Stochastic Logic Programs

1. Stochastic Logic Programs Stephen Muggleton

2. Outline • Stochastic automata • Stochastic context free grammars • Stochastic logic programs • Stochastic SLD-refutations

3. Deterministic Automata

4. Stochastic Automata

5. Stochastic Automata

6. SA Probabilities • Stochastic automata represent probability distributions • Probability of accepting u in L(A)

7. SA Productions • Stochastic automata can be represented by labelled productions becomes

8. SA Productions • Stochastic automata can be represented by labelled productions becomes

9. Stochastic CFGs • Straightforward extension of SA • Probability of a string is the sum of probabilities of its derivations

10. Stochastic Logic Programs • Set of weighted range restricted definite clauses • Require for each predicate q, sum of weights for clauses with q in their head is 1.

11. SSLD-refutations • Analogous to stochastic CFG productions • A SSLD-refutation is a sequence: • Probability of is • Probability of an atom is where is an SLD refutation

12. Learning in SLPs • Structure and parameters learned simultaneously • Requires existing ILP framework

13. Derivation Overview • Generalisation Model • Optimal parameter choice • The general case • Two example case • Numerical solutions

14. Generalisation Model • Add clauses one at a time • Given SLP S and positive examples E, choose x:H such that:

15. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

16. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

17. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

18. Optimal Parameter Choice • Since maximizing choose x such that • Derivative is messy, but since monotonicity preserves extrema we can maximize

19. General Derivation • Assuming S is non-recursive then has one of the forms: • c if p does not involve a q clause • cx if p involves x : H • c(1-x) if p involves a clause in q other than x : H

20. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

21. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

22. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

23. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

24. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

25. General Derivation • Assuming S is non-recursive then has one of the forms: • c if p does not involve a q clause • cx if p involves x : H • c(1-x) if p involves a clause in q other than x : H

26. General Derivation • Assuming S is non-recursive then has one of the forms: • c if p does not involve a q clause • cx if p involves x : H • c(1-x) if p involves a clause in q other than x : H

27. General Derivation • Assuming S is non-recursive then has one of the forms: • c if p does not involve a q clause • cx if p involves x : H • c(1-x) if p involves a clause in q other than x : H

28. General Derivation • Assuming S is non-recursive then has one of the forms: • c if p does not involve a q clause • cx if p involves x : H • c(1-x) if p involves a clause in q other than x : H

29. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

30. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

31. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

32. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

33. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

34. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x to maximize the likelihood:

35. Generalisation Model • If added to need to ensure weights sum to one. • Replace weight of each in by • Want to choose x such that:

36. Two Parameter Solution • Only two clauses, e1 and e2 • Then we can analytically derive:

37. Two Parameter Example

38. Two Parameter Example

39. Two Parameter Example

40. Two Parameter Example

41. Two Parameter Example

42. Two Parameter Example

43. Two Parameter Example

44. Numerical Solutions • Analytical solutions for more clauses involve solving higher order polynomials • Exponentially many terms in polynomial • Numerical solutions a good idea • Use iteration method

45. Iteration Method • Transform g(x) = 0 into x = f(x) • Iterate: • Converges so long as x0 close to root

46. Iteration Method • In our case, use:

47. Iteration Method Example

48. Iteration Method Example