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:
where
is an SLD refutation
• Probability of
is
• Probability of an atom is
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