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Algorithms for Probabilistic and Deterministic graphical Models
Class 1Rina Dechter
class1 828X-2018
Dechter-Morgan&claypool book (Dechter 1 book): Chapters 1-2
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Text Books
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Outline
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• Introduction: Constraint and probabilistic graphical models.
• Constraint networks: Graphs, modeling, Inference
• Inference in constraints: Adaptive consistency, constraint propagation, arc-conistency
• Graph properties: induced-width, tree-width, chordal graphs, hypertrees, join-trees
• Bayesian and Markov networks: Representing independencies by graphs
• Building Bayesian networks.
• Inference in Probabilistic models: Bucket-elimination (summation and optimization), Tree-decompositions, Join-tree/Junction-tree
algorithm
• Search in CSPs: Backtracking, pruning by constraint propagation,
backjumping and learning
• Search in Graphical models: AND/OR search Spaces for likelihood, optimization queries
• Approximate Bounded Inference: weighted Mini-bucket, belief-propagation,
generalized belief propagation
• Approximation by Sampling: MCMC schemes, Gibbs sampling, Importance sampling
• Causal Inference with causal graphs.
Class page
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Course Requirements/Textbook
• Homeworks : There will be 5-6 problem sets , graded 50% of the final
grades.
• A term project: paper presentation, a programming project (20%).
• Final (30%)
• Books:
• “Reasoning with probabilistic and deterministic graphical models”, R.
Dechter, Claypool, 2013
https://www.morganclaypool.com/doi/abs/10.2200/S00529ED1V01Y201
308AIM023
o “Modeling and Reasoning with Bayesian Networks”, A. Darwiche, MIT
Press, 2009.
o “Constraint Processing” , R. Dechter, Morgan Kauffman, 2003
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AI Renaissance
• Deep learning
– Fast predictions
– “Instinctive”
• Probabilistic models
– Slow reasoning
– “Logical / deliberative”
Tools: Tensorflow, PyTorch, …
Tools: Graphical Models,
Probabilistic programming,Markov Logic, …
5
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Outline of classes• Part 1: Introduction and Inference
• Part 2: Search
• Parr 3: Variational Methods and Monte-Carlo Sampling
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E K
F
LH
C
BA
M
G
J
DABC
BDEF
DGF
EFH
FHK
HJ KLM
0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0101010101010101010101010101010101010101010101010101010101010101
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1
EC
FD
BA 0 1
Context minimal AND/OR search graph
AOR0ANDBOR
0ANDOR E
OR F FAND 01
AND 0 1C
D D01
0 1
1EC
D D0 1
1B
0E
F F0 1
C1
EC
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• Basics of graphical models
– Queries
– Examples, applications, and tasks
– Algorithms overview
• Inference algorithms, exact
– Bucket elimination for trees
– Bucket elimination
– Jointree clustering
– Elimination orders
• Approximate elimination– Decomposition bounds
– Mini-bucket & weighted mini-bucket
– Belief propagation
• Summary and Part 2
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RoadMap: Introduction and Inference
ABC
BDEF
DGF
EFH
FHK
HJ KLM
A
D E
CBB C
ED
EK
F
L
H
C
BA
M
G
J
D
For Constraints first
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• Basics of graphical models
– Queries
– Examples, applications, and tasks
– Algorithms overview
• Inference algorithms, exact
– Bucket elimination for trees
– Bucket elimination
– Jointree clustering
– Elimination orders
• Approximate elimination
– Decomposition bounds
– Mini-bucket & weighted mini-bucket
– Belief propagation
• Summary and Class 2
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RoadMap: Introduction and Inference
ABC
BDEF
DGF
EFH
FHK
HJ KLM
A
D E
CBB C
ED
EK
F
L
H
C
BA
M
G
J
D
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Probabilistic Graphical models• Describe structure in large problems
– Large complex system
– Made of “smaller”, “local” interactions
– Complexity emerges through interdependence
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Probabilistic Graphical models• Describe structure in large problems
– Large complex system
– Made of “smaller”, “local” interactions
– Complexity emerges through interdependence
• Examples & Tasks– Maximization (MAP): compute the most probable configuration
[Yanover & Weiss 2002][Bruce R. Donald et. Al. 2016]
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• Protein Structure prediction: predicting the 3d structure from given sequences
• PDB: Protein design (backbone) algorithms enumerate a combinatorial number of candidate structures to compute the Global Minimum Energy Conformation (GMEC).
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Probabilistic Graphical models• Describe structure in large problems
– Large complex system
– Made of “smaller”, “local” interactions
– Complexity emerges through interdependence
• Examples & Tasks– Summation & marginalization
grass
plane
sky
grass
cow
Observation y Observation yMarginals p( xi | y ) Marginals p( xi | y )
and
“partition function”
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e.g., [Plath et al. 2009]
Image segmentation and classification:
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Graphical models• Describe structure in large problems
– Large complex system
– Made of “smaller”, “local” interactions
– Complexity emerges through interdependence
• Examples & Tasks– Mixed inference (marginal MAP, MEU, …)
Test
DrillOil sale
policy
Test
result
Seismic
structure
Oil
underground
Oil
produced
Test
cost
Drill
cost
Sales
cost
Oil
sales
Market
information
Influence diagrams &optimal decision-making
(the “oil wildcatter” problem)
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e.g., [Raiffa 1968; Shachter 1986]
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In more details…
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A Bred green
red yellow
green red
green yellow
yellow green
yellow red
Example: map coloring
Variables - countries (A,B,C,etc.)
Values - colors (red, green, blue)
Constraints: etc. ,ED D, AB,A
C
A
B
D
E
F
G
Constraint Networks
A
B
E
G
DF
C
Constraint graph
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Propositional Reasoning
• If Alex goes, then Becky goes:
• If Chris goes, then Alex goes:
• Question:Is it possible that Chris goes to the party but Becky does not?
Example: party problem
BA→
A C→
e?satisfiabl ,,
theIs
→→= C B, ACBA
theorynalpropositio
AB
C
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CELAR SCEN-06 n=100, d=44, m=350, optimum=3389
◼ CELAR SCEN-07r
n=162, d=44,
m=764, optimum=343592
Radio Link Frequency Assignment Problem (Cabon et al., Constraints 1999) (Koster et al., 4OR 2003)
Dechter, Flairs-2018
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Bayesian Networks (Pearl 1988)
P(S, C, B, X, D) = P(S) P(C|S) P(B|S) P(X|C,S) P(D|C,B)
lung Cancer
Smoking
X-ray
Bronchitis
Dyspnoea
P(D|C,B)
P(B|S)
P(S)
P(X|C,S)
P(C|S)
Θ) (G,BN=
CPD:
C B P(D|C,B)0 0 0.1 0.90 1 0.7 0.31 0 0.8 0.21 1 0.9 0.1
• Posterior marginals, probability of evidence, MPE
• P( D= 0) = σ𝑆,𝐿,𝐵,𝑋 P(S)· P(C|S)· P(B|S)· P(X|C,S)· P(D|C,B
MAP(P)= 𝑚𝑎𝑥𝑆,𝐿,𝐵,𝑋 P(S)· P(C|S)· P(B|S)· P(X|C,S)· P(D|C,B)
Combination: ProductMarginalization: sum/max
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An early exampleFrom medical diagnosis
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Alarm network• Bayes nets: compact representation of large joint distributions
PCWP CO
HRBP
HREKG HRSAT
ERRCAUTERHRHISTORY
CATECHOL
SAO2 EXPCO2
ARTCO2
VENTALV
VENTLUNG VENITUBE
DISCONNECT
MINVOLSET
VENTMACHKINKEDTUBEINTUBATIONPULMEMBOLUS
PAP SHUNT
ANAPHYLAXIS
MINOVL
PVSAT
FIO2
PRESS
INSUFFANESTHTPR
LVFAILURE
ERRBLOWOUTPUTSTROEVOLUMELVEDVOLUME
HYPOVOLEMIA
CVP
BP
The “alarm” network: 37 variables, 509 parameters (rather than 237 = 1011 !)
[Beinlich et al., 1989]
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Dechter, Flairs-2018
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Probabilistic reasoning (directed)
• Alex is-likely-to-go in bad weather• Chris rarely-goes in bad weather• Becky is indifferent but unpredictable
Questions:• Given bad weather, which group of individuals is most
likely to show up at the party? • What is the probability that Chris goes to the party
but Becky does not?
Party example: the weather effect
P(W,A,C,B) = P(B|W) · P(C|W) · P(A|W) · P(W)
P(A,C,B|W=bad) = 0.9 · 0.1 · 0.5
P(A|W=bad)=.9W A
P(C|W=bad)=.1W C
P(B|W=bad)=.5W B
W
P(W)
P(A|W)
P(C|W)P(B|W)
B CA
W A P(A|W)
good 0 .01
good 1 .99
bad 0 .1
bad 1 .9
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Mixed Probabilistic and Deterministic networks
P(C|W)P(B|W)
P(W)
P(A|W)
W
B A C
Query:Is it likely that Chris goes to the party if Becky does not but the weather is bad?
PN CN
),,|,( ACBAbadwBCP →→=
A→B C→A
B A CP(C|W)P(B|W)
P(W)
P(A|W)
W
B A C
A→B C→A
B A C
Alex is-likely-to-go in bad weatherChris rarely-goes in bad weatherBecky is indifferent but unpredictable
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Graphical models (cost networks)Example:
The combination operator defines an overall function from the individual factors,
e.g., “+” :
Notation:
Discrete Xi values called states
Tuple or configuration: states taken by a set of variables
Scope of f: set of variables that are arguments to a factor f
often index factors by their scope, e.g.,
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A graphical model consists of:
-- variables
-- domains
-- functions or “factors”
and a combination operator
(we’ll assume discrete)
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Graphical models (cost networks)
+= 0 + 6
A B f(A,B)
0 0 6
0 1 0
1 0 0
1 1 6
B C f(B,C)
0 0 6
0 1 0
1 0 0
1 1 6
A B C f(A,B,C)
0 0 0 12
0 0 1 6
0 1 0 0
0 1 1 6
1 0 0 6
1 0 1 0
1 1 0 6
1 1 1 12
=
For discrete variables, think of functions as “tables”
(though we might represent them more efficiently)
A graphical model consists of:
-- variables
-- domains
-- functions or “factors”
and a combination operator
Example:
(we’ll assume discrete)
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Graph Visualiization: Primal Graph
Primal graph:variables → nodesfactors → cliques
G
A
B C
D F
A graphical model consists of:
-- variables
-- domains
-- functions or “factors”
and a combination operator
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Example: Constraint networks
Overall function is “and” of individual constraints:
for adjacent regions i,j
“Tabular” form: X0 X1 f(X0 ,X1)
0 0 0
0 1 1
0 2 1
1 0 1
1 1 0
1 2 1
2 0 1
2 1 1
2 2 0Tasks: “max”: is there a solution?
“sum”: how many solutions?
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habits. smoking similar have Friends
cancer. causes Smoking
Markov logic, Markov networks
( ))()(),(,
)()(
ySmokesxSmokesyxFriendsyx
xCancerxSmokesx
1.1
5.1
Cancer(A)
Smokes(A)Friends(A,A)
Friends(B,A)
Smokes(B)
Friends(A,B)
Cancer(B)
Friends(B,B)
Two constants: Anna (A) and Bob (B)
SA CA f(SA,CA)
0 0 exp(1.5)
0 1 exp(1.5)
1 0 1.0
1 1 exp(1.5)
FAB SA SB f(.)
0 0 0 exp(1.1)
0 0 1 exp(1.1)
0 1 0 exp(1.1)
0 1 1 exp(1.1)
1 0 0 exp(1.1)
1 0 1 1.0
1 1 0 1.0
1 1 1 exp(1.1)
[Richardson & Domingos 2005]
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Graphical visualization
Primal graph:variables nodesfactors cliques
A graphical model consists of:
-- variables
-- domains
-- functions or “factors”
and a combination operator
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G
A
B C
D F
ABD
BCF AC
DFGD
B
C
A F
Dual graph:factor scopes nodesedges intersections (separators)
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Graphical visualization“Factor” graph: explicitly indicate the scope of each factor
variables circlesfactors squares
G
A
B C
D F
A
B
C
D
A
B
C
D
Useful for disambiguating factorization:
= vs.
O(d4) pairwise: O(d2)
A
B
C
D
?
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Graphical modelsA graphical model consists of:
-- variables
-- domains
-- functions or “factors”
Operators:
combination operator
(sum, product, join, …)
elimination operator
(projection, sum, max, min, ...)
Types of queries:
Marginal:
MPE / MAP:
Marginal MAP:
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)( : CAFfi +==
A
D
BC
E
F
• All these tasks are NP-hard
• exploit problem structure
• identify special cases
• approximate
A C F P(F|A,C)
0 0 0 0.14
0 0 1 0.96
0 1 0 0.40
0 1 1 0.60
1 0 0 0.35
1 0 1 0.65
1 1 0 0.72
1 1 1 0.68
Conditional Probability Table (CPT)
Primal graph(interaction graph)
A C F
red green blue
blue red red
blue blue green
green red blue
Relation
(𝐴⋁𝐶⋁𝐹)
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Graphical models/reasoning task
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Summary of graphical models types
• Constraint networks
• Cost networks
• Bayesian network
• Markov networks
• Mixed probability and constraint network
• Influence diagrams
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33
A Bred greenred yellowgreen redgreen yellowyellow greenyellow red
Map coloring
Variables: countries (A B C etc.)
Values: colors (red green blue)
Constraints: ... , ED D, AB,A
C
A
B
D
E
F
G
Constraint Networks
Constraint graph
A
BD
CG
F
E
Queries: Find one solution, all solutions, countingclass1 828X-2018
Combination = joinMarginalization = projection
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Example of a Cost Network
Combination: sumMarginalization:min/max
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A Bayesian Network
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Combination: productMarginalization: sum or min/max
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Markov Networks
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Example domains for graphical models• Natural Language processing
– Information extraction, semantic parsing, translation, topic models, …
• Computer vision– Object recognition, scene analysis, segmentation, tracking, …
• Computational biology– Pedigree analysis, protein folding and binding, sequence matching, …
• Networks– Webpage link analysis, social networks, communications, citations, ….
• Robotics– Planning & decision making
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Complexity of Reasoning Tasks
• Constraint satisfaction
• Counting solutions
• Combinatorial optimization
• Belief updating
• Most probable explanation
• Decision-theoretic planning
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Linear
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Exponential
Reasoning iscomputationally hard
Complexity is
Time and space(memory)class1 828X-2018
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Desired Properties: Guarantee, Anytime, Anyspace
• Anytime – valid solution at any point– solution quality improves with additional computation
• Anyspace– run with limited memory resources
39
time
Bounded error
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• Basics of graphical models
– Queries
– Examples, applications, and tasks
– Algorithms overview
• Inference algorithms, exact
– Bucket elimination for trees
– Bucket elimination
– Jointree clustering
– Elimination orders
• Approximate elimination
– Decomposition bounds
– Mini-bucket & weighted mini-bucket
– Belief propagation
• Summary and Class 2
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RoadMap: Introduction and Inference
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Tree-solving is easy
Belief updating
(sum-prod)
MPE (max-prod)
CSP – consistency
(projection-join)
#CSP (sum-prod)
P(X)
P(Y|X) P(Z|X)
P(T|Y) P(R|Y) P(L|Z) P(M|Z)
)(XmZX
)(XmXZ
)(ZmZM)(ZmZL
)(ZmMZ)(ZmLZ
)(XmYX
)(XmXY
)(YmTY
)(YmYT
)(YmRY
)(YmYR
Trees are processed in linear time and memory
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Transforming into a Tree • By Inference (thinking)
– Transform into a single, equivalent tree of sub-problems
• By Conditioning (guessing)
– Transform into many tree-like sub-problems.
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Inference and Treewidth
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treewidth = 4 - 1 = 3treewidth = (maximum cluster size) - 1
Inference algorithm:Time: exp(tree-width)
Space: exp(tree-width)
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Conditioning and Cycle cutset
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Cycle cutset = {A,B,C}
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Search over the Cutset
A=yellow A=green
B=red B=blue B=red B=blueB=green B=yellow
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• Inference may require too much memory
• Condition on some of the variables
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GraphColoringproblem
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Inference
exp(w*) time/space
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0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
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A 0 1 SearchExp(w*) timeO(w*) space
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Search+inference:Space: exp(q)Time: exp(q+c(q))
q: usercontrolled
Bird's-eye View of Exact Algorithms
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Inference
exp(w*) time/space
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0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
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Search+inference:Space: exp(q)Time: exp(q+c(q))
q: usercontrolled
Context minimal AND/OR search graph
18 AND nodes
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Bird's-eye View of Exact Algorithms
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A
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F0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
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Inference
Bounded Inference
Search
Sampling
Search + inference:
Sampling + bounded inference
Bird's-eye View of Approximate Algorithms
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Context minimal AND/OR search graph
18 AND nodes
AOR0ANDBOR
0AND
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OR F F
AND0 1
AND 0 1
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0 1
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End of slides