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
• Im 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
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 single SE Jeffreys 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 variables 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
Jeffreys Rule

• Jeffreys rule (J-conditioning) (R. Jeffrey 1983)
• Given SE R(a), any other variable c is updated by
• Extend Jeffreys 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?
• R1(educ) and R2(smoker) (constraints)
• How to make a minimum change to P(x) to satisfy
  ALL constraints?

7
IPFP

• We can try Jeffreys rule
• First on P(x) using R1 -gt Q1(x)
• Then on Q1(x) using R2 -gt 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
IPFP

• Problems with IPFP
• Very slow
• Each iteration (fitting step) has complexity of
  O(2x)
• Factorization -gt Bayesian network (BN)

• Inconsistent constraints
• No JPD satisfies all constraints
• IPFP wont converge (oscillating)

10
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(cA1, 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
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
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