Liftability of Probabilistic Inference: Upper and Lower Bounds
Manfred Jaeger and Guy Van den Broeck
Lifted Probabilistic Inference
–
[email protected], [email protected]
Formal Framework
• Inference in probabilistic logic models
Formulas in first-order predicate logic
Weights
• Weighted feature model KB:
• Query
for domain size n
• Informal definitions:
➔ “ … deals with groups of random variables at a first-order level”
➔ “The act of exploiting the high level structure in relational models is
called lifted inference”
➔ “The idea behind lifted inference is to carry out as much inference as
possible without propositionalizing”
➔ “lifted inference, which deals with groups of indistinguishable
variables, rather than individual ground atoms”
✗ Not very precise
• Classes of queries
and evidence
•
for single ground atoms
•
for terms of ground literals
•
for terms of ground literals of arity 0 or 1
•
for empty sets of evidence
• Classes of knowledge bases
•
for function-free first-order logic (no functions)
•
for relational first-order logic (no constants)
•
with quantifiers
•
with the equality predicate
• “2-” with two logical variables per formula
• Class of inference problems
Liftability – Definitions & Results
Definitions:
An algorithm is domain-lifted, iff for fixed KB, φ and ψ the computation of
is polynomial in n.
An algorithm is complete domain-lifted for
iff it is domain-lifted and solves this class.
A class of inference problems is liftable if there exists a complete domain-lifted algorithm for it.
• Typical empirical objective:
Polytime complexity in domain size
(number of objects in the world)
Liftability Results:
• Corresponding formal definition: Domain-lifted inference
Lower Complexity Bounds
Definition:
Let ψ be a sentence in first-order logic. The spectrum of ψ
is the set of integers n ∈ N for which ψ is satisfiable by an
interpretation of size n.
Theorems:
If NETIME≠ETIME, then there exists a first-order sentence
φ, such that {n | n ∈ spec(φ)} can not be recognized by a
deterministic algorithm in time polynomial in n.
If NETIME≠ETIME, then there does not exist an algorithm
that 0.25-approximately solves
in time
polynomial in the domain size. (Jaeger, 2000)
(Van den Broeck, 2011)
Positive Liftability Result
First-order knowledge compilation
compiles a first-order knowledge base into a
circuit language, FO d-DNNF.
● Evaluate weighted model count efficiently
in circuit for probabilsitic inference
●
First-order knowledge compilation is a
complete domain-lifted algorithm for the class
. (Van den Broeck, 2011)
Constructive proof that:
The class
is liftable.
Unbounded Queries and Evidence
Domain-lifted inference algorithms perform poorly with evidence
● Breaks symmetries in the first-order model
Definition:
An algorithm is domain-, query- and evidence- lifted, iff for fixed KB the
computation of
is polynomial in n and the size of φ and ψ.
●
#2SAT is #P-complete
● #2SAT reducible to computing
conditional probabilities in some KB Δ
Computing Δ
is #P-hard in
amount of evidence φ
●
(Van den Broeck and Davis, 2011)
If NETIME≠ETIME, then there does not exist an algorithm
that 0.25-approximately solves
.
If P≠NP, then
dqe-liftable.
?
M. Jaeger. On the complexity of inference about probabilistic relational models. Artificial Intelligence, 117:297–308, 2000.
G. Van den Broeck. On the completeness of first-order knowledge compilation for lifted probabilistic inference. In Proceedings of NIPS, 2011.
G. Van den Broeck and J. Davis. Conditioning in first-order knowledge compilation and lifted probabilistic inference. In Proceedings of AAAI, 2012.
First-order knowledge compilation is a
complete dqe-lifted algorithm for the
class
.
(Van den Broeck and Davis, 2011)
Δ has 2 logical variables per formula
We generalize to quantifier-free formulas without equality:
●
But certain evidence, of arity lower
than two is liftable:
Constructive proof that:
is not
is dqe-liftable.
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