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1 Department o Computer Science Undergraduate Eents More Masters o Management in Operations Research Ino Session Date: Tues. No 20 Time: 2 pm Location: Kaiser 2020/ Main Mall Grad School Panel Date: Thurs. No 22 Time: 2:30 2 pm Location: Rm X836 ICICS/CS Bldg. Volunteer or Experience Science Day Date: Fri. No 23 Time: 0 am or am 2 pm or 2 pm Location: ICICS/CS Bldg. RSVP: undergrad-ino@cs.ubc.ca i you are interested.

2 Reasoning Under Uncertainty: Variable elimination Computer Science cpsc322 Lecture 30 (Textbook Chpt 6.4) No Couple o questions with cards CPSC 322 Lecture 30 Slide 2

3 Lecture Oeriew Recap Intro Variable Elimination Variable Elimination Simpliications Example Independence Where are we? CPSC 322 Lecture 30 Slide 3

4 CPSC 322 Lecture 29 Slide 4 Bnet Inerence: General Suppose the ariables o the belie network are X X n. is the query ariable = = are the obsered ariables (with their alues) k are the remaining ariables What we want to compute: ) ( P P P P P P ) ( ) ( ) ( ) ( ) ( ) ( P We can actually compute:

5 CPSC 322 Lecture 29 Slide 5 Inerence with Factors We can compute P( = = ) by expressing the oint as a actor ( ) assigning = = and summing out the ariables k ).... ( ) ( k k P

6 CPSC 322 Lecture 29 Slide 6 Variable Elimination Intro () We can express the oint actor as a product o actors Using the chain rule and the deinition o a Bnet we can write P(X X n ) as n i X i px i P ) ( n i X i px i ) ( ) ( ) ( n i i i k px X P ( ) ).... ( ) ( k k P

7 Variable Elimination Intro (2) Inerence in belie networks thus reduces to computing the sums o products. P( n ) ( X i px i) k i. Construct a actor or each conditional probability. 2. In each actor assign the obsered ariables to their obsered alues. 3. Multiply the actors 4. For each o the other ariables i { k } sum out i CPSC 322 Lecture 29 Slide 7

8 Lecture Oeriew Recap Intro Variable Elimination Variable Elimination Simpliications Example Independence Where are we? CPSC 322 Lecture 30 Slide 8

9 CPSC 322 Lecture 30 Slide 9 How to simpliy the Computation? (arsxi) n i k Assume we hae turned the CPTs into actors and perormed the assignments (arsx ) ) ( i px X i i? ) ( t D G G C ) ( n i i i k px X Let s ocus on the basic case or instance ) ( ) ( ) ( ) ( D A E D B A D C A

10 How to simpliy: basic case Let s ocus on the basic case. A n i How can we compute eiciently? (arsxi) ( C D) ( A B D) ( E A) ( D) Factor out those terms that don't inole! (arsx ) i (arsxi) i arsxi i arsxi CPSC 322 Lecture 30 Slide 0

11 CPSC 322 Lecture 30 Slide General case: Summing out ariables eiciently 2 ) ( h i i h k k 2... i k Now to sum out a ariable 2 rom a product i o actors again partition the actors into two sets F: those that F: those that

12 Analogy with Computing sums o products This simpliication is similar to what you can do in basic algebra with multiplication and addition It takes 4 multiplications or additions to ealuate the expression a b + a c + a d + a e h + a h + a g h. This expression be ealuated more eiciently. CPSC 322 Lecture 30 Slide 2

13 Variable elimination ordering Is there only one way to simpliy? P(GD=t) = ABC (AG) (BA) (CG) (BC) P(GD=t) = A (AG) B (BA) C (CG) (BC) P(GD=t) = A (AG) C (CG) B (BC) (BA) CPSC 322 Lecture 30 Slide 3

14 Variable elimination algorithm: Summary P( ) To compute P( = = ) :. Construct a actor or each conditional probability. 2. Set the obsered ariables to their obsered alues. 3. Gien an elimination ordering simpliy/decompose sum o products 4. Perorm products and sum out i = 5. Multiply the remaining actors (all in? ) 6. Normalize: diide the resulting actor () by (). 2 2 CPSC 322 Lecture 0 Slide 4

15 Variable elimination algorithm: Summary P( ) To compute P( = = ) :. Construct a actor or each conditional probability. 2. Set the obsered ariables to their obsered alues. 3. Gien an elimination ordering simpliy/decompose sum o products 4. Perorm products and sum out i 5. Multiply the remaining actors (all in? ) 6. Normalize: diide the resulting actor () by (). CPSC 322 Lecture 0 Slide 5

16 Lecture Oeriew Recap Intro Variable Elimination Variable Elimination Simpliications Example Independence Where are we? CPSC 322 Lecture 30 Slide 6

17 Variable elimination example Compute P(G H=h ). P(GH) = ABCDEFI P(ABCDEFGHI) CPSC 322 Lecture 30 Slide 7

18 Variable elimination example Compute P(G H=h ). P(GH) = ABCDEFI P(ABCDEFGHI) Chain Rule + Conditional Independence: P(GH) = ABCDEFI P(A)P(B A)P(C)P(D BC)P(E C)P(F D)P(G FE)P(H G)P(I G) CPSC 322 Lecture 30 Slide 8

19 Variable elimination example (step) Compute P(G H=h ). P(GH) = ABCDEFI P(A)P(B A)P(C)P(D BC)P(E C)P(F D)P(G FE)P(H G)P(I G) Factorized Representation: P(GH) = ABCDEFI 0 (A) (BA) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) 0 (A) (BA) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) CPSC 322 Lecture 30 Slide 9

20 Variable elimination example (step 2) Compute P(G H=h ). Preious state: P(GH) = ABCDEFI 0 (A) (BA) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) Obsere H : P(GH=h ) = ABCDEFI 0 (A) (BA) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 9 (G) 8 (IG) 0 (A) (BA) 9 (G) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) CPSC 322 Lecture 30 Slide 20

21 Variable elimination example (steps 3-4) Preious state: Compute P(G H=h ). P(GH) = ABCDEFI 0 (A) (BA) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 9 (G) 8 (IG) Elimination ordering A C E I B D F : P(GH=h ) = 9 (G) F D 5 (F D) B I 8 (IG) E 6 (GFE) C 2 (C) 3 (DBC) 4 (EC) A 0 (A) (BA) 0 (A) 9 (G) (BA) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) CPSC 322 Lecture 30 Slide 2

22 Variable elimination example(steps 3-4) Compute P(G H=h ). Elimination ordering A C E I B D F. Preious state: P(GH=h ) = 9 (G) F D 5 (F D) B I 8 (IG) E 6 (GFE) C 2 (C) 3 (DBC) 4 (EC) A 0 (A) (BA) Eliminate A: P(GH=h ) = 9 (G) F D 5 (F D) B 0 (B) I 8 (IG) E 6 (GFE) C 2 (C) 3 (DBC) 4 (EC) 0 (A) (BA) 2 (C) 9 (G) 0 (B) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) CPSC 322 Lecture 30 Slide 22

23 Variable elimination example(steps 3-4) Compute P(G H=h ). Elimination ordering A C E I B D F. Preious state: P(GH=h ) = 9 (G) F D 5 (F D) B 0 (B) I 8 (IG) E 6 (GFE) C 2 (C) 3 (DBC) 4 (EC) Eliminate C: P(GH=h ) = 9 (G) F D 5 (F D) B 0 (B) I 8 (IG) E 6 (GFE) 2 (BDE) 0 (A) (BA) 2 (C) 3 (DBC) 9 (G) 0 (B) 2 (BDE) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) CPSC 322 Lecture 30 Slide 23

24 Variable elimination example(steps 3-4) Compute P(G H=h ). Elimination ordering A C E I B D F. Preious state: P(GH=h ) = 9 (G) F D 5 (F D) B 0 (B) I 8 (IG) E 6 (GFE) 2 (BDE) Eliminate E: P(GH=h ) = 9 (G) F D 5 (F D) B 0 (B) 3 (BDFG) I 8 (IG) 0 (A) (BA) 2 (C) 3 (DBC) 4 (EC) 9 (G) 0 (B) 2 (BDE) 3 (BDFG) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) CPSC 322 Lecture 30 Slide 24

25 Variable elimination example(steps 3-4) Compute P(G H=h ). Elimination ordering A C E I B D F. Preious state: P(GH=h ) = 9 (G) F D 5 (F D) B 0 (B) 3 (BDFG) I 8 (IG) Eliminate I: P(GH=h ) = 9 (G) 4 (G) F D 5 (F D) B 0 (B) 3 (BDFG) 0 (A) (BA) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 9 (G) 0 (B) 2 (BDE) 3 (BDFG) 4 (G) 6 (GFE) 7 (HG) 8 (IG) CPSC 322 Lecture 30 Slide 25

26 Variable elimination example(steps 3-4) Compute P(G H=h ). Elimination ordering A C E I B D F. Preious state: P(GH=h ) = 9 (G) 4 (G) F D 5 (F D) B 0 (B) 3 (BDFG) Eliminate B: P(GH=h ) = 9 (G) 4 (G) F D 5 (F D) 5 (DFG) 0 (A) (BA) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 9 (G) 0 (B) 2 (BDE) 3 (BDFG) 4 (G) 5 (DFG) 8 (IG) CPSC 322 Lecture 30 Slide 26

27 Variable elimination example(steps 3-4) Compute P(G H=h ). Elimination ordering A C E I B D F. Preious state: P(GH=h ) = 9 (G) 4 (G) F D 5 (F D) 5 (DFG) Eliminate D: P(GH=h ) = 9 (G) 4 (G) F 6 (F G) 0 (A) (BA) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) 9 (G) 0 (B) 2 (BDE) 3 (BDFG) 4 (G) 5 (DFG) 6 (F G) CPSC 322 Lecture 30 Slide 27

28 Variable elimination example(steps 3-4) Compute P(G H=h ). Elimination ordering A C E I B D F. Preious state: P(GH=h ) = 9 (G) 4 (G) F 6 (F G) Eliminate F: P(GH=h ) = 9 (G) 4 (G) 7 (G) 0 (A) (BA) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) 9 (G) 0 (B) 2 (BDE) 3 (BDFG) 4 (G) 5 (DFG) 6 (F G) 7 (G) CPSC 322 Lecture 30 Slide 28

29 Variable elimination example (step 5) Compute P(G H=h ). Elimination ordering A C E I B D F. Preious state: P(GH=h ) = 9 (G) 4 (G) 7 (G) Multiply remaining actors: P(GH=h ) = 8 (G) 0 (A) (BA) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) 9 (G) 0 (B) 2 (BDE) 3 (BDFG) 4 (G) 5 (DFG) 6 (F G) 7 (G) 8 (G) CPSC 322 Lecture 30 Slide 29

30 Variable elimination example (step 6) Compute P(G H=h ). Elimination ordering A C E I B D F. Preious state: P(GH=h ) = 8 (G) Normalize: 9 (G) P(G H=h ) = 8 (G) / g dom(g) 8 (G) 0 (A) (BA) 0 (B) 2 (BDE) 2 (C) 3 (DBC) 4 (EC) 5 (F D) 6 (GFE) 7 (HG) 8 (IG) 3 (BDFG) 4 (G) 5 (DFG) 6 (F G) 7 (G) 8 (G) CPSC 322 Lecture 30 Slide 30

31 Lecture Oeriew Recap Intro Variable Elimination Variable Elimination Simpliications Example Independence Where are we? CPSC 322 Lecture 30 Slide 3

32 Complexity (not required) The complexity o the algorithm depends on a measure o complexity o the network. The size o a tabular representation o a actor is exponential in the number o ariables in the actor. The treewidth o a network gien an elimination ordering is the maximum number o ariables in a actor created by summing out a ariable gien the elimination ordering. The treewidth o a belie network is the minimum treewidth oer all elimination orderings. The treewidth depends only on the graph structure and is a measure o the sparseness o the graph. The complexity o VE is exponential in the treewidth and linear in the number o ariables. Finding the elimination ordering with minimum treewidth is NP-hard but there is some good elimination ordering heuristics. CPSC 322 Lecture 30 Slide 32

33 Variable elimination and conditional independence Variable Elimination looks incredibly painul or large graphs? We used conditional independence.. Can we use it to make ariable elimination simpler? es all the ariables rom which the query is conditional independent gien the obserations can be pruned rom the Bnet CPSC 322 Lecture 30 Slide 33

34 VE and conditional independence: Example All the ariables rom which the query is conditional independent gien the obserations can be pruned rom the Bnet e.g. P(G H= F= 2 C= 3 ). B D E E D D I B D A CPSC 322 Lecture 0 Slide 34

35 VE and conditional independence: Example All the ariables rom which the query is conditional independent gien the obserations can be pruned rom the Bnet e.g. P(G H= F= 2 C= 3 ). CPSC 322 Lecture 0 Slide 35

36 VE and conditional independence: Example All the ariables rom which the query is conditional independent gien the obserations can be pruned rom the Bnet e.g. P(G H= F= 2 C= 3 ). CPSC 322 Lecture 30 Slide 36

37 VE and conditional independence: Example All the ariables rom which the query is conditional independent gien the obserations can be pruned rom the Bnet e.g. P(G H= F= 2 C= 3 ). CPSC 322 Lecture 30 Slide 37

38 ou can: Learning Goals or today s class Carry out ariable elimination by using actor representation and using the actor operations. Use techniques to simpliy ariable elimination. CPSC 322 Lecture 4 Slide 38

39 Big Picture: R&R systems Problem Static Query Sequential Planning Deterministic Arc Consistency Constraint Search Vars + Satisaction Constraints SLS Representation Reasoning Technique Logics Search STRIPS Search Enironment Stochastic Belie Nets Var. Elimination Decision Nets Var. Elimination Marko Processes Value Iteration CPSC 322 Lecture 2 Slide 39

40 Answering Query under Uncertainty Probability Theory Static Belie Network & Variable Elimination Dynamic Bayesian Network Hidden Marko Models Monitoring (e.g credit cards) Diagnostic Systems (e.g. medicine) BioInormatics spam ilters Natural Language Processing Student Tracing in tutoring Systems CPSC 322 Lecture 8 Slide 40

41 Next Class Probability and Time (TextBook 6.5) Course Elements Work on Practice Exercise 6.C on ariable elimination. Assignment 4 will be aailable on Wednesday and due on No the 28th (last class). Fill out teaching ealuations. ou should hae receied an about this. CPSC 322 Lecture 29 Slide 4