Conservative Updating with Incomplete Observations in Bayesian Networks

1. IDSIA Updating with incomplete observations(UAI-2003) Gert de Cooman Marco Zaffalon “Dalle Molle” Institute for Artificial Intelligence SWITZERLAND http://www.idsia.ch/~zaffalon zaffalon@idsia.ch SYSTeMS research group BELGIUM http://ippserv.ugent.be/~gert gert.decooman@ugent.be

2. What are incomplete observations?A simple example • C (class) and A (attribute) are Boolean random variables • C = 1 is the presence of a disease • A = 1 is the positive result of a medical test • Let us do diagnosis • Good point: you know that • p(C = 0, A = 0) = 0.99 • p(C = 1, A = 1) = 0.01 • Whence p(C = 0 | A = a) allows you to make a sure diagnosis • Bad point: the test result can be missing • This is an incomplete, or set-valued, observation {0,1} for A What is p(C = 0 | A is missing)?

3. Example ctd • Kolmogorov’s definition of conditional probability seems to say • p(C = 0 | A  {0,1}) = p(C = 0) = 0.99 • i.e., with high probability the patient is healthy • Is this right? • In general, it is not • Why?

4. Why? • Because A can be selectively reported • e.g., the medical test machine is broken;it produces an output  the test is negative (A = 0) • In this case p(C = 0 | A is missing) = p(C = 0 | A = 1) = 0 • The patient is definitely ill! • Compare this with the former naive application ofKolmogorov’s updating (or naive updating, for short)

5. (c,a) o IM p(C,A) Incompleteness Mechanism (IM) Actual observation (o) about A Distribution generating pairs for (C,A) Complete pair (not observed) Modeling it the right way • Observations-generating model • o is a generic value for O, another random variable • o can be 0, 1, or * (i.e., missing value for A) • IM = p(O | C,A) should not be neglected! The correct overall model we need is p(C,A)p(O | C,A)

6. (V)isit to Asia (S)moking = y (T)uberculosis = n Lung (C)ancer? Bronc(H)itis Abnorma(L) X-rays = y (D)yspnea What about Bayesian nets (BNs)? • Asia net • Let us predict C on the basis of the observation (L,S,T) = (y,y,n) • BN updating instructs us to use p(C | L = y,S = y,T = n) to predict C

7. Asia ctd • Should we really use p(C | L = y,S = y,T = n) to predict C? (V,H,D) is missing (L,S,T,V,H,D) = (y,y,n,*,*,*) is an incomplete observation • p(C | L = y,S = y,T = n) is just the naive updating • By using the naive updating, we are neglecting the IM! Wrong inference in general

8. New problem? • Problems with naive updating were already clear since 1985 at least (Shafer) • Practical consequences were not so clear • How often does naive updating make problems? • Perhaps it is not a problem in practice?

9. Grünwald & Halpern (UAI-2002) on naive updating • Three points made strongly • naive updating works  CAR holds • i.e., neglecting the IM is correct  CAR holds • With missing data:CAR (coarsening at random) = MAR (missing at random) =p(A is missing | c,a) is the same for all pairs (c,a) • CAR holds rather infrequently • The IM, p(O | C,A), can be difficult to model 2 & 3 = serious theoretical & practical problem How should we do updating given 2 & 3?

10. What this paper is about • Have a conservative (i.e., robust) point of view • Deliberately worst case, as opposed to the MAR best case • Assume little knowledge about the IM • You are not allowed to assume MAR • You are not able/willing to model the IM explicitly • Derive an updating rule for this important case • Conservative updating rule

11. (c,a) o IM p(C,A) Unknown Incompleteness Mechanism Actual observation (o) about A Known prior distribution Complete pair (not observed) 1st step: plug ignorance into your model • Fact: the IM is unknown • p(O{0,1,*} | C,A) = 1 • a constraint on p(O | C,A) • i.e. any distribution p(O | C,A) is possible • This is too conservative;to draw useful conclusionswe need a little less ignorance • Consider the set of all p(O | C,A) s.t. p(O | C,A) = p(O | A) • i.e., all the IMs which do not depend on what you want to predict • Use this set of IMs jointly with prior information p(C,A)

12. 2nd step: derive the conservative updating • Let E = evidence = observed variables, in state e • Let R = remaining unobserved variables (except C) • Formal derivation yields: • All the values for R should be considered • In particular, updating becomes: Conservative Updating Rule(CUR) minrRp(c | E = e,R = r) p(c | o)  maxrRp(c | E = e,R = r)

13. (V)isit to Asia (S)moking = y (T)uberculosis = n Lung (C)ancer? Bronc(H)itis Abnorma(L) X-rays = y (D)yspnea CUR & Bayesian nets • Evidence: (L,S,T) = (y,y,n) • What is your posterior confidence on C = y? • Consider all the jointvalues of nodes in RTake min & max of p(C = y | L = y,S = y,T = n,v,h,d) Posterior confidence  [0.42,0.71] • Computational note: only Markov blanket matters!

14. A few remarks • The CUR… • is based only on p(C,A), like the naive updating • produces lower & upper probabilities • can produce indecision

15. CUR & decision-making • Decisions • c’ dominates c’’ (c’,c’’ C) if for all r R , p(c’ | E = e, R = r) > p(c’’ | E = e, R = r) • Indecision? • It may happen that r’,r’’ R so that: p(c’ | E = e, R = r’) > p(c’’ | E = e, R = r’) and p(c’ | E = e, R = r’’) < p(c’’ | E = e, R = r’’) There is no evidence that you should prefer c’ to c’’ and vice versa (= keep both)

16. (V)isit to Asia (S)moking = y (T)uberculosis Lung (C)ancer? Bronc(H)itis Abnorma(L) X-rays = y (D)yspnea Decision-making example • Evidence: E = (L,S,T) = (y,y,n) = e • What is your diagnosis for C? • p(C = y | E = e, H = n, D = y) > p(C = n | E = e, H = n, D = y) • p(C = y | E = e, H = n, D = n) < p(C = n | E = e, H = n, D = n) • Both C = y and C = n are plausible • Evidence:E = (L,S,T) = (y,y,y) = e • C = n dominates C = y: “cancer” is ruled out

17. Algorithmic facts • CUR  restrict attention to Markov blanket • State enumeration still prohibitive in some cases • e.g., naive Bayes • Dominance test based on dynamic programming • Linear in the number of children of class node C However: decision-making possible in linear time, by provided algorithm, even on some multiply connected nets!

18. On the application side • Important characteristics of present approach • Robust approach, easy to implement • Does not require changes in pre-existing BN knowledge bases • based on p(C,A) only! • Markov blanket  favors low computational complexity • If you can write down the IM explicitly, your decisions/inferences will be contained in ours • By-product for large networks • Even when naive updating is OK, CUR can serve as a useful preprocessing phase • Restricting attention to Markov blanket may produce strong enough inferences and decisions

19. What we did in the paper • Theory of coherent lower previsions (imprecise probabilities) • Coherence • Equivalent to a large extent to sets of probability distributions • Weaker assumptions • CUR derived in quite a general framework

20. Concluding notes • There are cases when: • IM is unknown/difficult to model • MAR does not hold • Serious theoretical and practical problem • CUR applies • Robust to the unknown IM • Computationally easy decision-making with BNs • CUR works with credal nets, too • Same complexity • Future: how to make stronger inferences and decisions • Hybrid MAR/non-MAR modeling?