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Computational Complexity of Inference Probabilistic AI Srihari Computational Complexity of Inference Sargur Srihari [email protected] 1 Probabilistic AI Srihari Topics in Complexity of Inference 1. Inferring a conditional probability from a BN 2. NP Complete Decision Problem 3. Inference decision problem is NP-complete 4. Exact Inference is #P-complete 5. Approximate Inference is NP-hard 6. Types of Inference algorithms: exact and approximate 2 Probabilistic AI Srihari Common BN Inference Problem • PGMs represent joint probability distributions over a set of variables χ • They are used to answer queries of interest • The most common query type is a conditional probability query – E: evidence variables, whose known value is e – Y: query variables, whose distribution we wish to know – Conditional probability query is P(Y|E=e) 3 Probabilistic AI Srihari Computing conditional probability • Assume: variables χ, evidence E=e, query Y • Conditional probability query P(Y|E=e) is: P(Y,e) P(Y | E = e) = From product rule P( ) e – Evaluation of Numerator P(Y,e) • If W=χ-Y-E, ie, variables not query or evidence P(y,e) = ∑P(y,e,w) (1) Each term in summation over w is simply w an entry in the distribution – Evaluation of Denominator P(e) P(e) = ∑P(y,e) Rather than marginalizing over P(y,e,w) this y allows reusing computation of (1) – This process corresponds to taking a vector of marginal probabilities P(y1,e),..P(yk,e) and renormalizing the entries by P(e) to sum to 1 Probabilistic AI Srihari Example Query with Cancer BN • χ={A,Ge,E,Sm,C,Gd,Se,L} – Evidence variable is Se • Serum Calcium is calcium in blood – Query variable is C cancer • We can evaluate P(C|Se )as: P(C,Se) P(C | Se) = P(Se) – With W = χ-C-Se={A,Ge,E,Sm,Gd,L}: P(C,Se) = ∑ P(A,Ge,E,Sm,Se,C,L,Gd) A,Ge,E,Sm,L,Gd = ∑ P(A)P(Ge)P(Gd)P(E | A),P(Sm | A,G)P(Se |C)P(C | E,Sm)P(L |C,Gd) A,Ge,E,Sm,L,Gd P(Se) = ∑P(C,Se) C We take the marginal probabilities P(c1,Se), P(c2,Se) and renormalize the entries by P(Se) to sum to 1 Probabilistic AI Srihari Analysis of Complexity • Approach of summing out the variables in the joint distribution is unsatisfactory P(y,e) = ∑P(y,e,w) w – Returns us to exponential blow-up • PGM was precisely designed to avoid this! • We now show that problem of inference in PGMs is NP-hard – Requires exponential time in the worst case except if P=NP – Even worse, approximate inference is NP - hard • Discussion for BNs applies to MNs also 6 Probabilistic AI Srihari What is a decision problem? • Subset sum decision problem – Given a set of integers, is there a subset that sums to zero? • No polynomial time algorithm to decide this – Given an instance, say set ω={−2, −3, 15, 14, 7, −10} • The guess {−2, −3, −10, 15} can be verified in poly time • Definition of Decision Problem Π: – LΠ defines a precise set of instances • LΠ consists of integer sets which have a subset that sums to zero – Decision problem Π: Is instance ω in LΠ ? • Is ω a set of integers with a subset that sums to zero? Probabilistic AI Srihari P and NP decision problems • Definition of Decision Problem Π: – LΠ defines a precise set of instances – Decision problem: Is instance ω in LΠ ? • Decision problem Π is in – P if there is an algorithm that decides in poly time – NP if a guess can be verified in poly time • Guess is produced non-deterministically • Hence the name non-deterministic polynomial time – Subset sum decision problem is in NP • Whether a given subset sums to zero verified in poly time • But not in P – No poly algorithm to determine whether there exists any subset that sums to zero Probabilistic AI Srihari 3-SAT (Satisfiability) decision problem • 3-SAT formula over binary variables q1,.,,qn – has the form C1 C2 ..Cm Ci is a clause of form li,1 li,2 li,3 ; li,j i =1,..,m; j =1,2,3 are literals which are either qk or ~qk e.g., with n=3 m=2 and clauses (q1 ~q2 ~q3) (~q1 q2 ~q3) there is a satisfying assignment (assigns true to formula): q1=q2=q3= true with clauses (~q1 q2 ) ( q2 q3) (~q1 q3) there is no satisfying assignment Each assignment verified in polynomial time • Decision problem Π: Given a 3-SAT formula of size n, is there a satisfying assignment? – To answer this we need to check n binary variables with 2n assignments 9 •LΠ is the set of 3-SAT formulas that have a satisfying assignment Probabilistic AI Srihari What is P=NP ? • Input is a formula of size n – A particular assignment γ can be verified in polynomial time, e.g., q1=q2=q3= true – Suppose generate guess γ and verify if it satisfies • Since guess verified in polynomial time, decision problem Π is in NP • Deterministic problems are subset of nondeterministic ones. So P ⊆ NP . – Converse is biggest problem in complexity • If you can verify in polynomial time, can you decide in polynomial time? – Eg., is there a prime greater than n? 10 Probabilistic AI Srihari P=NP intuition • Creating great art – Complexity is NP • Appreciating art: – Complexity is P • Is P=NP ? – Most mathematicians don’t think so 11 Probabilistic AI Srihari NP-hard and NP-complete • Hardest problems in NP are called NP-complete – If poly time solution exists, can solve any in NP • NP-hard problems need not have polynomial time verification • If Π is NP-hard it can be transformed into Π’ in NP • 3-SAT is NP-complete 12 Probabilistic AI Srihari BN for 3-SAT • Propositional variables q1,..,qn – Return true if C1 C2 ..Cm , where Ci is a DNF of 3 binary variables qk , has a satisfying assignment, • e.g., return true for 3-SAT formula (q1 V ~q2 V ~q3) (~q1 V q2~q3) since q1=q2=q3=true is a satisfying assignment and return false for (~q1 q2 ~q3) ( q2 q3) (~q1 q3) which has no satisfying assignments BN to infer this: Q1 Q2 Q3 Q4 Qn 1 P(qk )=0.5 Ci are deterministic OR C1 C2 C3 . Cm –1 Cm Ai are deterministic AND X is output (has value 1 A A . A 1 2 m –2 X 13 iff all of the Ci’s are 1 Probabilistic AI Srihari #P-complete Problem • Counting the no. of satisfying assignments • E.g., Propositional variables q1,.,,qn Return true if C1 C2 ..Cm , where Ci is a DNF of 3 binary variables qk , has a satisfying assignment, 14 Probabilistic AI Srihari Analysis of Exact Inference • Worst case: CPD is a table of size |Val({Xi} ∨ PaXi)| • Most analyses of complexity are stated as decision-problems • Consider decision problem first, then numerical one • Natural version of conditional probability task is given next Probabilistic AI Srihari Conditional Probability Decision Task • Bayesian Network – Probability - Decision task: – BN-Pr-DP: Bayesian Network Decision Problem • Given a BN B over χ, a variable X ε χ, and a value x εVal(X) decide PB (X=x)>0 • This decision problem can be shown to be NP- complete as seen next Probabilistic AI Srihari Proof of BN-Pr-DP is NP-complete • Whether in NP : – Guess assignment ξ to network variables. Check whether X=x and P(ξ)>0 – One such guess succeeds iff P(X=x)>0. – Done in linear time • Is NP-hard: – Answer for instances in BN-Pr-DP can be used to answer an NP-hard problem – Show a reduction from 3-SAT problem 17 Probabilistic AI Srihari Reduction of 3-SAT to BN inference • Given a 3-SAT formula φ create BN Bφwith variable X such that φ is satisfiable iff PBφ(X=x1)>0 • If BN inference is solved in poly time we can also solve 3-SAT in poly time Q1 Q2 Q3 Q4 Qn BN to infer this: 1 P(qk )=0.5 C1 C2 C3 . Cm –1 Cm Ci are deterministic OR Ai are deterministic AND X is output A1 A2 . Am –2 X 18 Probabilistic AI Srihari Original Inference Problem • It is a numerical problem – rather than a decision problem • Define BN-Pr – Given a BN B over χ, a variable X ε χ, and a value x εVal(X) compute PB (X=x) – Task is to compute the total probability of instantiations that are consistent with X=x • Weighted count of instantiations, with weight being the probability • This problem is #P-complete 19 Probabilistic AI Srihari Analysis of Approximate Inference • Metrics for quality of approximation • Absolute Error – Estimate ρ has absolute error ε for P(y|e) if |P(y|e)-ρ| ≤ ε – A weak definition of error. If a rare disease has probability 0.0001 then error of 0.0001 is unacceptable. If the probability is 0.3 then error of 0.0001 is fine • Relative Error – Estimate ρ has relative error ε for P(y|e) if ρ/(1+ε) ≤ P(y|e) ≤ ρ(1+ε) • ε=4 means P(y|e) is at least 20% of ρ and at most 600% of ρ. For low values much better than absolute error Probabilistic AI Srihari Approximate Inference is NP-hard • The following problem is NP-hard – Given a BN B over χ, a variable X εχ and a value x ε Val(X), find a number ρ that has relative error ε for PB(X=x) • Proof: 1 – It is NP-hard to decide if PB (x ) >0 1 – Assume algorithm returns estimate ρ to PB(x ) which has relative error ε for some ε > 0 1 – ρ>0 if and only if PB(x )>0 – This achieving relative error is NP-hard 21 Probabilistic AI Srihari Inference Algorithms • Worst case is exponential • Two types of inference algorithms – Exact • Variable Elimination • Clique trees – Approximate • Optimization – Propagation with approximate messages – Variational (analytical approximations) • Particle-based (sampling) 22.
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