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Performing Bayesian Inference by Weighted Model Counting. Tian Sang, Paul Beame, and Henry Kautz Department of Computer Science & Engineering University of Washington Seattle, WA. Goal.

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performing bayesian inference by weighted model counting

Performing Bayesian Inference by Weighted Model Counting

Tian Sang, Paul Beame, and Henry Kautz

Department of Computer Science & Engineering

University of Washington

Seattle, WA

slide2
Goal
  • Extend success of “compilation to SAT” work for NP-complete problems to “compilation to #SAT” for #P-complete problems
    • Leverage rapid advances in SAT technology
    • Example: Computing permanent of a 0/1 matrix
    • Inference in Bayesian networks (Roth 1996, Dechter 1999)
  • Provide practical reasoning tool
  • Demonstrate relationship between #SAT and conditioning algorithms
    • In particular: compilation to DNNF (Darwiche 2002, 2004)
contributions
Contributions
  • Simple encoding of Bayesian networks into weighted model counting
  • Techniques for extending state-of-the-art SAT algorithms for efficient weighted model counting
  • Evaluation on computationally challenging domains
    • Outperforms join-tree methods on problems with high tree-width
    • Competitive with best conditioning methods on problems with high degree of determinism
outline
Outline
  • Model counting
  • Encoding Bayesian networks
  • Related Bayesian inference algorithms
  • Experiments
    • Grid networks
    • Plan recognition
  • Conclusion
sat and sat
SAT and #SAT
  • Given a CNF formula,
    • SAT: find a satisfying assignment n
    • #SAT: count satisfying assignments
  • Example: (x  y)  (y  z)
    • 5 models:

(0,1,0), (0,1,1), (1,1,0), (1,1,1), (1, 0, 0)

    • Equivalently: satisfying probability = 5/23
      • Probability that formula is satisfied by a random truth assignment
  • Can modify Davis-Putnam-Logemann-Loveland to calculate this value
slide6
DPLL for SAT

DPLL(F)

if F is empty, return 1

if F contains an empty clause, return 0

else choose a variable x to branch

return (DPLL(F|x=1) V DPLL(F|x=0))

#DPLL for #SAT

#DPLL(F) // computes satisfying probability of F

if F is empty, return 1

if F contains an empty clause, return 0

else choose a variable x to branch

return 0.5*#DPLL(F|x=1 )+ 0.5*#DPLL(F|x=0)

weighted model counting
Weighted Model Counting
  • Each literal has a weight
    • Weight of a model = Product of weight of its literals
    • Weight of a formula = Sum of weight of its models

WMC(F)

if F is empty, return 1

if F contains an empty clause, return 0

else choose a variable x to branch

return weight(x) * WMC(F|x=1) +

weight(x) * WMC(F|x=0)

cachet
Cachet
  • State of the art model counting program (Sang, Bacchus, Beame, Kautz, & Pitassi 2004)
  • Key innovation: sound integration of component caching and clause learning
    • Component analysis(Bayardo & Pehoushek 2000): if formulas C1 and C2 share no variables,

BWMC (C1 C2) = BWMC (C1) * BWMC (C2)

    • Caching (Majercik & Littman 1998; Darwiche 2002; Bacchus, Dalmao, & Pitassi 2003; Beame, Impagliazzo, Pitassi, & Segerland 2003): save and reuse values of internal nodes of search tree
    • Clause learning(Marquis-Silva 1996; Bayardo & Shrag 1997; Zhang, Madigan, Moskewicz, & Malik 2001): analyze reason for backtracking, store as a new clause
cachet1
Cachet
  • State of the art model counting program (Sang, Bacchus, Beame, Kautz, & Pitassi 2004)
  • Key innovation: sound integration of component caching and clause learning
    • Naïve combination of all three techniques is unsound
    • Can resolve by careful cache management (Sang, Bacchus, Beame, Kautz, & Pitassi 2004)
    • New branching strategy (VSADS) optimized for counting (Sang, Beame, & Kautz SAT-2005)
computing all marginals
Computing All Marginals
  • Task: In one counting pass,
    • Compute number of models in which each literal is true
    • Equivalently: compute marginal satisfying probabilities
  • Approach
    • Each recursion computes a vector of marginals
    • At branch point: compute left and right vectors, combine with vector sum
    • Cache vectors, not just counts
  • Reasonable overhead: 10% - 40% slower than counting
encoding bayesian networks to weighted model counting1

B

B

A

0.2

0.8

A

0.6

0.4

Encoding Bayesian Networks to Weighted Model Counting

A

A

0.1

Chance variable P added with weight(P)=0.2

B

encoding bayesian networks to weighted model counting2

B

B

A

0.2

0.8

A

0.6

0.4

Encoding Bayesian Networks to Weighted Model Counting

A

A

0.1

and weight(P)=0.8

B

encoding bayesian networks to weighted model counting3

B

B

A

0.2

0.8

A

0.6

0.4

Encoding Bayesian Networks to Weighted Model Counting

A

A

0.1

Chance variable Q added with weight(Q)=0.6

B

encoding bayesian networks to weighted model counting4

B

B

A

0.2

0.8

A

0.6

0.4

Encoding Bayesian Networks to Weighted Model Counting

A

A

0.1

and weight(Q)=0.4

B

main theorem
Main Theorem
  • Let:
    • F = a weighted CNF encoding of a Bayes net
    • E = an arbitrary CNF formula, the evidence
    • Q = an arbitrary CNF formula, the query
  • Then:
exact bayesian inference algorithms
Exact Bayesian Inference Algorithms
  • Junction tree algorithm (Shenoy & Shafer 1990)
    • Most widely used approach
    • Data structure grows exponentially large in tree-width of underlying graph
  • To handle high tree-width, researchers developed conditioning algorithms, e.g.:
    • Recursive conditioning (Darwiche 2001)
    • Value elimination (Bacchus, Dalmao, Pitassi 2003)
    • Compilation to d-DNNF (Darwiche 2002; Chavira, Darwiche, Jaeger 2004; Darwiche 2004)
  • These algorithms become similar to DPLL...
experiments
Experiments
  • Our benchmarks: Grid, Plan Recognition
    • Junction tree - Netica
    • Recursive conditioning – SamIam
    • Value elimination – Valelim
    • Weighted model counting – Cachet
  • ISCAS-85 and SATLIB benchmarks
    • Compilation to d-DNNF – timings from (Darwiche 2004)
    • Weighted model counting - Cachet
experiments grid networks

S

T

Experiments: Grid Networks
  • CPT’s are set randomly.
  • A fraction of the nodes are deterministic, specified as a parameter ratio.
  • T is the query node
results of ratio 0 5
Results of ratio=0.5

10 problems of each size, X=memory out or time out

plan recognition
Plan Recognition
  • Task:
    • Given a planning domain described by STRIPS operators, initial and goal states, and time horizon
    • Infer the marginal probabilities of each action
  • Abstraction of strategic plan recognition: We know enemy’s capabilities and goals, what will it do?
  • Modified Blackbox planning system (Kautz & Selman 1999) to create instances
summary
Summary
  • Bayesian inference by translation to model counting is competitive with best known algorithms for problems with
    • High tree-width
    • High degree of determinism
  • Recent conditioning algorithms already make use of important SAT techniques
    • Most striking: compilation to d-DNNF
  • Translation approach makes it possible to quickly exploit future SAT algorithms and implementations
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