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part ii − application to markov structures. a motivating example ✕✕✕✕✕✕✕✕

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Part II − Application to Markov Structures

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Page 1: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Part II − Application to Markov

Structures

Page 2: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

A Motivating Example

✕ ✕✕ ✕✕ ✕✕ ✕

Page 3: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Conditional Mutual Independence

Page 4: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Example 1

Page 5: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Example 2

Page 6: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Example 3

Page 7: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

____________________________

_________

Page 8: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Full Conditional Mutual Independence

(FCMI)

Page 9: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 10: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Notations

Page 11: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Notations

Page 12: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Notations

Page 13: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Notations

Page 14: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 15: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 16: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 17: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 18: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Illustration of Im(K)

Page 19: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Illustration of Im(K)

Page 20: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 21: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 22: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 23: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 24: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 25: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 26: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 27: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 28: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 29: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

The Implication Problem

Page 30: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

The Implication Problem

Page 31: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

The Implication Problem

Page 32: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

The Implication Problem

Page 33: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

The Implication Problem

Page 34: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 35: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 36: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 37: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 38: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 39: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 40: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 41: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Markov Random Field

Page 42: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Graph Theory Terminologies

Page 43: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Graph Theory Terminologies

Page 44: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Graph Theory Terminologies

Page 45: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Graph Theory Terminologies

Page 46: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Global Markov Property

Page 47: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Global Markov Property

Page 48: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Global Markov Property

Page 49: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Global Markov Property

Page 50: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

FCMIs Specified by a MRF

Page 51: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

FCMIs Specified by a MRF

Page 52: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

FCMIs Specified by a MRF

Page 53: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

FCMIs Specified by a MRF

Page 54: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

FCMIs Specified by a MRF

Page 55: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Type I and Type II Atoms

Page 56: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Type I and Type II Atoms

Page 57: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Type I and Type II Atoms

Page 58: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Type I and Type II Atoms

Page 59: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 60: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

A Graph-Theoretic Analog

Page 61: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 62: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 63: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Markov Chain

Page 64: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Special Properties of a Markov Chain

Page 65: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Special Properties of a Markov Chain

Page 66: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Special Properties of a Markov Chain

Page 67: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕
Page 68: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Construction of Information Diagram

Page 69: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Construction of Information Diagram

Page 70: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Construction of Information Diagram

Page 71: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Construction of Information Diagram

Page 72: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Construction of Information Diagram

Page 73: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Nonnegativity of µ*

Page 74: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Nonnegativity of µ*

Page 75: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

Nonnegativity of µ*

Page 76: Part II − Application to Markov Structures. A Motivating Example ✕✕✕✕✕✕✕✕

A Formal Justification


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