Probabilistic Ontologies: Logical Approach

Transcription

1 Probabilistic Ontologies: Logical Approach Pavel Klinov Applied Artificial Intelligence Lab ECE Department University of Cincinnati

2 Agenda Why do we study ontologies? Uncertainty Probabilistic ontologies One relevant medical problem Demonstration

3 Why do we need ontologies? Many domains experience explosive data growth Biomedicine Web Human understanding is hampered by volumes Machine understanding is hampered by the lack of semantics Conclusion: data should be made machine understandable in order to be useful

4 Roles of Ontologies Ontology explicit and formal specification of terms and relationships in a domain [Gruber, 1993] Ontologies supply semantics (meaning) to data They help to: Share common understanding of the domain among people or machines Allow reusing of domain knowledge Make domain assumptions explicit Separate domain knowledge from the operational (factual) knowledge Analyze domain knowledge

5 Some examples Some medical ontologies: NCI Thesaurus UMLS SNOMED GALEN

6 One typical scenario (NCBO) Open biomedical ontologies are learned or constructed by domain experts Ontologies are used to annotate open biomedical data Annotated data is analyzed/visualized through BioPortal Ontologies are revised

7 Ontologies Are Different Bodies of Expert Knowledge NCI Cancer Ontology Integrated Vocabularies Unified Medical Language System Upper-Level Ontologies Cyc Linguistic Resources WordNet

8 Ontologies can be classified by: Level of syntactic formality Highly informal (glossaries) Structured informal (thesauri, taxonomies) Formal (expressed in a formal language) Level of semantic formality Implicit, informal Explicit, informal Explicit and formal OWL ontologies for the Semantic Web

9 Logical Foundations Why logic? Logic is a formal language Logical statements have well-defined formal and explicit semantics Logic supports automated reasoning Logics for ontologies First Order Logic Very expressive but only semidecidable Description Logic Expressive and decidable

10 DL Basics Concepts (terms, classes) E.g., Person, Doctor, HappyParent, (Doctor Lawyer) Roles (relationships) E.g., haschild, loves Individuals E.g., John, Mary, Italy Operators (for forming concepts and roles) restricted so that: Satisfiability/subsumption is decidable and, if possible, of low complexity No need for explicit use of variables Restricted form of and Features such as counting succinctly expressed

11 DL Family (1) Smallest interesting DL is ALC (equivalent to K (m) ) Concepts constructed using booleans,,, plus restricted quantifiers, Only atomic roles E.g., Person all of whose children are either Doctors or have a child who is a Doctor: Person haschild.(doctor haschild.doctor)

12 DL Family (2) S often used for ALC extended with transitive roles i.e., the union of K (m) and K4 (m) Additional letters indicate other extensions, e.g.: H for role hierarchy (e.g., hasdaughter haschild) O for nominals/singleton classes (e.g., {Italy}) I for inverse roles (converse modalities) Q for qualified number restrictions N for number restrictions S + role hierarchy (H) + nominals (O) + inverse (I) + NR (N) = SHOIN SHOIN is the basis for W3C s OWL Web Ontology Language

13 DL Knowledge Base A TBox is a set of schema axioms (sentences), e.g.: {Doctor Person, HappyParent Person haschild.(doctor haschild.doctor)} i.e., a background theory An ABox is a set of data axioms (ground facts), e.g.: {John:HappyParent, John haschild Mary} i.e., factual knowledge TBox + ABox = DL Knowledge Base

14 Uncertainty

15 Imprecision is ubiquitous Uncertainty 90% of birds fly. A bird is likely to fly Ambiguity Washington. State? City? Team? President? Vagueness Retrieve all inexpensive hotels nearby Subjectivity I believe that not all birds can fly

16 Imprecision in the Semantic Web Important for: Knowledge representation Data integration Example: ontology alignment is usually uncertain Information and knowledge retrieval Query processing To what degree an object (Web page, text passage) matches my information need? Annotations Intelligent agent dialogue

17 Probabilistic Uncertainty Probability theory is the best studied theory of managing uncertainty Well suited for capturing: Statistics Degrees of belief Conditionals Natural question: can semantics of uncertain statements in ontologies be captured probabilistically?

18 Probabilistic Ontologies Classical ontologies augmented with uncertain assertions: Uncertain subclass relationships. Birds are flying objects with probability >0.9 Uncertain individual statement. Tweety is a flying object with probability <0.05 Classical ontological languages (DL-based) do not provide built-in constructs for the representation of such statements Language requirements: Syntactic constructs for representing conditional statements Well-defined formal semantics Plausible entailment relations

19 Approaches to probabilistic ontologies Bayesian approaches Mapping to classical Bayesian networks Severe loss of expressivity! Multi-Entity Bayesian Networks (MEBN) The formalism based on instantiations of BN Logical approaches P-SHOIN(D) generalization of default probabilistic propositional logic

20 Propositional Probabilistic Logic Integration of propositional logic and probabilistic calculus KB include both, logical formulas and interval restrictions on conditional probabilities (conditional constraints) Semantics is provided using probability distributions over classical models Entailments are logical (model-theoretic)!

21 Syntax of Probabilistic Knowledge Bases Finite set of basic events Φ={p 1,..,p n } Event φ: Boolean combination of basic events Conditional constraint (ψ φ)[l,u]: ψ,φ events, [l,u] [0,1] Probabilistic knowledge base PKB=(L,P): Finite set of events L Finite set of conditional constraints P

22 Some Examples Probabilistic KB: L = {penguin bird} All penguins are birds P = {(fly bird)[0.9,0.95]; (wings bird)[0.99,1.0]; (fly penguin)[0,0.05]} Most of birds fly Almost all birds have wings Penguins almost never fly

23 Semantics of Probabilistic KBs World: truth assignment to all basic events Possible world I: world that satisfies all propositional formulas (events) I Φ : all possible worlds Probabilistic interpretation Pr: probability distribution on I Φ Pr(φ): sum over all Pr(I) such that I satisfies φ Pr(ψ φ) = Pr(ψ φ)/pr(φ) Truth under Pr: Pr =(ψ φ) [l,u] either Pr(φ) = 0 or l Pr(ψ φ) u Given φ, the probability of ψ is within [l,u]

24 Example again Basic events Φ={bird,fly} I Φ ={I 1,I 2,I 3,I 4 } Possible probabilistic distributions: So: Pr 1 (fly bird) = 19/40 and Pr 1 (bird) = 20/40 Pr 2 (fly bird) = 0 and Pr 2 (bird) = 1/3 Pr 1 (bird -> fly) < 1 but Pr 2 (bird -> fly) = 1 Pr 1 entails (fly bird)[0.95,1] but Pr 2 does not

25 Satisfiability and Logical Entailment Pr is a model of PKB=(L,P) Pr is built over possible worlds of L and Pr satisfies all conditional constraints in P PKB is satisfiable model exists Logical consequence: PKB =(ψ φ) [l,u] every model of PKB satisfies (ψ φ) [l,u] Tight logical consequence: PKB = tight (ψ φ) [l,u] l (resp. u) is min (resp. max) over all Pr(ψ φ) s.t. Pr(φ) > 0

26 Example PKB = (L,P) L = {penguin -> bird} P= {(fly bird)[0.9,0.95]; (wings bird)[0.99,1.0]; (fly penguin)[0,0.05]} PKB is satisfiable (believe me!) Some logical consequences: PKB =(wings bird)[0.8,1]; PKB =(fly bird)[0.5,1] Trivial tight logical consequences: PKB =(wings bird)[0.99,1]; PKB =(fly bird)[0.9,0.95]

27 Is there a model? Satisfiability can be decided by solving a system of linear inequalities Each conditional constraint imposes two inequalities on probabilistic models If all inequalities can be satisfied => model exists

28 Computing Tight Logical Consequence Goal: compute [l,u] s.t. PKB = tight (ψ φ) [l,u] Produce the same linear system as for satisfiability + extra constraint (φ T)[1,1] Add objective function sum over all worlds that satisfy ψ Minimize (resp. maximize) the function to compute l (resp. u) This is linear programming problem => can use existing LP solvers

29 The Problem Logical entailment is unsatisfactory PKB = (L,P) L = {penguin -> bird} P= {(fly bird)[0.9,0.95]; (wings bird)[0.99,1.0]; (fly penguin)[0,0.05]} PKB = tight (fly penguin) [l,u] fails (linear system unsolvable) Why?!

30 Conflicts The problem is that constraints (fly bird)[0.9,0.95] and (fly penguin)[0,0.05] are in conflict given penguin -> bird So for any model: Pr(penguin) = 0, so the extra constraint (penguin T)[1,1] makes the system unsolvable Solution? We need stronger notion of entailment to plausibly resolve such conflicts

31 Default Reasoning Constraints express default knowledge, i.e., it s assumed to be true in general but might fail in specific cases (like for penguins) Overriding: When general statements are in conflicts with more specific ones, the latter should be preferred Penguins don t fly in spite of being birds Inheritance: If there re no conflicts, general statements are assumed to hold Penguins have wings because of being birds Reasoning becomes non-monotonic!

32 Lexicographic Entailment(1) Pr verifies (ψ φ) [l,u] Pr(φ) = 1 and Pr =(ψ φ)[l,u] P tolerates (ψ φ) [l,u] under L L P has a model that verifies (ψ φ) [l,u] {(fly penguin)[0,0.05]} tolerates (fly bird)[0.9,0.95] under {penguin->bird} but not vice versa PKB = (L,P) is consistent there exists an ordered partition s.t. all constraints are tolerated by those with higher index P = {(fly bird)[0.9,0.95] } {(fly penguin)[0,0.05]} This is known as z-partition

33 Lexicographic Entailment(2) Given z-partition it s possible to build preference relation on models The most preferred models are called lexicographically minimal Informally: lex. minimal models satisfy most specific constraints PKB = lex (ψ φ) [l,u] (ψ φ) [l,u] is satisfied by every lex minimal model of PKB

34 Example PKB = (L,P) L = {penguin -> bird} P= {(fly bird)[0.9,0.95]; (wings bird)[0.99,1.0]; (fly penguin)[0,0.05]} z-partition: 0: {(fly bird)[0.9,0.95]; (wings bird)[0.99,1.0]} 1: {(fly penguin)[0,0.05]} Deriving (fly penguin)[l,u]: lex minimal models those satisfying {(fly penguin)[0,0.05]; (wings bird)[0.99,1.0]; (penguin T)[1.0,1.0]} Result: (fly penguin)[0,0.05] as it should be Also, PKB = tlex (wings penguin)[0.99,1] inheritance at work

35 Probabilistic DL(1) Generalization is easy because of variable-free, fu nction-free syntax of DLs: Basic events <-> atomic concepts Formulas <-> complex concepts Conditional constraints: (D C)[l,u] where C,D arbitrary concepts (FlyingObject Penguin)[0,0.05] All generic constraints (TBox) are default There are also strict individual constraints: (C T)[l,u] for given individual a (Penguin T)[0.9,0.9] for Tweety Probabilistic ontology: PKB = (T,PT,(PA) a ) where T classical TBox, PT defaults,,(pa) a - collections of strict constraints indexed by individuals.

36 Probabilistic DL(2) Semantics is defined in a syntactic way World I: complete characterization of some fresh individual the world specifies which concepts the individual belongs to. I 1 ={Bird,Penguin} This is provably equivalent to the traditional DL semantics Allows all definitions and notions (e.g., satisfiability, entailments, etc.) to migrate from probabilistic propositional logic to probabilistic DL

37 Example Probabilistic ontology PO={T,PT,(PA) a } T = {Penguin Bird} PT = {(FlyingObject Bird)[0.9,0.95]; (WingedObject Bird)[0.99,1.0]; (FlyingObject Penguin)[0,0.05]; (PA) Tweety = {(Penguin T)[0.9,0.9]} (PA) Sam = {(Bird T)[1,1]} Lexicographic entailments: (FlyingObject T)[0.9,0.95] for Sam (FlyingObject T)[0,0.05] for Tweety (WingedObject T)[0.99,1.0] for Tweety, Sam

38 Pronto: Probabilistic DL Reasoner

39 Overview Pronto provides representation and reasoning in P-SHIN(D) probabilistic DL formalism Pronto is built on top of Pellet classical DL reasoner Features: Capturing uncertainty in terminological and individual axioms Reasoning. Entailing both, TBox and ABox conditional constraints Probabilistic explanations

40 Knowledge Representation Probabilistic axioms are represented using axiom annotations (OWL 1.1 feature) This allows attaching uncertainty to any TBox or ABox axioms Example (RDF/XML syntax): <owl11:axiom> <rdf:subject rdf:resource= #Bird"/> <rdf:predicate rdf:resource="&rdfs;subclassof"/> <rdf:object rdf:resource= #FlyingObject"/> <pronto:certainty>0.9;0.95</pronto:certainty> </owl11:axiom> Probabilistic part can import classical part of the ontology or they can be mixed

41 Reasoning Main inference tasks: Probabilistic satisfiability (PSAT) Consistency G-consistency (z-partition) Overall consistency Tight logical entailment Tight lexicographic entailment Subsumption entailment. What s the probability that penguins fly? Membership entailment. What s the probability that Tweety flies? Pronto uses Lehmann s lexicographic entailment

42 Architecture(1)

43 Architecture(2) Loader: Loads classical and probabilistic parts of ontology Lexicographic reasoner: Makes inferences PSAT solver: uses LP techniques to solve PSAT/TLogEnt Consistency checker: decides probabilistic consistency (generates z-partition) Query processor: Accepts queries and reduces them to main inference problems Explanator: Generates explanations of inference results

44 Explanations Why a particular inference has been made? The bigger is KB, the more obscure are the inferences Goal produce explanation set: minimal fragment of KB that supports the inference (may not be unique)

45 Probabilistic Explanations Now two components: Classical (subset of classical ontology) Probabilistic (subset of conditional constraints) Pronto computes all probabilistic components The main difficulty non-monotonicity What does minimal mean now? Solution restore reasoning process exactly Pronto: Explains preference relation on models using specificity relation on constraints Uses pin-pointing to filter out irrelevant constraints

46 Breast Cancer Risk Assessment (amateur view!)

47 Introduction Goal: estimate risk of developing breast cancer given set of factors: Age Ethnicity (e.g., being an Ashkenazi Jew is a risk) Medical history (whether immediate relatives have B RC) Genetics (e.g., BRCA1, BRCA2 gene mutations) Risks Absolute Short-term (5-10 years) Lifetime Relative Increased Decreased

48 BRC and Uncertainty The domain is highly uncertain: Not all risk factors are known Not all risk factors have been sufficiently investigated (e.g., gene mutations) Not all relationships between risk factors are sufficiently studied Presence of certain factors cannot always be precisely determined => errors (e.g., unreliable medical history) This issues complicate use of NCI thesaurus one of the largest formal ontologies in the world!

49 Statistical models(1) One approach is to use a black-box statistical modeling Assumption: risk is a function of presence/absence of risk factors

50 Statistical models(2) Advantages: Fast calculation Can handle large number of parameters Can be statistically validated Disadvantages Non-transparent (results aren t easily interpretable) Adding new factors isn t always easy Multiple models are hard to integrate

51 Gail Model Used by the NCI risk calculator ( Extensively uses medical history Based on extensive statistics Study involved > 280,000 women Produces adequate results on most of women Underestimates risk of African American women Unsatisfactory for some ethnic groups

52 Ontological modeling One alternative is to re-use existing medical ontologies (NCI thesaurus) Key idea: reduce risk assessment to ontological reasoning What is probability that Amy belongs to the concept WomenToDevelopBRCInNext10Years In terms of probabilistic DL, this is simply membership entailment Of course, we need a probabilistic reasoner

53 Pronto and BRCA The approach is the following Create probabilistic ontology where: Classical part = NCI thesaurus + extra classes to model risks, categories of women and risk factors Probabilistic part = set of conditional constraints that represent statistical knowledge* (WomenWithBRCInLongTerm Women)[0;0.13] Individual women can supply knowledge about their factors as ABox assertions: Certain. Amy:WomenAfter50 Uncertain. Genes, bone density, etc Pronto reasoner computes risks as interval probabilities PKB = tlex (WomenWithBRCInShortTerm T)[0.027;0.041] for Amy * Taken from:

54 BRC Model: Classical part Risks Absolute Short-term Long-term Relative Risk factors Known Inferred Categories of women Women having particular risk factors evidence classes Women at particular risk conclusion classes

55 BRC Model. Probabilistic Part Statistics captured in the form of TBox default conditional constraints WomenWithBRCFactors -> WomenUnderBRCRisk They can be overridden By more specific generic constraints. Useful for capturing combinations of factors: What if Amy is over 50 and has BRC history? By strict individual constraints Beliefs captured in the form of ABox strict conditional constraints It is assumed that statistics and beliefs can be combined (debatable!)

56 BRCA Explanations Main goal: Explain why one s risk is in some interval. How? Filter out all irrelevant risk factors Example: Helen is between 50 and 60 Her mother was BRC affected Pronto computes: Helen has >90% probability of being in the highest risk category. Why? Explanation set: (MediumRisk WomenOver50)[0.7,1.0] (HighestRisk WomenWithMotherBRCAffected)[0.31,0.31] (HighestRisk SeniorWomenWithMotherBRCAffected)[0.9,1] The last constraint overrides the previous one! All other constraints (risk factors) are ignored as not relev ant to Helen

57 Short Demo

58 Conclusion Handling uncertainty is important as it allows to conceptualize more domains Logics can be extended to handle uncertainty Existing ontologies can be reused and augmented with probabilistic axioms Challenges: Performance and scalability. Pronto can currently handle constraints Expressivity Different types of imprecision. Probability + Fuzzy? Query answering Debugging and explanations