 Z( ) exp (s,t) E(G) st st (X s,x t ) st(x s,x t ) : arbitrary potential functions Ising x s x t Potts I(x s = x t ) Indicator I(x s,x t = j, k)

145

146

147

148

149

150 Learning Graphical Models

151 Learning Graphical Models Two Step Procedures:

152 Learning Graphical Models Two Step Procedures: 1. Model Selection; estimate graph structure

153 Learning Graphical Models Two Step Procedures: 1. Model Selection; estimate graph structure 2. Parameter Inference given graph structure

154 Learning Graphical Models Two Step Procedures: 1. Model Selection; estimate graph structure 2. Parameter Inference given graph structure Score Based Approaches: search over space of graphs, with a score for graph based on parameter inference

155 Learning Graphical Models Two Step Procedures: 1. Model Selection; estimate graph structure 2. Parameter Inference given graph structure Score Based Approaches: search over space of graphs, with a score for graph based on parameter inference Constraint-based Approaches: estimate individual edges by hypothesis tests for conditional independences

156 Learning Graphical Models Two Step Procedures: 1. Model Selection; estimate graph structure 2. Parameter Inference given graph structure Score Based Approaches: search over space of graphs, with a score for graph based on parameter inference Constraint-based Approaches: estimate individual edges by hypothesis tests for conditional independences Caveats: (a) difficult to provide guarantees for estimators; (b) estimators are NP- Hard

157 Learning Graphical Models State of the art methods are based on estimating neighborhoods Via high-dimensional statistical model estimation Via high-dimensional hypothesis tests

Lecture 3: Conditional Independence - Undirected

Lecture 3: Conditional Independence - Undirected CS598: Graphical Models, Fall 2016 Lecture 3: Conditional Independence - Undirected Lecturer: Sanmi Koyejo Scribe: Nate Bowman and Erin Carrier, Aug. 30, 2016 1 Review for the Bayes-Ball Algorithm Recall

More information

Graphical Models. David M. Blei Columbia University. September 17, 2014

Graphical Models. David M. Blei Columbia University. September 17, 2014 Graphical Models David M. Blei Columbia University September 17, 2014 These lecture notes follow the ideas in Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan. In addition,

More information

The Basics of Graphical Models

The Basics of Graphical Models The Basics of Graphical Models David M. Blei Columbia University September 30, 2016 1 Introduction (These notes follow Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan.

More information

CS242: Probabilistic Graphical Models Lecture 3: Factor Graphs & Variable Elimination

CS242: Probabilistic Graphical Models Lecture 3: Factor Graphs & Variable Elimination CS242: Probabilistic Graphical Models Lecture 3: Factor Graphs & Variable Elimination Instructor: Erik Sudderth Brown University Computer Science September 11, 2014 Some figures and materials courtesy

More information

Loopy Belief Propagation

Loopy Belief Propagation Loopy Belief Propagation Research Exam Kristin Branson September 29, 2003 Loopy Belief Propagation p.1/73 Problem Formalization Reasoning about any real-world problem requires assumptions about the structure

More information

Computer Vision Group Prof. Daniel Cremers. 4. Probabilistic Graphical Models Directed Models

Computer Vision Group Prof. Daniel Cremers. 4. Probabilistic Graphical Models Directed Models Prof. Daniel Cremers 4. Probabilistic Graphical Models Directed Models The Bayes Filter (Rep.) (Bayes) (Markov) (Tot. prob.) (Markov) (Markov) 2 Graphical Representation (Rep.) We can describe the overall

More information

2. Graphical Models. Undirected graphical models. Factor graphs. Bayesian networks. Conversion between graphical models. Graphical Models 2-1

2. Graphical Models. Undirected graphical models. Factor graphs. Bayesian networks. Conversion between graphical models. Graphical Models 2-1 Graphical Models 2-1 2. Graphical Models Undirected graphical models Factor graphs Bayesian networks Conversion between graphical models Graphical Models 2-2 Graphical models There are three families of

More information

Lecture 5: Exact inference. Queries. Complexity of inference. Queries (continued) Bayesian networks can answer questions about the underlying

Lecture 5: Exact inference. Queries. Complexity of inference. Queries (continued) Bayesian networks can answer questions about the underlying given that Maximum a posteriori (MAP query: given evidence 2 which has the highest probability: instantiation of all other variables in the network,, Most probable evidence (MPE: given evidence, find an

More information

D-Separation. b) the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, are in the set C.

D-Separation. b) the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, are in the set C. D-Separation Say: A, B, and C are non-intersecting subsets of nodes in a directed graph. A path from A to B is blocked by C if it contains a node such that either a) the arrows on the path meet either

More information

Lecture 5: Exact inference

Lecture 5: Exact inference Lecture 5: Exact inference Queries Inference in chains Variable elimination Without evidence With evidence Complexity of variable elimination which has the highest probability: instantiation of all other

More information

Probabilistic Graphical Models

Probabilistic Graphical Models School of Computer Science Probabilistic Graphical Models Theory of Variational Inference: Inner and Outer Approximation Eric Xing Lecture 14, February 29, 2016 Reading: W & J Book Chapters Eric Xing @

More information

Part II. C. M. Bishop PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS

Part II. C. M. Bishop PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS Part II C. M. Bishop PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS Converting Directed to Undirected Graphs (1) Converting Directed to Undirected Graphs (2) Add extra links between

More information

Computer Vision Group Prof. Daniel Cremers. 4. Probabilistic Graphical Models Directed Models

Computer Vision Group Prof. Daniel Cremers. 4. Probabilistic Graphical Models Directed Models Prof. Daniel Cremers 4. Probabilistic Graphical Models Directed Models The Bayes Filter (Rep.) (Bayes) (Markov) (Tot. prob.) (Markov) (Markov) 2 Graphical Representation (Rep.) We can describe the overall

More information

Graphical models and message-passing algorithms: Some introductory lectures

Graphical models and message-passing algorithms: Some introductory lectures Graphical models and message-passing algorithms: Some introductory lectures Martin J. Wainwright 1 Introduction Graphical models provide a framework for describing statistical dependencies in (possibly

More information

FMA901F: Machine Learning Lecture 6: Graphical Models. Cristian Sminchisescu

FMA901F: Machine Learning Lecture 6: Graphical Models. Cristian Sminchisescu FMA901F: Machine Learning Lecture 6: Graphical Models Cristian Sminchisescu Graphical Models Provide a simple way to visualize the structure of a probabilistic model and can be used to design and motivate

More information

Expectation Propagation

Expectation Propagation Expectation Propagation Erik Sudderth 6.975 Week 11 Presentation November 20, 2002 Introduction Goal: Efficiently approximate intractable distributions Features of Expectation Propagation (EP): Deterministic,

More information

3 : Representation of Undirected GMs

3 : Representation of Undirected GMs 0-708: Probabilistic Graphical Models 0-708, Spring 202 3 : Representation of Undirected GMs Lecturer: Eric P. Xing Scribes: Nicole Rafidi, Kirstin Early Last Time In the last lecture, we discussed directed

More information

Undirected Graphical Models. Raul Queiroz Feitosa

Undirected Graphical Models. Raul Queiroz Feitosa Undirected Graphical Models Raul Queiroz Feitosa Pros and Cons Advantages of UGMs over DGMs UGMs are more natural for some domains (e.g. context-dependent entities) Discriminative UGMs (CRF) are better

More information

Bipartite Roots of Graphs

Bipartite Roots of Graphs Bipartite Roots of Graphs Lap Chi Lau Department of Computer Science University of Toronto Graph H is a root of graph G if there exists a positive integer k such that x and y are adjacent in G if and only

More information

Recall from last time. Lecture 4: Wrap-up of Bayes net representation. Markov networks. Markov blanket. Isolating a node

Recall from last time. Lecture 4: Wrap-up of Bayes net representation. Markov networks. Markov blanket. Isolating a node Recall from last time Lecture 4: Wrap-up of Bayes net representation. Markov networks Markov blanket, moral graph Independence maps and perfect maps Undirected graphical models (Markov networks) A Bayes

More information

ECE 6504: Advanced Topics in Machine Learning Probabilistic Graphical Models and Large-Scale Learning

ECE 6504: Advanced Topics in Machine Learning Probabilistic Graphical Models and Large-Scale Learning ECE 6504: Advanced Topics in Machine Learning Probabilistic Graphical Models and Large-Scale Learning Topics Bayes Nets: Inference (Finish) Variable Elimination Graph-view of VE: Fill-edges, induced width

More information

6 : Factor Graphs, Message Passing and Junction Trees

6 : Factor Graphs, Message Passing and Junction Trees 10-708: Probabilistic Graphical Models 10-708, Spring 2018 6 : Factor Graphs, Message Passing and Junction Trees Lecturer: Kayhan Batmanghelich Scribes: Sarthak Garg 1 Factor Graphs Factor Graphs are graphical

More information

Chapter 8 of Bishop's Book: Graphical Models

Chapter 8 of Bishop's Book: Graphical Models Chapter 8 of Bishop's Book: Graphical Models Review of Probability Probability density over possible values of x Used to find probability of x falling in some range For continuous variables, the probability

More information

CS242: Probabilistic Graphical Models Lecture 2B: Loopy Belief Propagation & Junction Trees

CS242: Probabilistic Graphical Models Lecture 2B: Loopy Belief Propagation & Junction Trees CS242: Probabilistic Graphical Models Lecture 2B: Loopy Belief Propagation & Junction Trees Professor Erik Sudderth Brown University Computer Science September 22, 2016 Some figures and materials courtesy

More information

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS 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

More information

OSU CS 536 Probabilistic Graphical Models. Loopy Belief Propagation and Clique Trees / Join Trees

OSU CS 536 Probabilistic Graphical Models. Loopy Belief Propagation and Clique Trees / Join Trees OSU CS 536 Probabilistic Graphical Models Loopy Belief Propagation and Clique Trees / Join Trees Slides from Kevin Murphy s Graphical Model Tutorial (with minor changes) Reading: Koller and Friedman Ch

More information

These notes present some properties of chordal graphs, a set of undirected graphs that are important for undirected graphical models.

These notes present some properties of chordal graphs, a set of undirected graphs that are important for undirected graphical models. Undirected Graphical Models: Chordal Graphs, Decomposable Graphs, Junction Trees, and Factorizations Peter Bartlett. October 2003. These notes present some properties of chordal graphs, a set of undirected

More information

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Algorithms For Inference Fall 2014

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Algorithms For Inference Fall 2014 Suggested Reading: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.438 Algorithms For Inference Fall 2014 Probabilistic Modelling and Reasoning: The Junction

More information

COS 513: Foundations of Probabilistic Modeling. Lecture 5

COS 513: Foundations of Probabilistic Modeling. Lecture 5 COS 513: Foundations of Probabilistic Modeling Young-suk Lee 1 Administrative Midterm report is due Oct. 29 th. Recitation is at 4:26pm in Friend 108. Lecture 5 R is a computer language for statistical

More information

Lecture 4: Undirected Graphical Models

Lecture 4: Undirected Graphical Models Lecture 4: Undirected Graphical Models Department of Biostatistics University of Michigan zhenkewu@umich.edu http://zhenkewu.com/teaching/graphical_model 15 September, 2016 Zhenke Wu BIOSTAT830 Graphical

More information

STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 5 Inference

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Overview of Part One Probabilistic Graphical Models Part One: Graphs and Markov Properties Christopher M. Bishop Graphs and probabilities Directed graphs Markov properties Undirected graphs Examples Microsoft

More information

Exact Inference: Elimination and Sum Product (and hidden Markov models)

Exact Inference: Elimination and Sum Product (and hidden Markov models) Exact Inference: Elimination and Sum Product (and hidden Markov models) David M. Blei Columbia University October 13, 2015 The first sections of these lecture notes follow the ideas in Chapters 3 and 4

More information

Machine Learning. Sourangshu Bhattacharya

Machine Learning. Sourangshu Bhattacharya Machine Learning Sourangshu Bhattacharya Bayesian Networks Directed Acyclic Graph (DAG) Bayesian Networks General Factorization Curve Fitting Re-visited Maximum Likelihood Determine by minimizing sum-of-squares

More information

arxiv: v1 [cs.dm] 21 Dec 2015

arxiv: v1 [cs.dm] 21 Dec 2015 The Maximum Cardinality Cut Problem is Polynomial in Proper Interval Graphs Arman Boyacı 1, Tinaz Ekim 1, and Mordechai Shalom 1 Department of Industrial Engineering, Boğaziçi University, Istanbul, Turkey

More information

4 Factor graphs and Comparing Graphical Model Types

4 Factor graphs and Comparing Graphical Model Types Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.438 Algorithms for Inference Fall 2014 4 Factor graphs and Comparing Graphical Model Types We now introduce

More information

V,T C3: S,L,B T C4: A,L,T A,L C5: A,L,B A,B C6: C2: X,A A

V,T C3: S,L,B T C4: A,L,T A,L C5: A,L,B A,B C6: C2: X,A A Inference II Daphne Koller Stanford University CS228 Handout #13 In the previous chapter, we showed how efficient inference can be done in a BN using an algorithm called Variable Elimination, that sums

More information

Computer Vision Group Prof. Daniel Cremers. 4a. Inference in Graphical Models

Computer Vision Group Prof. Daniel Cremers. 4a. Inference in Graphical Models Group Prof. Daniel Cremers 4a. Inference in Graphical Models Inference on a Chain (Rep.) The first values of µ α and µ β are: The partition function can be computed at any node: Overall, we have O(NK 2

More information

Computer vision: models, learning and inference. Chapter 10 Graphical Models

Computer vision: models, learning and inference. Chapter 10 Graphical Models Computer vision: models, learning and inference Chapter 10 Graphical Models Independence Two variables x 1 and x 2 are independent if their joint probability distribution factorizes as Pr(x 1, x 2 )=Pr(x

More information

Ch9: Exact Inference: Variable Elimination. Shimi Salant, Barak Sternberg

Ch9: Exact Inference: Variable Elimination. Shimi Salant, Barak Sternberg Ch9: Exact Inference: Variable Elimination Shimi Salant Barak Sternberg Part 1 Reminder introduction (1/3) We saw two ways to represent (finite discrete) distributions via graphical data structures: Bayesian

More information

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 7, JULY A New Class of Upper Bounds on the Log Partition Function

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 7, JULY A New Class of Upper Bounds on the Log Partition Function IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 7, JULY 2005 2313 A New Class of Upper Bounds on the Log Partition Function Martin J. Wainwright, Member, IEEE, Tommi S. Jaakkola, and Alan S. Willsky,

More information

1 : Introduction to GM and Directed GMs: Bayesian Networks. 3 Multivariate Distributions and Graphical Models

1 : Introduction to GM and Directed GMs: Bayesian Networks. 3 Multivariate Distributions and Graphical Models 10-708: Probabilistic Graphical Models, Spring 2015 1 : Introduction to GM and Directed GMs: Bayesian Networks Lecturer: Eric P. Xing Scribes: Wenbo Liu, Venkata Krishna Pillutla 1 Overview This lecture

More information

Statistical and Learning Techniques in Computer Vision Lecture 1: Markov Random Fields Jens Rittscher and Chuck Stewart

Statistical and Learning Techniques in Computer Vision Lecture 1: Markov Random Fields Jens Rittscher and Chuck Stewart Statistical and Learning Techniques in Computer Vision Lecture 1: Markov Random Fields Jens Rittscher and Chuck Stewart 1 Motivation Up to now we have considered distributions of a single random variable

More information

Workshop report 1. Daniels report is on website 2. Don t expect to write it based on listening to one project (we had 6 only 2 was sufficient

Workshop report 1. Daniels report is on website 2. Don t expect to write it based on listening to one project (we had 6 only 2 was sufficient Workshop report 1. Daniels report is on website 2. Don t expect to write it based on listening to one project (we had 6 only 2 was sufficient quality) 3. I suggest writing it on one presentation. 4. Include

More information

Discrete Wiskunde II. Lecture 6: Planar Graphs

Discrete Wiskunde II. Lecture 6: Planar Graphs , 2009 Lecture 6: Planar Graphs University of Twente m.uetz@utwente.nl wwwhome.math.utwente.nl/~uetzm/dw/ Planar Graphs Given an undirected graph (or multigraph) G = (V, E). A planar embedding of G is

More information

Finding Non-overlapping Clusters for Generalized Inference Over Graphical Models

Finding Non-overlapping Clusters for Generalized Inference Over Graphical Models 1 Finding Non-overlapping Clusters for Generalized Inference Over Graphical Models Divyanshu Vats and José M. F. Moura arxiv:1107.4067v2 [stat.ml] 18 Mar 2012 Abstract Graphical models use graphs to compactly

More information

Information Processing Letters

Information Processing Letters Information Processing Letters 112 (2012) 449 456 Contents lists available at SciVerse ScienceDirect Information Processing Letters www.elsevier.com/locate/ipl Recursive sum product algorithm for generalized

More information

Recitation 4: Elimination algorithm, reconstituted graph, triangulation

Recitation 4: Elimination algorithm, reconstituted graph, triangulation Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.438 Algorithms For Inference Fall 2014 Recitation 4: Elimination algorithm, reconstituted graph, triangulation

More information

Faster parameterized algorithms for Minimum Fill-In

Faster parameterized algorithms for Minimum Fill-In Faster parameterized algorithms for Minimum Fill-In Hans L. Bodlaender Pinar Heggernes Yngve Villanger Technical Report UU-CS-2008-042 December 2008 Department of Information and Computing Sciences Utrecht

More information

CS 343: Artificial Intelligence

CS 343: Artificial Intelligence CS 343: Artificial Intelligence Bayes Nets: Inference Prof. Scott Niekum The University of Texas at Austin [These slides based on those of Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley.

More information

Collective classification in network data

Collective classification in network data 1 / 50 Collective classification in network data Seminar on graphs, UCSB 2009 Outline 2 / 50 1 Problem 2 Methods Local methods Global methods 3 Experiments Outline 3 / 50 1 Problem 2 Methods Local methods

More information

Bayesian Networks Inference (continued) Learning

Bayesian Networks Inference (continued) Learning Learning BN tutorial: ftp://ftp.research.microsoft.com/pub/tr/tr-95-06.pdf TAN paper: http://www.cs.huji.ac.il/~nir/abstracts/frgg1.html Bayesian Networks Inference (continued) Learning Machine Learning

More information

Regularization and Markov Random Fields (MRF) CS 664 Spring 2008

Regularization and Markov Random Fields (MRF) CS 664 Spring 2008 Regularization and Markov Random Fields (MRF) CS 664 Spring 2008 Regularization in Low Level Vision Low level vision problems concerned with estimating some quantity at each pixel Visual motion (u(x,y),v(x,y))

More information

Lecture 9: Undirected Graphical Models Machine Learning

Lecture 9: Undirected Graphical Models Machine Learning Lecture 9: Undirected Graphical Models Machine Learning Andrew Rosenberg March 5, 2010 1/1 Today Graphical Models Probabilities in Undirected Graphs 2/1 Undirected Graphs What if we allow undirected graphs?

More information

5 Minimal I-Maps, Chordal Graphs, Trees, and Markov Chains

5 Minimal I-Maps, Chordal Graphs, Trees, and Markov Chains Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.438 Algorithms for Inference Fall 2014 5 Minimal I-Maps, Chordal Graphs, Trees, and Markov Chains Recall

More information

Learning Bayesian Networks (part 3) Goals for the lecture

Learning Bayesian Networks (part 3) Goals for the lecture Learning Bayesian Networks (part 3) Mark Craven and David Page Computer Sciences 760 Spring 2018 www.biostat.wisc.edu/~craven/cs760/ Some of the slides in these lectures have been adapted/borrowed from

More information

2. Graphical Models. Undirected pairwise graphical models. Factor graphs. Bayesian networks. Conversion between graphical models. Graphical Models 2-1

2. Graphical Models. Undirected pairwise graphical models. Factor graphs. Bayesian networks. Conversion between graphical models. Graphical Models 2-1 Graphical Models 2-1 2. Graphical Models Undirected pairwise graphical models Factor graphs Bayesian networks Conversion between graphical models Graphical Models 2-2 Graphical models Families of graphical

More information

(67686) Mathematical Foundations of AI July 30, Lecture 11

(67686) Mathematical Foundations of AI July 30, Lecture 11 (67686) Mathematical Foundations of AI July 30, 2008 Lecturer: Ariel D. Procaccia Lecture 11 Scribe: Michael Zuckerman and Na ama Zohary 1 Cooperative Games N = {1,...,n} is the set of players (agents).

More information

Algorithms for Markov Random Fields in Computer Vision

Algorithms for Markov Random Fields in Computer Vision Algorithms for Markov Random Fields in Computer Vision Dan Huttenlocher November, 2003 (Joint work with Pedro Felzenszwalb) Random Field Broadly applicable stochastic model Collection of n sites S Hidden

More information

An Introduction to Graph Theory

An Introduction to Graph Theory An Introduction to Graph Theory CIS008-2 Logic and Foundations of Mathematics David Goodwin david.goodwin@perisic.com 12:00, Friday 17 th February 2012 Outline 1 Graphs 2 Paths and cycles 3 Graphs and

More information

10708 Graphical Models: Homework 2

10708 Graphical Models: Homework 2 10708 Graphical Models: Homework 2 Due October 15th, beginning of class October 1, 2008 Instructions: There are six questions on this assignment. Each question has the name of one of the TAs beside it,

More information

Treewidth and graph minors

Treewidth and graph minors Treewidth and graph minors Lectures 9 and 10, December 29, 2011, January 5, 2012 We shall touch upon the theory of Graph Minors by Robertson and Seymour. This theory gives a very general condition under

More information

Markov Random Fields and Gibbs Sampling for Image Denoising

Markov Random Fields and Gibbs Sampling for Image Denoising Markov Random Fields and Gibbs Sampling for Image Denoising Chang Yue Electrical Engineering Stanford University changyue@stanfoed.edu Abstract This project applies Gibbs Sampling based on different Markov

More information

Lecture 2 - Introduction to Polytopes

Lecture 2 - Introduction to Polytopes Lecture 2 - Introduction to Polytopes Optimization and Approximation - ENS M1 Nicolas Bousquet 1 Reminder of Linear Algebra definitions Let x 1,..., x m be points in R n and λ 1,..., λ m be real numbers.

More information

Bayesian Classification Using Probabilistic Graphical Models

Bayesian Classification Using Probabilistic Graphical Models San Jose State University SJSU ScholarWorks Master's Projects Master's Theses and Graduate Research Spring 2014 Bayesian Classification Using Probabilistic Graphical Models Mehal Patel San Jose State University

More information

Bayesian Networks Inference

Bayesian Networks Inference Bayesian Networks Inference Machine Learning 10701/15781 Carlos Guestrin Carnegie Mellon University November 5 th, 2007 2005-2007 Carlos Guestrin 1 General probabilistic inference Flu Allergy Query: Sinus

More information

Algorithms for Nearest Neighbors

Algorithms for Nearest Neighbors Algorithms for Nearest Neighbors Classic Ideas, New Ideas Yury Lifshits Steklov Institute of Mathematics at St.Petersburg http://logic.pdmi.ras.ru/~yura University of Toronto, July 2007 1 / 39 Outline

More information

10 Sum-product on factor tree graphs, MAP elimination

10 Sum-product on factor tree graphs, MAP elimination assachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.438 Algorithms For Inference Fall 2014 10 Sum-product on factor tree graphs, AP elimination algorithm The

More information

Applied Mathematics Letters. Graph triangulations and the compatibility of unrooted phylogenetic trees

Applied Mathematics Letters. Graph triangulations and the compatibility of unrooted phylogenetic trees Applied Mathematics Letters 24 (2011) 719 723 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Graph triangulations and the compatibility

More information

STAT 598L Probabilistic Graphical Models. Instructor: Sergey Kirshner. Exact Inference

STAT 598L Probabilistic Graphical Models. Instructor: Sergey Kirshner. Exact Inference STAT 598L Probabilistic Graphical Models Instructor: Sergey Kirshner Exact Inference What To Do With Bayesian/Markov Network? Compact representation of a complex model, but Goal: efficient extraction of

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models Raquel Urtasun and Tamir Hazan TTI Chicago April 22, 2011 Raquel Urtasun and Tamir Hazan (TTI-C) Graphical Models April 22, 2011 1 / 22 If the graph is non-chordal, then

More information

1) Give decision trees to represent the following Boolean functions:

1) Give decision trees to represent the following Boolean functions: 1) Give decision trees to represent the following Boolean functions: 1) A B 2) A [B C] 3) A XOR B 4) [A B] [C Dl Answer: 1) A B 2) A [B C] 1 3) A XOR B = (A B) ( A B) 4) [A B] [C D] 2 2) Consider the following

More information

Learning decomposable models with a bounded clique size

Learning decomposable models with a bounded clique size Learning decomposable models with a bounded clique size Achievements 2014-2016 Aritz Pérez Basque Center for Applied Mathematics Bilbao, March, 2016 Outline 1 Motivation and background 2 The problem 3

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models Raquel Urtasun and Tamir Hazan TTI Chicago April 25, 2011 Raquel Urtasun and Tamir Hazan (TTI-C) Graphical Models April 25, 2011 1 / 17 Clique Trees Today we are going to

More information

Exact Algorithms Lecture 7: FPT Hardness and the ETH

Exact Algorithms Lecture 7: FPT Hardness and the ETH Exact Algorithms Lecture 7: FPT Hardness and the ETH February 12, 2016 Lecturer: Michael Lampis 1 Reminder: FPT algorithms Definition 1. A parameterized problem is a function from (χ, k) {0, 1} N to {0,

More information

Small Survey on Perfect Graphs

Small Survey on Perfect Graphs Small Survey on Perfect Graphs Michele Alberti ENS Lyon December 8, 2010 Abstract This is a small survey on the exciting world of Perfect Graphs. We will see when a graph is perfect and which are families

More information

Part I: Sum Product Algorithm and (Loopy) Belief Propagation. What s wrong with VarElim. Forwards algorithm (filtering) Forwards-backwards algorithm

Part I: Sum Product Algorithm and (Loopy) Belief Propagation. What s wrong with VarElim. Forwards algorithm (filtering) Forwards-backwards algorithm OU 56 Probabilistic Graphical Models Loopy Belief Propagation and lique Trees / Join Trees lides from Kevin Murphy s Graphical Model Tutorial (with minor changes) eading: Koller and Friedman h 0 Part I:

More information

CS 188: Artificial Intelligence

CS 188: Artificial Intelligence CS 188: Artificial Intelligence Bayes Nets: Inference Instructors: Dan Klein and Pieter Abbeel --- University of California, Berkeley [These slides were created by Dan Klein and Pieter Abbeel for CS188

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models Raquel Urtasun and Tamir Hazan TTI Chicago April 8, 2011 Raquel Urtasun and Tamir Hazan (TTI-C) Graphical Models April 8, 2011 1 / 19 Factor Graphs H does not reveal the

More information

Building Classifiers using Bayesian Networks

Building Classifiers using Bayesian Networks Building Classifiers using Bayesian Networks Nir Friedman and Moises Goldszmidt 1997 Presented by Brian Collins and Lukas Seitlinger Paper Summary The Naive Bayes classifier has reasonable performance

More information

Chordal deletion is fixed-parameter tractable

Chordal deletion is fixed-parameter tractable Chordal deletion is fixed-parameter tractable Dániel Marx Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany. dmarx@informatik.hu-berlin.de Abstract. It

More information

CS 534: Computer Vision Segmentation II Graph Cuts and Image Segmentation

CS 534: Computer Vision Segmentation II Graph Cuts and Image Segmentation CS 534: Computer Vision Segmentation II Graph Cuts and Image Segmentation Spring 2005 Ahmed Elgammal Dept of Computer Science CS 534 Segmentation II - 1 Outlines What is Graph cuts Graph-based clustering

More information

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Algorithms For Inference Fall 2014

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Algorithms For Inference Fall 2014 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.438 Algorithms For Inference Fall 2014 Recitation-6: Hardness of Inference Contents 1 NP-Hardness Part-II

More information

Graphs: Introduction. Ali Shokoufandeh, Department of Computer Science, Drexel University

Graphs: Introduction. Ali Shokoufandeh, Department of Computer Science, Drexel University Graphs: Introduction Ali Shokoufandeh, Department of Computer Science, Drexel University Overview of this talk Introduction: Notations and Definitions Graphs and Modeling Algorithmic Graph Theory and Combinatorial

More information

An Effective Upperbound on Treewidth Using Partial Fill-in of Separators

An Effective Upperbound on Treewidth Using Partial Fill-in of Separators An Effective Upperbound on Treewidth Using Partial Fill-in of Separators Boi Faltings Martin Charles Golumbic June 28, 2009 Abstract Partitioning a graph using graph separators, and particularly clique

More information

Recap: Gaussian (or Normal) Distribution. Recap: Minimizing the Expected Loss. Topics of This Lecture. Recap: Maximum Likelihood Approach

Recap: Gaussian (or Normal) Distribution. Recap: Minimizing the Expected Loss. Topics of This Lecture. Recap: Maximum Likelihood Approach Truth Course Outline Machine Learning Lecture 3 Fundamentals (2 weeks) Bayes Decision Theory Probability Density Estimation Probability Density Estimation II 2.04.205 Discriminative Approaches (5 weeks)

More information

COMP90051 Statistical Machine Learning

COMP90051 Statistical Machine Learning COMP90051 Statistical Machine Learning Semester 2, 2016 Lecturer: Trevor Cohn 21. Independence in PGMs; Example PGMs Independence PGMs encode assumption of statistical independence between variables. Critical

More information

Edge Colorings of Complete Multipartite Graphs Forbidding Rainbow Cycles

Edge Colorings of Complete Multipartite Graphs Forbidding Rainbow Cycles Theory and Applications of Graphs Volume 4 Issue 2 Article 2 November 2017 Edge Colorings of Complete Multipartite Graphs Forbidding Rainbow Cycles Peter Johnson johnspd@auburn.edu Andrew Owens Auburn

More information

Decomposition of log-linear models

Decomposition of log-linear models Graphical Models, Lecture 5, Michaelmas Term 2009 October 27, 2009 Generating class Dependence graph of log-linear model Conformal graphical models Factor graphs A density f factorizes w.r.t. A if there

More information

MC 302 GRAPH THEORY 10/1/13 Solutions to HW #2 50 points + 6 XC points

MC 302 GRAPH THEORY 10/1/13 Solutions to HW #2 50 points + 6 XC points MC 0 GRAPH THEORY 0// Solutions to HW # 0 points + XC points ) [CH] p.,..7. This problem introduces an important class of graphs called the hypercubes or k-cubes, Q, Q, Q, etc. I suggest that before you

More information

Matching Algorithms. Proof. If a bipartite graph has a perfect matching, then it is easy to see that the right hand side is a necessary condition.

Matching Algorithms. Proof. If a bipartite graph has a perfect matching, then it is easy to see that the right hand side is a necessary condition. 18.433 Combinatorial Optimization Matching Algorithms September 9,14,16 Lecturer: Santosh Vempala Given a graph G = (V, E), a matching M is a set of edges with the property that no two of the edges have

More information

Sum-Product Networks. STAT946 Deep Learning Guest Lecture by Pascal Poupart University of Waterloo October 15, 2015

Sum-Product Networks. STAT946 Deep Learning Guest Lecture by Pascal Poupart University of Waterloo October 15, 2015 Sum-Product Networks STAT946 Deep Learning Guest Lecture by Pascal Poupart University of Waterloo October 15, 2015 Introduction Outline What is a Sum-Product Network? Inference Applications In more depth

More information

Generative and discriminative classification techniques

Generative and discriminative classification techniques Generative and discriminative classification techniques Machine Learning and Category Representation 013-014 Jakob Verbeek, December 13+0, 013 Course website: http://lear.inrialpes.fr/~verbeek/mlcr.13.14

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models Lecture 8 Junction Trees CS/CNS/EE 155 Andreas Krause Announcements Homework 2 due next Wednesday (Nov 4) in class Start early!!! Project milestones due Monday (Nov 9) 4

More information

Cheng Soon Ong & Christian Walder. Canberra February June 2018

Cheng Soon Ong & Christian Walder. Canberra February June 2018 Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression

More information

Graphical Modeling for High Dimensional Data

Graphical Modeling for High Dimensional Data Journal of Modern Applied Statistical Methods Volume 11 Issue 2 Article 17 11-1-2012 Graphical Modeling for High Dimensional Data Munni Begum Ball State University, Muncie, IN Jay Bagga Ball State University,

More information

Markov Logic: Representation

Markov Logic: Representation Markov Logic: Representation Overview Statistical relational learning Markov logic Basic inference Basic learning Statistical Relational Learning Goals: Combine (subsets of) logic and probability into

More information

Module 7. Independent sets, coverings. and matchings. Contents

Module 7. Independent sets, coverings. and matchings. Contents Module 7 Independent sets, coverings Contents and matchings 7.1 Introduction.......................... 152 7.2 Independent sets and coverings: basic equations..... 152 7.3 Matchings in bipartite graphs................

More information

K 4 C 5. Figure 4.5: Some well known family of graphs

K 4 C 5. Figure 4.5: Some well known family of graphs 08 CHAPTER. TOPICS IN CLASSICAL GRAPH THEORY K, K K K, K K, K K, K C C C C 6 6 P P P P P. Graph Operations Figure.: Some well known family of graphs A graph Y = (V,E ) is said to be a subgraph of a graph

More information

Faster parameterized algorithms for Minimum Fill-In

Faster parameterized algorithms for Minimum Fill-In Faster parameterized algorithms for Minimum Fill-In Hans L. Bodlaender Pinar Heggernes Yngve Villanger Abstract We present two parameterized algorithms for the Minimum Fill-In problem, also known as Chordal

More information