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Automated Reasoning Group

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### Automated Reasoning Group

PI:

Adnan Darwiche, UCLA

http://www.cs.ucla.edu/~darwiche

Collaborators:

David Allen

Keith Cascio

Hei Chan

James Park

Key Results/Publications

KR’02: A logical approach to factoring belief networksAdnan Darwiche

AAAI’02: A distance measure for bounding probabilistic belief changeHei Chan and Adnan Darwiche

AAAI’02: A compiler for deterministic decomposable negation normal formAdnan Darwiche

AAAI’02: Using weighted MAX-SAT to approximate MPEJames Park

UAI’02: MAP complexity results and approximation methodsJames Park

TR-118: A differential semantics for jointree algorithmsJames Park and Adnan Darwiche

TR-130: Optimal time-space tradeoffs in probabilistic inferenceDavid Allen and Adnan Darwiche

Key Results

Factoring belief networks for exact inference:

- Exact inference with networks of treewidth > 60
- A new perspective on factoring belief networks
Bounding probabilistic belief change:

- New distance measure
- Applications to sensitivity analysis, belief revision and uncertain evidence

Key Results

MAP/MPE advances:

- New complexity results
- Most efficient MAP/MPE engines
Time-Space tradeoffs:

- Optimal utilization of space given time constraints
- Time-space tradeoff curves for real-world networks
SamIam Demo:

- Sensitivity engine
- MAP/MPE
- Time-Space tradeoffs

Recursive Conditioning

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Case-Analysis

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Case I

Case II

Decomposition

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Case I

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DecompositionCase I

Case II

Recursive Decomposition

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Time: O(n2w log n)

Space: O(n)

A

C

D

Decomposition TreeA

B

C

D

E

Time: O(n2w)

B

Space: O(n2w)

B

C

A

E

D

128

8

64

512

8

1024

32

1728 cache entries

Time-Space Tradeoffs

64 cache entries

rc(T)=cutset#(Tp)[cf(Tp)context#(Tp)+(1-cf(Tp))rc(Tp)]

Results

- Networks
- Barley
- Mildew
- Water
- Random

- Graphs
- Optimal time-space curves
- 8 byte cache values
- 3.5 million calls to RC per second

Random Network

- 40 nodes, 86 edges, width of 14 (non-binary nodes)
- Full Caching would require 767 MB
- Netica cannot compile network: needs ~6 GB
- Hugin cannot compile network: needs ~11 GB

Key Results

MAP/MPE advances:

- New complexity results
- Most efficient MAP/MPE engines
Time-Space tradeoffs:

- Optimal utilization of space given time constraints
- Time-space tradeoff curves for real-world networks
SamIam Demo:

- Sensitivity engine
- MAP/MPE
- Time-Space tradeoffs

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.99

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OFF

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.99

.01

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.80

.20

WEAK

0

1

DEAD

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Bayesian NetworkPr(Lights=ON | Battery-Power=OK) = .99

Query Types

- Pr: Posterior marginals
- MPE: Most probable instantiation
- MAP: Maximum a posteriori hypothesis

Pr: Posterior Marginals

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MPE: Most Probable Explanation

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MPE: Most Probable Explanation

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MAP: Maximum a Posteriori Hypothesis

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MAP: Maximum a Posteriori Hypothesis

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MAP: Maximum a Posteriori Hypothesis

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Complexity Results

- MPE is effectively an optimization problem
- MPE is NP-complete
- MPE is usually solved using counting algorithms!

- Pr is effectively a counting problem
- Pr is PP-complete (Roth 96)

- MAP requires both optimization and counting
- MAP is NPPP-complete
- MAP is NP-complete for polytrees

- NP PP NPPP PHNPPP

Local Search +BP

- Previous work focused on: local search + exact inferenceApplicable when inference is tractable.
- Local search + approximate inference (BP)Both optimization and inference problems are intractable.

Experimental Results

- Tested on random networks
- 100 variables, 20-25 map variables, width about 13.

- Also real world networks
- Pigs
- Barley

# solved Exactly of 59

Worst found/actual

MPE

9

.015

MPE-Hill

41

.06

MPE-Shill

43

.21

ML

31

.34

ML-Hill

38

.46

ML-Shill

42

.72

Random NetworksMedian

Mean

Max

MPE-Hill

1

8.4

1.3x1011

3.1x1012

MPE-SHill

1

8.4

1.3x1011

3.1x1012

ML-Hill

1.0x104

3.6x107

3.4x1015

8.4x1016

ML-SHill

7.7x103

3.6x107

3.4x1015

8.4x1016

BarleyMin

Median

Mean

Max

MPE-Hill

1.0

1.7x105

1.5x107

3.3x108

MPE-SHill

1.0

2.5x105

4.5x1011

1.1x1013

ML-Hill

13.0

2.0x103

3.3x105

4.5x106

ML-SHill

13.0

1.2x104

8.2x105

8.2x106

PigsReducing MPE to MAXSAT

- MPE can be reduced to MAXSAT
- Compared 3 algorithms:
- Discrete Lagrangian Multipliers (DLM):MAXSAT algorithm
- Guided Local Search (GLS):MAXSAT algorithm
- Stochastic Local Search (SLS):A direct MPE solution technique based on stochastic local search

Big Networks

- The third set is not amenable to exact solution so we compare relative solution quality

Key Results

MAP/MPE advances:

- New complexity results
- Most efficient MAP/MPE engines
Time-Space tradeoffs:

- Optimal utilization of space given time constraints
- Time-space tradeoff curves for real-world networks
SamIam Demo:

- Sensitivity engine
- MAP/MPE
- Time-Space tradeoffs

Key Results/Publications

KR’02: A logical approach to factoring belief networksAdnan Darwiche

AAAI’02: A distance measure for bounding probabilistic belief changeHei Chan and Adnan Darwiche

AAAI’02: A compiler for deterministic decomposable negation normal formAdnan Darwiche

AAAI’02: Using weighted MAX-SAT to approximate MPEJames Park

UAI’02: MAP complexity results and approximation methodsJames Park

TR-118: A differential semantics for jointree algorithmsJames Park and Adnan Darwiche

TR-130: Optimal time-space tradeoffs in probabilistic inferenceDavid Allen and Adnan Darwiche

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