Approximate Inference for Logic Programs with Annotated Disjunctions (RCRA 2009)

Approximate Inference for Logic Programs with Annotated Disjunctions (RCRA 2009)

• 1.
  Stefano Bragaglia DEIS –University of Bologna [email protected] Fabrizio Riguzzi ENDIF – University of Ferrara [email protected]
• 2.
    Increasing interest incombining logic and probability • C1: sneezing(X): 0.7 v not_sneezing(X): 0.3 ← flu(X). C2: sneezing(X): 0.6 v not_sneezing(X): 0.4 ← hayfever(X). C3: flu(andrew). C4: flu(david). C5: hayfever(david). C6: hayfever(robert). Many formalism:  Markov Logic Networks  ProbLog  Logic Programs with Annotated Disjunctions (LPADs)  etc.  LPADs can easily express: • Queries: ?- sneezing(andrew). ?- sneezing(robert). ?- sneezing(david).  cause-effect relationships among events  possible effects of a single cause  contemporary contribution of more causes to the same effect Example: simple medical diagnosis • 0.7 0.6 0.88 Probability: Instances where query is true: 3 out of 4 P = 0.7×0.6 + 0.7×0.4 + 0.3×0.6 = 0.88
• 3.
   Syntax      Program: set ofdisjunctive clauses Head: set of mutually exclusive and exhaustive logical atoms annotated with probability values between 0 and 1 whose sum is 1 Body: event whose effects are represented by the atoms of the corresponding head 1. 2. Semantic    Inference: in 2 steps (by cplint) Explanations: a meta-interpreter performs resolution keeping current set of choices Probability of a query: a dynamic algorithm converts explanations into a Binary Decision Diagram (BDD) and traverse it; this makes explanations mutually exclusive ?- sneezing(david). (C1;{X1/david};0) (C2;{X2/david};0) :- flu(david). :- hayfever(david). Instance: normal logic program obtained by choosing a logical atom from the head of each grounding of every clause Probability of an instance: product of the probability values of all the atoms chosen for that instance Probability of a query: sum of the probabilities of each instance where the query is true   0.4 0.3 0 XC2 XC1 0.7 0.6 1 P = 0.3×0.6 + 0.7 = 0.88  Not possible in some domains
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   Best K ▪Deterministic algorithm based on branch and bound technique ▪ Explanations built incrementally by keeping only the k most probable ones (k = 64) ▪ Probability of the query computed on chosen explanations trough BDD (lower bound) ▪ Note: limiting explanations allows better control on complexity  Monte Carlo ▪ Stochastic algorithm based on Monte Carlo approach ▪ Instances sampled repeatedly from LPAD by considering only relevant clauses ▪ Head atoms of resolving clauses chosen stochastically by a meta-interpreter ▪ Note: the probability of the query is the fraction of sampled instances where the query is true (no BDD is needed)
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   Real datasets: 10incremental samples of a graph describing biological entities responsible of Alzheimer’s disease Biological Graph HGNC_1505 PubMed_12653 0.643 0.493 0.665 EntrezGene_11803 PubMed_15529 0.523   Artificial datasets: 3 procedurally built graphs of increasing complexity Queries: compute the probability that a path exists between two given nodes of each graph 0.567 EntrezProtein_7891032 Tests: performed on Linux machines equipped with Intel Core 2 Duo E6550 (2333 MHz) and 4 Gb RAM with a 24 hours time limit PubMed_1741124 0.602 0.621 PubMed_157782 0.713 EntrezProtein_8147603 Artificial Graphs 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 “lanes” 0.3 0.3 0.3 0.3  HGNC_620 0.3 0.3 0.3 0.3 0.3 0.3 0.3 “branches” “parachutes”
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  12 10 8 6 4 2 0 Biological Graphs: CPU timesaveraged on successes 100000 Standard Best K Monte Carlo Size (edges) Time (log s) Answers Biological Graphs: number of successes 1000 Standard 10 Best K Monte Carlo 0,1 0,001 Size (edges)
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  Lanes Graphs: CPUtimes Parachutes Graphs: CPU times 100 Standard Best K Monte Carlo 0,01 Monte Carlo 0,000001 Size (steps) Size (steps) Branches Graphs: CPU times Time (log s) 100 1 1 3 5 7 9 11 13 15 17 19 0,01 Standard Best K Monte Carlo 0,0001 0,000001 Size (steps) Standard Best K 0,0001 0,0001 0,000001 1 1 23 45 67 89 111 133 155 177 199 221 243 265 287 0,01 Time (log s) 1 1 23 45 67 89 111 133 155 177 199 221 243 265 287 Time (log s) 100