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Performing Bayesian Inference by Weighted Model Counting

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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

- 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

- 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

- Model counting
- Encoding Bayesian networks
- Related Bayesian inference algorithms
- Experiments
- Grid networks
- Plan recognition
- Conclusion

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

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

- 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

- 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

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

- 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

B

A

0.2

0.8

A

0.6

0.4

Encoding Bayesian Networks to Weighted Model CountingA

A

0.1

Chance variable P added with weight(P)=0.2

B

B

A

0.2

0.8

A

0.6

0.4

Encoding Bayesian Networks to Weighted Model CountingA

A

0.1

Chance variable Q added with weight(Q)=0.6

B

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

- 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

- 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

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

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

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

- 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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