Integrating Probability Constraints into Bayesian Nets

This paper presents a formal convergence proof for EIPFP, an algorithm that integrates low dimensional probabilistic constraints into a Bayesian network (BN) based on the mathematical procedure IPFP. It also extends E-IPFP to deal with constraints that are inconsistent with each other or with the BN structure. 12 1 CONVERGENCE OF E-IPFP Let ( , ) s P G G G  denote the given BN of n variables ( , ) i n x x x   , where {( , )} S i i G x   gives the network structure and { ( | )} P i i G P x   is the set of conditional probability tables (CPTs). Denote JPD of x defined by G as P(x). Let 1 1 { ( ), R R y  2 2 ( ), , ( )} m m R y R y  be a set of probabilistic constraints, where ( ) j j R y x  . Our objective is to construct a new BN ' G  ( ' , ' ) s P G G with its JPD '( ) P x meeting the following conditions: C1: Constraint satisfaction: '( ) ( ) j j j P y R y   ( ) j j R y R  ; C2: Structural invariance: ' S S G G  ; C3: Minimality: '( ) P x is as close to ( ) P x as possible. E-IPFP [1] is based on the mathematical procedure IPFP (iterative proportional fitting procedure) [2] which iteratively modifies the JPD by the constraints until convergence. It has been shown that the converging JPD satisfies all constraints in R (C1) and is closest to the original JPD measured by the I-divergence (C3). To satisfy the structural invariance (C2), E-IPFP extends IPFP by making the BN structure ( S G ) an additional constraint 1 1 1 ( ) ( | ) n m i k i i R x Q x       . (1) E-IPFP( ( , ) s P G G G  , } , , { 2 1 m R R R R   ) { 1. 0 1 ( ) ( | ) n i i i Q x P x     where ( | ) i i P P x G   ; 2. Starting with k = 1, repeat the following procedure until convergence { 2.1. j = ((k-1) mod (m+1)) + 1; 2.2. if j < m+1 1 1 ( ) ( ) ( ) / ( ) j j k k j k Q x Q x R y Q y    2.3. else {extract ( | ) k i i Q x  from ( ) k Q x according to S G ; 1 ( ) ( | ) n k i k i i Q x Q x     ;} 2.4. k = k+1;} 3. return ' ' ( , ) S P G G G  with ' { ( | )} P k i i G Q x   ;} E-IPFP is exactly the same as standard IPFP except in Step 2.3 where the structural constraint applies. However, convergence proofs for IPFP’s [2,3] do not apply to E-IPFP because 1) 1  m R changes its value in every iteration and 2) the set of all JPD satisfying S G is not convex. We have shown in [4] IPFP with 1 1 { ( ), ( )} m m R R y R y   is equivalent to IPFP with a single composite constraint 1 2 '( ) m R y y y y     , which is computed by applying IPFP to 0 ( ) Q y with 1 1 { ( ), ( )} m m R R y R y   . So it suffices to prove the convergence of E-IPFP with a single constraint R(y). 1 Univ. Maryland, Baltimore County, USA, email: ypeng@umbc. edu 2 University of Science and Technology of China, China, email: [email protected] Denote the set of JPD of x that satisfy R(y) as ( ) R y P and the set of JPD that satisfy structural constraint as S G P . Let 0 ( ) Q x  ( | ) i x x i i P x    be the JPD of the given BN; 1( ) Q x  0 0 ( ) ( ) / ( ) Q x R y Q y the I-Projection of 0 ( ) Q x to ( ) R y P ; 2 ( ) Q x  1( | ) i x x i i Q x    the structural constraint; and 3( ) Q x  2 2 ( ) ( ) / ( ) Q x R y Q y be the I-Projection of 2 ( ) Q x back to ( ) R y P . Points of Q0 through Q3 are depicted in Figure 1 below. Note that Q1 is obtained from Q0 by Step 2.2, Q2 from Q1 by Step 2.3, and Q3 from Q2 by Step 2.2 in the next iteration of E-IPFP. Figure 1. Successive JPDs from E-IPFP The convergence of E-IPFP can be established by showing 1 0 3 2 ( || ) ( || ) I Q Q I Q Q  , i.e., the I-divergence between the two endpoints of I-projection to ( ) R y P is monotonically decreasing in successive iterations. Since 1 3 ( ) , R y Q Q P , and 3 Q is an I-Projection of 2 Q , we have 1 2 3 2 ( || ) ( || ) I Q Q I Q Q  . So E-IPFP converges if 1 0 1 2 ( ) ( || ) ( || ) x I Q Q I Q Q    (2) Is non-negative Theorem 1. For any given BN ( , ) s P G G G  and R(y), ( ) 0 x   . Proof. By induction on |x|, the number of variables in G. Base case: |x| = 1, 1 ( ) x x  , the constraint is 1 ( ) R x . It is trivial that 2 1 1 1 1 ( ) ( ) ( ) Q x Q x R x   . Then by (8) 1 1 1 0 1 0 1 ( ) ( ) ( ) log ( ( ) || ( )) 0, ( ) R x x R x I R x Q x Q x      Inductive assumption: 1 2 ( , ,..., ) 0 n x x x   for any 1 n  . Inductive proof: show that 0 1 2 ( , , ,..., ) 0 n x x x x   . Without loss of generality, let 0 x be a root node of the BN. For clarity, let 1 2 ( , ,..., ) n x x x x  . By (2),