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.

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Performing Bayesian Inference by Weighted Model Counting

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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


Performing bayesian inference by weighted model counting

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


Performing bayesian inference by weighted model counting

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 counting

B

B

A

0.2

0.8

A

0.6

0.4

Encoding Bayesian Networks to Weighted Model Counting

A

A

0.1

B


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


Encoding bayesian networks to weighted model counting5

B

B

A

0.2

0.8

A

0.6

0.4

Encoding Bayesian Networks to Weighted Model Counting

A

A

0.1

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...


Techniques

Techniques


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


Results of ratio 0 75

Results of ratio=0.75


Results of ratio 0 9

Results of ratio=0.9


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


Iscas satlib benchmarks

ISCAS/SATLIB Benchmarks


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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