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

• Stochastic automata represent probability distributions

• Probability of accepting u in L(A)

SA Productions

• Stochastic automata can be represented by labelled productions

becomes

SA Productions

• Stochastic automata can be represented by labelled productions

becomes

Stochastic CFGs

• Straightforward extension of SA• Probability of a string is the sum of

probabilities of its derivations

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.

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

Learning in SLPs

• Structure and parameters learned simultaneously

• Requires existing ILP framework

Derivation Overview

• Generalisation Model• Optimal parameter choice

– The general case– Two example case– Numerical solutions

Generalisation Model

• Add clauses one at a time

• Given SLP S and positive examples E, choose x:H such that:

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:

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:

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:

Optimal Parameter Choice

• Since maximizing choose x such that

• Derivative is messy, but since monotonicity preserves extrema we can maximize

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

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:

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:

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:

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:

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:

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

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

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

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

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:

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:

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:

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:

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:

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:

Generalisation Model

• If added to need to ensure weights sum to one.

• Replace weight of each in by• Want to choose x such that:

Two Parameter Solution

• Only two clauses, e1 and e2

• Then we can analytically derive:

Two Parameter Example

Two Parameter Example

Two Parameter Example

Two Parameter Example

Two Parameter Example

Two Parameter Example

Two Parameter Example

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

Iteration Method

• Transform g(x) = 0 into x = f(x)• Iterate:

• Converges so long as x0 close to root

Iteration Method

• In our case, use:

Iteration Method Example

Iteration Method Example