1. Axioms and Algorithms for Inferences Involving Probabilistic Independence Dan Geiger, Azaria Paz, and Judea Pearl, Information and Computation 91(1), March 1991, 128-141. Presentation by Guy Moses & Omer Weissbrod for the course 236372 - Bayesian Networks Computer Science Faculty, Technion – winter 2009 partially based on the presentation by Ilan Gronau

2. 2 What’s ahead?  Introduction  Introduction - some definitions, notations and reminders.  Proof of Completeness  Proof of Completeness. - “if it’s true – it can be proved”.  Preparations for the Membership Algorithm  Preparations for the Membership Algorithm – more definitions, and some theoretical groundwork.  The Membership Algorithm  The Membership Algorithm – description, proof of correctness, complexity analysis.

3. 3 Definitions U (Universe) – set of random variables with probability distribution P. X,Y – finite sets of random variables: X =  x 1,…,x n , Y =  y 1,…,y m  P(X,Y) = P(X)·P(Y) - a short-hand notation for the equality: Pr{x 1 =a 1,…, x n =a n, y 1 =b 1, …, y m =b m } = Pr{x 1 =a 1,…, x n =a n } · Pr{y 1 =b 1, …, y m =b m } for every choice of a 1, …, a n, b 1, …, b m (X,Y) – short-hand for P(X,Y) = P(X)·P(Y) This is called an independence statement. *note that X, Y are disjoint sets of variables (X  Y =  ).

4. 4 Notations  - a specific independence statement of the form (X,Y)  - a set of independence statements of the form (X,Y):  =  1, …,  k  XY - short-hand notation for the union X  Y P satisfies  = (X,Y) means: P(X,Y) = P(X)·P(Y) for that specific P.

5. 5 Soundness and Completeness Definitions:    iff every distribution that satisfies  also satisfies .    iff   cl(  ), i.e. there exists a derivation chain  1,…,  n =  s.t. for each  j, either  j   or  j is derived by an axiom from the previous statements. For a set of axioms A : Soundness: A is sound iff for every  and  :        Completeness: A is complete iff for every  and  :        Completeness - Alternative definition: A is complete iff for every  and every   cl(  ) there exists a distribution P  that satisfies cl)  ( and does not satisfy .

6. 6 Independence Axioms We saw (in 1st lecture) that axioms 1a-1d are sound (always infer correctly). Today we’ll show they are complete (can derive every true statement).

7. 7 What’s ahead?  Introduction  Introduction - some definitions, notations and reminders.  Proof of Completeness  Proof of Completeness. - “if it’s true – it can be proved”.  Preparations for the Membership Algorithm  Preparations for the Membership Algorithm – more definitions, and some theoretical groundwork.  The Membership Algorithm  The Membership Algorithm – description, proof of correctness, complexity analysis.

8. 8 Minimal Statement Definition:  =(X,Y)  cl(  ) is minimal if for every non-empty X’,Y’ s.t. X’  X, Y’  Y, X’Y’  XY we have (X’,Y’)  cl(  ). For every  =(X,Y)  cl(  ) we can find an appropriate minimal  ’=(X’,Y’)  cl(  ) through iterative decomposition. Observation: P  satisfies   P  satisfies  ’ (decomposition soundness), Therefore: P  doesn’t satisfy  ’  P  doesn’t satisfy . Our plan: Given an arbitrary   cl(  ), We will find a distribution P  that satisfies cl(  ) but doesn’t satisfy  ’. This will prove completeness (using the alternative completeness definition and the observation above). To simplify annotation, we will assume WLOG that  =(X,Y) is already minimal.

9. 9 Let  =(X,Y)  cl(  ) be a minimal statement where: X={x 1,…,x n }, Y={y 1,…,y m }, and Z={z 1,z 2,…,z k } stand for the rest of the variables in U. We will construct P  as follows: All variables, except x 1, are fair coins (probability  for each of their two values) x 1 is defined thus: Part 1: P  does not satisfy  We will inspect the following scenario: x 1 =1, all other variables are 0. P  (x 1, …, x n, y 1, …, y m )  P  (x 1, …, x n )·P  (y 1, …, y m ) Therefore, P  does not satisfy , as required. Completeness Proof =0=0.5 n =0.5 m

10. 10 Completeness Proof – cont’d Part 2: P  satisfies cl(  ) Let (V,W)  cl(  ). We will show that P  (V,W)=P  (V)·P  (W). This is done by inspecting different scenarios: Scenario 1: either V or W contains only elements of Z. We will assume WLOG that W contains only elements of Z. all variables in Z are independent under P  and therefore: W V X Z Y Y Y Y Y Y X X X Z Z Z Z Z Z Z Z Z Z Z Z Z

11. 11 Completeness Proof – cont’d Part 2: P  satisfies cl(  ) Let (V,W)  cl(  ). We will show that P  (V,W)=P  (V)·P  (W). This is done by inspecting different scenarios: Scenario 2: Both V and W contain elements of X  Y, but V  W doesn’t contain all elements of X  Y. Without full information about the assignments of the variables in X  Y, x 1 could turn out to be 0 or 1 with probability , and therefore: W V X Z Y Y Y Y Y Y X X X Z Z Z Z Z Z Z Z Z Z Z Z Z

12. 12 W X Z Y Y Y Y Y Y X X X Z Z Z Z Z Z Z Z Z Z Z Z Z V Completeness Proof – cont’d Part 2: P  satisfies cl(  ) - continued Scenario 3: Both V and W contain elements of X  Y, and (X  Y)  (V  W). We will show a derivation chain for  =(X,Y), contradicting our original assumption that   cl(  ) : Mark: (V,W)=(X V Y V Z V, X W Y W Z W )  cl(  ) where: Y=Y V Y W, X=X V X W, Z V Z W  Z, V=X V Y V Z V, W=X W Y W Z W Remove all z ’s by decomposition: (X V Y V, X W Y W )  cl(  ) Due to minimality of  =(X,Y) : (X V,Y V )  cl(  ) and (X W,Y)  cl(  ) (X V,Y V )  (X V Y V, X W Y W ) (X V,Y V X W Y W ) = (X V,X W Y) (X W,Y)  (X W Y, X V ) (Y, X V X W ) = (Y,X) =  mix 

13. 13 Completeness Proof – Summary Reminder: Completeness - Alternative definition: A is complete iff for every  and every   cl(  ) there exists a distribution P  that satisfies cl)  ( and does not satisfy . We’ve shown: given a minimal   cl(  ), there exists a distribution P  that obeys: 1. P  does not satisfy . 2. P  satisfies . Given a non-minimal   cl(  ), we will derive its minimal statement  ’, and devise a distribution P  ’ that satisfies  but does not satisfy  ’. Due to soundness of decomposition, P  ’ cannot satisfy  as well.

14. 14 Scope of Completeness The proof uses P   - a binary p.d. (probability distribution function) therefore: all p.d.’s over U discrete p.d.’s binary p.d.’s normal p.d.’s however, for normal p.d.’s, the axiom set a1-d1 is not complete. a stronger axiom is required: replace: with: P 

15. 15 What’s ahead?  Introduction  Introduction - some definitions, notations and reminders.  Proof of Completeness  Proof of Completeness. - “if it’s true – it can be proved”.  Preparations for the Membership Algorithm  Preparations for the Membership Algorithm – more definitions, and some theoretical groundwork.  The Membership Algorithm  The Membership Algorithm – description, proof of correctness, complexity analysis.

16. 16 Some more Definitions and Tools Definition: Span span(  ) : the set of elements represented in statement . Example: span(x 1 x 2,x 3, x 4 ) = {x 1,x 2,x 3, x 4 } span(  ) : the set of elements represented in all statements of . Example: span({(x 1,x 2 ),(x 1,x 3 )}) = {x 1,x 2,x 3 }

17. 17 Some more Definitions and Tools Definition: Projection The projection of  on X, denoted  (X), is the statement derived from  by removing all elements not in X from . Example: if  =(x 1 x 2 x 3, x 4 x 5 ) and X={x 2,x 3,x 4 } then  (X)=(x 2 x 3, x 4 ). The projection of  on X, denoted  (X), is {  (X) |    }.

18. 18 Some more Definitions and Tools Projection Lemma:    iff  ‘  , where  ’ =  (span(  ))  ) if  '   then clearly    because all the statements in  ‘ can be derived from the statements in  by decomposition.

19. 19 Some more Definitions and Tools Projection Lemma:    iff  ’  , where  ’ =  (span(  )), s = span(  )  ) if    then there is a derivation chain for  :  1,  2, …,  k. For each  j : if  k   j, k<j, (by symmetry or decomposition) then  k (s)   j (s) by symmetry or decomposition respectively. Similarly, if  j is derived from  k and  l by mixing, then  j (s) is derived from  k (s),  l (s) by mixing. 

20. 20 Some more Definitions and Tools Projection Lemma:    iff  ’  , where  ’ =  (span(  )), s = span(  ) Observations from projection lemma: Variables not in  are unnecessary for determining whether    . The problem of verifying whether     can be simplified to the problem of verifying whether  '  , where  '=  (span(  )). This problem can be solved with a possibly reduced time and space complexity.

21. 21 Conditions for Inference of Independence Maim claim: for a given ,  we have  ’   iff: 1.  is trivial:  = (X,  ) (up to symmetry) OR 2.  is in  ’:  ’ (up to symmetry) OR 3.  is derivable from  ’: there exists  ’  ’ s.t. span(  ) = span(  ’) and for  ’ = (AP,BQ)  = (AQ,BP) (A,B,Q,P may be empty)  ’  (A,P),  ’  (B,Q) (up to symmetry)

22. 22 Proof of Main Claim Maim claim: for a given ,  we have  ’   iff: 1.  is trivial*:  = (X,  ) *up to symmetry 2.  is in*  ’:  ’ 3.  is derivable* from  ’:   ’  ’ s.t. span(  ) = span(  ’) and for  ’=(AP,BQ)  =(AQ,BP) :  ’  (A,P),  ’  (B,Q)  ) if 1.  is trivial* OR 2.  is in*  ’. than the proof is immediate. otherwise, 3. there exists  ’  ’ s.t. span(  ) = span(  ’) and for  ’=(AP,BQ)  =(AQ,BP) :  ’  (A,P),  ’  (B,Q) we will show a constructive proof under these conditions

23. 23 Proof of Main Claim Maim claim: for a given ,  we have  ’   iff: 1.  is trivial*:  = (X,  ) *up to symmetry 2.  is in*  ’:  ’ 3.  is derivable* from  ’:   ’  ’ s.t. span(  ) = span(  ’) and for  ’=(AP,BQ)  =(AQ,BP) :  ’  (A,P),  ’  (B,Q)  ) (contd.) given that  ’  (AP,BQ),  ’  (A,P),  ’  (B,Q). 1. (A,P)  (AP,BQ) (A,PBQ) 2. (B,Q)  (AP,BQ) (APB,Q) (PB,Q) 3. (PB,Q)  (A,PBQ) (AQ,PB) = (AQ, BP) =  We’ve proven this direction. mix dec.

24. 24 Proof of Main Claim Maim claim: for a given ,  we have  ’   iff: 1.  is trivial*:  = (X,  ) *up to symmetry 2.  is in*  ’:  ’ 3.  is derivable* from  ’:   ’  ’ s.t. span(  ) = span(  ’) and for  ’=(AP,BQ)  =(AQ,BP) :  ’  (A,P),  ’  (B,Q)  ) Given  ’  , if 1.  is trivial* OR 2.  is in*  ’, than the proof is immediate. Otherwise, since no axiom can add new variables to a statement, there must exist  ’  ’ s.t. span(  ) = span(  ’) in the derivation chain of . also:  = (AQ,BP) (A,P)  = (AQ,BP) (Q,B)  dec.

25. 25 Conclusions from Claim We’ve seen that, after discarding unneeded variables, it is possible to tell whether  ’   (when it’s not immediately obvious) by: a.Finding another statement  ’  ’ for which span(  ) = span(  ’), b.Verifying that  ’  (A,P),  ’  (B,Q) when  ’=(AP,BQ)  =(AQ,BP). This suggests using a recursive “divide and conquer” approach.

26. 26 What’s ahead?  Introduction  Introduction - some definitions, notations and reminders.  Proof of Completeness  Proof of Completeness. - “if it’s true – it can be proved”.  Preparations for the Membership Algorithm  Preparations for the Membership Algorithm – more definitions, and some theoretical groundwork.  The Membership Algorithm  The Membership Algorithm – description, proof of correctness, complexity analysis.

27. 27 The Membership Algorithm Procedure Find( ,  ): 1.set  ’ :=  (span(  )). 2.if  is trivial, or  ’ (up to symmetry) then Find( ,  ) := TRUE. 3.elseif for all non-trivial  ’  ’ : span(  )  span(  ’), then Find( ,  ) := FALSE. 4.else there exists  ’  ’ : span(  ) = span(  ’), and  ’=(AP,BQ)  =(AQ,BP), set  1 := (A,P),  2 := (B,Q). Find( ,  ) := ( Find(  ’,  1 )  Find(  ’,  2 ) )

28. 28 Algorithm Correctness Proof We will prove that Find( ,  ) := TRUE   cl(  ) by induction on k= . Induction base: if k=1 then  is trivial, therefore the algorithm will return TRUE in step 2 and  cl(  ).

29. 29 Algorithm Correctness Proof Induction assumption: Find( ,  ) := TRUE   cl(  ) for each  ’  <k. Induction step: Find( ,  ) := TRUE iff either: 1. Step 2 returns TRUE   is trivial or  ’   cl(  ). 2. Step 4 returns TRUE iff Find(  ’,  1 ) := TRUE  Find(  ’,  2 ) := TRUE iff  1  cl(  ’ )   2  cl(  ’ ) iff  cl(  ’ ) (according to algorithm’s definition) (according to induction assumption) (according to main claim) (according to projection lemma) iff  cl(  ) 

30. 30 Complexity Analysis Definitions: n = the number of distinct variables in   {  }. k = the number of distinct variables in {  }. First projection cost: O (|  |·n) – happens only once. Recursive step: T)k)  |  |·k + T(k 1 ) + T(k 2 ) where k 1 +k 2 =k, k 1 =|  1 |, k 2 =|  2 | Can be shown by induction: T)k)  |  |·k·( depth of recursion) Worst case analysis: T)k)  |  |·k·k= |  |·k 2 Total run time is bounded by: O (|  |·n + |  |·k 2 ) which is also: O (|  |·n 2 ) since k  n.

31. 31 Improvements and Variations Instead of arbitrarily choosing  ’, find one whose sub- statements { A, B, P, Q } have balanced size (can improve run-time complexity). Using the derivation chain presented in the constructive proof, the algorithm can also return a derivation chain for  with a length of O (k).

32. 32 Variations (contd.) The algorithm can be expanded into a polynomial algorithm for the following problems: Given two sets  and , is cl(  )  cl(  ) ? is cl(  ) = cl(  ) ? Minimize the size of  while preserving cl(  ) : Start with a maximal-size statement and remove from  all statements derivable from it. Repeat with the next largest statement etc.

33. Thank you!