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Stochastic Logic Programs

Stochastic Logic Programs. Stephen Muggleton. Outline. Stochastic automata Stochastic context free grammars Stochastic logic programs Stochastic SLD-refutations. Deterministic Automata. Stochastic Automata. Stochastic Automata. SA Probabilities.

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Stochastic Logic Programs

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  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

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