Independence Alice Gao Lecture 13 Based on work by K. - PowerPoint PPT Presentation

1/17 Independence Alice Gao Lecture 13 Based on work by K. Leyton-Brown, K. Larson, and P. van Beek

2/17 Outline Learning Goals Unconditional and Conditional Independence Revisiting the Learning goals

3/17 Learning Goals By the end of the lecture, you should be able to the domain, determine whether two variables are independent. the domain, determine whether two variables are conditionally independent given a third variable. ▶ Given a description of a domain or a probabilistic model for ▶ Given a description of a domain or a probabilistic model for

4/17 The Holmes Scenario Mr. Holmes lives in a high crime area and therefore has installed a burglar alarm. He relies on his neighbors to phone him when they hear the alarm sound. Mr. Holmes has two neighbors, Dr. Watson and Mrs. Gibbon. Unfortunately, his neighbors are not entirely reliable. Dr. Watson is known to be a tasteless practical joker and Mrs. Gibbon, while more reliable in general, has occasional drinking problems. Mr. Holmes also knows from reading the instruction manual of his alarm system that the device is sensitive to earthquakes and can be triggered by one accidentally. He realizes that if an earthquake has occurred, it would surely be on the radio news.

5/17 Learning Goals Unconditional and Conditional Independence Revisiting the Learning goals

6/17 (Unconditional) Independence Defjnition ((unconditional) independence) X and Y are (unconditionally) independent ifg Learning Y ’s value doesn’t afgect your belief about X . ∀ x , ∀ y , P ( X = x | Y = y ) = P ( X = x ) ∀ x , ∀ y , P ( Y = y | X = x ) = P ( Y = y ) ∀ x , ∀ y , P ( X = x ∧ Y = y ) = P ( X = x ) P ( Y = y )

7/17 Conditional Independence Defjnition (conditional independence) X and Y are conditionally independent given Z if Learning Y ’s value doesn’t afgect your belief about X , knowing the value of Z . ∀ x , ∀ y , ∀ z , P ( X = x | Y = y ∧ Z = z ) = P ( X = x | Z = z ) . ∀ x , ∀ y , ∀ z , P ( Y = y | X = x ∧ Z = z ) = P ( Y = y | Z = z ) . ∀ x , ∀ y , ∀ z , P ( Y = y ∧ X = x | Z = z ) = P ( Y = y | Z = z ) P ( X = x | Z = z ) .

8/17 Burglary, Alarm and Watson Burglary Alarm Watson P ( B ) = 0 . 1 P ( A | B ) = 0 . 9 P ( A |¬ B ) = 0 . 1 P ( W | B ∧ A ) = 0 . 8 P ( W | B ∧ ¬ A ) = 0 . 4 P ( W |¬ B ∧ A ) = 0 . 8 P ( W |¬ B ∧ ¬ A ) = 0 . 4

9/17 CQ Unconditional Independence CQ: Are Burglary and Watson independent? Burglary Alarm Watson (A) Yes (B) No (C) I don’t know. P ( B ) = 0 . 1 P ( A | B ) = 0 . 9 P ( A |¬ B ) = 0 . 1 P ( W | B ∧ A ) = 0 . 8 P ( W | B ∧ ¬ A ) = 0 . 4 P ( W |¬ B ∧ A ) = 0 . 8 P ( W |¬ B ∧ ¬ A ) = 0 . 4

10/17 Watson (B) No (A) Yes CQ: Conditional Independence (C) I don’t know. Alarm Burglary Alarm? CQ: Are Burglary and Watson conditionally independent given P ( B ) = 0 . 1 P ( A | B ) = 0 . 9 P ( A |¬ B ) = 0 . 1 P ( W | B ∧ A ) = 0 . 8 P ( W | B ∧ ¬ A ) = 0 . 4 P ( W |¬ B ∧ A ) = 0 . 8 P ( W |¬ B ∧ ¬ A ) = 0 . 4

11/17 Alarm, Watson and Gibbon Alarm Watson Gibbon P ( A ) = 0 . 1 P ( W | A ) = 0 . 8 P ( W |¬ A ) = 0 . 4 P ( G | W ∧ A ) = 0 . 4 P ( G | W ∧ ¬ A ) = 0 . 1 P ( G |¬ W ∧ A ) = 0 . 4 P ( G |¬ W ∧ ¬ A ) = 0 . 1

12/17 CQ Unconditional Independence CQ: Are Watson and Gibbon independent? Alarm Watson Gibbon (A) Yes (B) No (C) I don’t know. P ( A ) = 0 . 1 P ( W | A ) = 0 . 8 P ( W |¬ A ) = 0 . 4 P ( G | W ∧ A ) = 0 . 4 P ( G | W ∧ ¬ A ) = 0 . 1 P ( G |¬ W ∧ A ) = 0 . 4 P ( G |¬ W ∧ ¬ A ) = 0 . 1

13/17 Gibbon (B) No (A) Yes CQ Conditional Independence (C) I don’t know. Watson Alarm Alarm? CQ: Are Watson and Gibbon conditionally independent given P ( A ) = 0 . 1 P ( W | A ) = 0 . 8 P ( W |¬ A ) = 0 . 4 P ( G | W ∧ A ) = 0 . 4 P ( G | W ∧ ¬ A ) = 0 . 1 P ( G |¬ W ∧ A ) = 0 . 4 P ( G |¬ W ∧ ¬ A ) = 0 . 1

14/17 Earthquake, Burglary, and Alarm Alarm Earthquake Burglary P ( E ) = 0 . 1 P ( B | E ) = 0 . 2 P ( B |¬ E ) = 0 . 2 P ( A | B ∧ E ) = 0 . 9 P ( A | B ∧ ¬ E ) = 0 . 8 P ( A |¬ B ∧ E ) = 0 . 2 P ( A |¬ B ∧ ¬ E ) = 0 . 1

15/17 CQ Unconditional Independence CQ: Are Earthquake and Burglary independent? Alarm Earthquake Burglary (A) Yes (B) No (C) I don’t know. P ( E ) = 0 . 1 P ( B | E ) = 0 . 2 P ( B |¬ E ) = 0 . 2 P ( A | B ∧ E ) = 0 . 9 P ( A | B ∧ ¬ E ) = 0 . 8 P ( A |¬ B ∧ E ) = 0 . 2 P ( A |¬ B ∧ ¬ E ) = 0 . 1

16/17 Burglary (B) No (A) Yes CQ: Conditional Independence (C) I don’t know. Earthquake Alarm Alarm? CQ: Are Earthquake and Burglary conditionally independent given P ( E ) = 0 . 1 P ( B | E ) = 0 . 2 P ( B |¬ E ) = 0 . 2 P ( A | B ∧ E ) = 0 . 9 P ( A | B ∧ ¬ E ) = 0 . 8 P ( A |¬ B ∧ E ) = 0 . 2 P ( A |¬ B ∧ ¬ E ) = 0 . 1

17/17 Revisiting the Learning Goals By the end of the lecture, you should be able to the domain, determine whether two variables are independent. the domain, determine whether two variables are conditionally independent given a third variable. ▶ Given a description of a domain or a probabilistic model for ▶ Given a description of a domain or a probabilistic model for