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

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 = a 1 – 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)

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 A ve A ve B B

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 SE: Jeffrey’s rule ► Reasoning with a single SE: Jeffrey’s rule  For the given se A = R(A) for evidence variable A for evidence variable A for the rest of variables  For BN: convert SE to VE: calculate likelihood ratio

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 se A se B ► Solution: IPFP  A procedure that modify a distribution by one or more distributions over subsets of variables B

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 smallest KL 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? –R 1 (educ) and R 2 (smoker) (constraints) –How to make a minimum change to P(x) to satisfy ALL constraints?

IPFP We can try Jeffrey’s rule –First on P(x) using R 1 -> Q 1 (x) –Then on Q 1 (x) using R 2 -> Q 2 (x) –Q 2 (x) satisfies R 2 but not R 1 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

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

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

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 a b a ve a ve b b Virtual evidence –Uncertainty of the HE (observation) –Represented as a likelihood ratio –Virtual node ve a, with conditional probability table calculated from L(a) –When ve a is clamped to “true”, P(a) on a is updated to have its likelihood ratio = L(a)

BN Belief Update with SE Convert SE to VE – –Belief update with yields Q(a) = R 1 (a) se a se b a b Solution: combine VE with IPFP Not work with multiple SE –When apply both se a and se b, Q(a) != R 1 (a); Q(b) != R 2 (b)

BN Belief Update with SE V-IPFP: at k th iteration – Pick up a se i, say R 1 (a), create a new ve i,j, with likelihood ratio –Apply ve i,j to update the entire network se a,1 se b,1 se a,2 … … Convergence –Converges to the I-projection on all constraints Cost –Space: small –Time: large for large BN

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 current constraint new constraint with influences from other constraints

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)

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), , 2010Bayesian Network Reasoning with Uncertain Evidences [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 Belief Update in Bayesian Networks Using Uncertain Evidenc [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, 2005Modifying Bayesian Networks by Probability Constraints