Probabilistic Reasoning with Uncertain Data

1. Probabilistic Reasoning with Uncertain Data Yun Peng and Zhongli Ding, Rong Pan, Shenyong Zhang

2. Uncertain Evidences • Causes for uncertainty of evidence • Observation error • Unable to observe the precise state the world is in • Two types of uncertain evidences • Virtual evidence: evidence with uncertainty I’m not sure about my observation that A = a1 • Soft evidence: evidence of uncertainty I cannot observe the state of A but have observed the distribution of A as P(A) = (0.7, 0.3)

3. A veA veB Virtual Evidences • Represent uncertainty in VE by likelihood ratio • This ratio shall be preserved (invariant) in belief update • Implemented by adding a VE node • It is a leaf node, with A as its only parent • Its CPT conform the likelihood ratio • Many BN engine accept likelihood ratio directly • Multiple VE is not a problem B

4. Soft Evidences • Represent uncertainty in SE by distribution • itself is to be believed without uncertainty and must be preserved (invariant) in belief update • Reasoning with a singleSE: Jeffrey’s rule • For the given seA = R(A) for evidence variable A for the rest of variables • For BN: convert SE to VE: calculate likelihood ratio

5. Multiple Soft Evidences • Problem: cannot satisfy all SE • update one variable’s distribution to its target value (the observed distribution) can make those of others’ off their targets A B seA seB • Solution: IPFP • A procedure that modify a distribution by one or more distributions over subsets of variables

6. Jeffrey’s Rule • Jeffrey’s rule (J-conditioning) (R. Jeffrey 1983) • Given SE R(a), any other variable c is updated by • Extend Jeffrey’s rule to the entire distribution • Q(a) = R(a) • Among all JPD sayisfying R(a), Q(x) has the smallestKL distance (I-divergence) to the original P(x) • Q(x) is called an I-projection of P(x) on R(B) • What if we have more than one SE? • R1(educ) and R2(smoker) (constraints) • How to make a minimum change to P(x) to satisfy ALL constraints?

7. IPFP • We can try Jeffrey’s rule • First on P(x) using R1->Q1(x) • Then on Q1(x) using R2->Q2(x) • Q2(x) satisfies R2 but not R1 • Iterative Proportional Fitting Procedure (IPFP) • Proposed by R. Kruithof (1937); convergence proved by I. Csiszar (1975) • Loop over the set of constraints, each step tries to fit one constraint • Converges to Q*(x), which is the I-projection of P(x) on the set of given constraints

8. IPFP All JPD satisfying R1 R1 Q1 Q3 Q2 P Q* R2 All JPD satisfying R2

9. oscillating IPFP • Problems with IPFP • Very slow • Each iteration (fitting step) has complexity of O(2|x|) • Factorization -> Bayesian network (BN) • Inconsistent constraints • No JPD satisfies all constraints • IPFP won’t converge (oscillating)

10. a b a b vea veb BN Belief Update with SE • BN belief update with hard evidence • HE a = A1; b = B3 • Clamp node a to A1 and b to B3 • Calculate P(c|A1, B3) for all c • Virtual evidence • Uncertainty of the HE (observation) • Represented as a likelihood ratio • Virtual node vea, with conditional probability table calculated from L(a) • When vea is clamped to “true”, P(a) on a is updated to have its likelihood ratio = L(a)

11. a b BN Belief Update with SE • Convert SE to VE • Belief update with yields Q(a) = R1(a) • Not work with multiple SE • When apply both sea and seb, • Q(a) != R1(a); Q(b) != R2(b) sea seb • Solution: combine VE with IPFP

12. BN Belief Update with SE • V-IPFP: at kth iteration • Pick up a sei, say R1(a), create a new vei,j, with likelihood ratio • Apply vei,j to update the entire network sea,1 … • Convergence • Converges to the I-projection on all constraints … sea,2 seb,1 sea,2 • Cost • Space: small • Time: large for large BN

13. current constraint with influences from other constraints new constraint Inconsistent Constraints • Smooth: • Phase I: apply IPFP until oscillation is detected • Pull Q to the neighborhood of the solution • Phase II: continue IPFP, but each time the constraint is modified • A new constraint is generated at each step, • Original constraints gradually phased out • Serialized GEMA • New constraints are generated only based on and • Incorporate into V-IPFP for BN reasoning is straightforward

14. BN Learning with Uncertain Data • Modify BN by a set of low dimensional PD (constraints) • Approach 1: • Compute the JPD P(x) from BN, • Modify P(x) to Q*(x) by constraints using IPFP • Construct a new BN from Q*(x) (it may have different structure that the original BN • Our approach: • Keep BN structure unchanged, only modify the CPTs • Developed a localized version of IPFP • Next step: • Dealing with inconsistency • Change structure (minimum necessary) • Learning both structure and CPT with mixed data (samples as low dimensional PDs)

15. Remarks • Wide potential applications • Probabilistic resources are all over the places (survey data, databases, probabilistic knowledge bases of different kinds) • This line of research may lead to effective ways to connect them • Problems with the IPFP based approaches • Computationally expensive • Hard to do mathematical proofs References: [1] Peng, Y., Zhang, S., Pan, R.: “Bayesian Network Reasoning with Uncertain Evidences”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 18 (5), 539-564, 2010 [2] Pan, R., Peng, Y., and Ding, Z: “Belief Update in Bayesian Networks Using Uncertain Evidence”, in Proceedings of the IEEE International Conference on Tools with Artificial Intelligence (ICTAI-2006), Washington, DC,13 – 15, Nov. 2006. [3] Peng, Y. and Ding, Z.: “Modifying Bayesian Networks by Probability Constraints”, in Proceedings of 21st Conference on Uncertainty in Artificial Intelligence (UAI-2005), Edinburgh, Scotland, July 26-29, 2005