A Language and Execution Model for the Inference of Uncertain Events in Active Systems

1. stockQuote stockPurchase illegalStockTrading stockValue stockSell Technion – Israel Institute of Technology A Language and Execution Model for the Inference of Uncertain Events in Active Systems Segev Wasserkrug Advisors: Avigdor Gal, Opher Etzion

2. Active Systems • Systems that contains active (event-driven) components illegalStockTrading stockPurchase stockSell • Problem: not all events of interest are signaled. Some must be inferred • Solution: Deterministic event composition languages defined • Problem: Uncertainty associated with events. Examples: • Inherent uncertainty associated with inference. • Noisy/unreliable sensors in a sensor network • Illegal stock trading – inherently uncertain inference

3. Research Goal • A language and execution model for the inference of uncertain events in active systems: • Language: • Represent uncertainty regarding events • Define uncertain inference rules • Define semantics • Execution Model: • Algorithm(s) for event occurrence probability quantification • Calculate correct probabilities as defined by rule semantics • Based on rules and incoming information regarding event occurrence

4. E6 Er E4 E2 Probabilistic Logics (Representation) E1 E3 Existing Works • Deterministic Event Composition Systems (e.g. ODE, SMRL) • No uncertainty handling mechanism • Uncertainty regarding the occurrence time of events in distributed systems (Liebig Et. Al.) • Budding works in the area of sensor networks (Garofalakis Et. Al, Wang Et. Al.) • Bayesian Networks for the representation of noisy/unreliable sensor readings • Deterministic inference language – translated to individual probabilistic queries Builds Upon KBMC Bayesian Networks (Quantification)

5. e1 e2 1 2 3 4 5 e2 e1 1 2 3 4 5 6 Events and Event Histories • Event: An actual occurrence that is significant, instantaneous and atomic • Example: Illegal Stock Trading of Intel Corporation Stock at 10AM carried out by Customer1 • Can be represented by a tuple of values, e.g. illegalStockQuote1=<10,INTC, $1,000,000, Customer1> • Event History(eh): • The set of all events between two points in time • Set of ordered tuples • Event history between time t1 and t2 denoted by

6. Event Uncertainty Representation • Event Instance Data (EID): The information the system has about the event • May be uncertain • Uncertainty about the attributes as well as the actual occurrence • Represented by sets of tuples and associated probability, e.g. • {notOccurred} w. p. 0.3 • <5, INTC, $1,000,000, Customer1> w.p. 0.3 • <10,INTC, $1,000,000, Customer1> w.p. 0.4 • Equivalent representation: RV with marginal distribution

7. e1 e2 1 2 3 4 5 e2 e1 1 2 3 4 5 6 Probability Space • Different probability space Ptfor every point in time t • Defining the occurrence probability of events until t • Intuitive Definition of Pt – Possible worlds semantics: • Every event history the system considers possible is a “possible world” • where: • is the set of possible worlds (event histories) • is a s-algebra over the possible worlds • is a probability measure over Ft • Computationally convenient definition of Pt in terms of RVs • Each RV represents the uncertainty associated with a specific event (EID) • Semantically equivalent when overall number of events is finite • Possible event histories represented by a system event history EH - set of EIDs(RVs) E1,…,Em

8. Rule Example • Semantics: • Quantitative: • If event history is such that condition holds • New event considered possible with defined probability. • Attribute values of event defined by mapping expressions • If condition does not hold – new event not possible.

9. Rule Example (Cont.) • Semantics: • Qualitative (independence): • Given state of relevant events – inferred event is probabilistically independent of all other events • For the above rule, only EIDs corresponding to events of either class1 or class2 are relevant • Used to increase inference efficiency

10. Rule Examples

11. Inference Example • EID E1 (corresponding to event e1 of type class1) reaches the system at time 5 • {notOccurred} w. p. 0.3 • {Occurred} w.p. 0.7 • EID E3 (class3)reaches the system at time 5: • {notOccurred} w. p. 0.5 • {Occurred} w.p. 0.5 • EID E2(class2)corresponding to event e2 reaches the system at time 5 • {notOccurred} w. p. 0.4 • {Occurred} w.p. 0.6

12. Inference Issues • Indeterminism: May different EIDs be inferred in different triggerings of the algorithm? • Resolved by defining an order on the rules • Termination: Can infinite rule triggering cycles occur? • A rule may be triggered at most once • Correctness: How to ensure that overall probability space adheres to specified semantics? • Create a Bayesian Network based on semantics • Efficiency: How inference efficiency be maintained while constantly updating Bayesian Network structure? • Smart updating of Bayesian Network Structure • Sampling algorithm that bypasses Bayesian Network construction

13. Bayesian Network Construction (Cont.) • r1: e1 and e2 both occur Þ e4 considered possible • Semantics + rule definition:

14. Bayesian Network Construction • r2: e2 and e3 both occur Þ e5 considered possible • Semantics + rule definition:

15. Bayesian Network Construction • r3: e4 occurs and e5 does not occur Þ e6 considered possible • Semantics + rule definition:

16. Example Probabilities • Updates to data: • e2 occurs with certainty 1 • e3 does not occur with certainty 1

17. Summary • Need and theory explained • Demonstrated by simple example • Solution contains: • Detailed underlying probabilistic theory • Advanced language • Three inference algorithms (language agnostic) • Infer from scratch • Updated inference • Sampling

18. Future Work • Enhance languages to include probabilistic predicates • Create learning algorithms to generate rules • Structure of rules – based on experience/expert knowledge • Probabilities inferred from historical data