Bayesian Networks presentation

lettuceescargatoireAI and Robotics

Nov 7, 2013 (4 years and 1 day ago)

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A. Darwiche

Bayesian Networks


A. Darwiche

Bayesian Network

Battery Age

Alternator

Fan Belt

Battery

Charge Delivered

Battery Power

Starter

Radio

Lights

Engine Turn Over

Gas Gauge

Gas

Fuel Pump

Fuel Line

Distributor

Spark Plugs

Engine Start

A. Darwiche

Bayesian Network

Battery Age

Alternator

Fan Belt

Battery

Charge Delivered

Battery Power

Starter

Radio

Lights

Engine Turn Over

Gas Gauge

Gas

Fuel Pump

Fuel Line

Distributor

Spark Plugs

Engine Start

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

ON

OFF

OK

WEAK

DEAD

Lights

Battery Power

.99

.01

.20

.80

0

1

.99

θ
1

+ θ
2
= 1

A. Darwiche

A. Darwiche

A. Darwiche

A. Darwiche



No children
0
1
62.9
37.1
0.37 ± 0.48
Children ages 12-17
0
1
82.3
17.7
0.18 ± 0.38
Children ages 6-11
0
1
80.6
19.4
0.19 ± 0.4
Education
Under K12
K12
Some college
Bachelor degree
Master or PhD
1.34
10.9
35.1
33.0
19.6
3.59 ± 0.97
Marital Status
Single never married
Married
Divorced or separated
Widowed
Domestic partnership
24.4
60.2
10.2
1.40
3.79
2 ± 0.86
Income
0 to 20000
20000 to 30000
30000 to 40000
40000 to 50000
50000 to 60000
60000 to 75000
75000 to 1e5
1e5 to 1.5e5
1.5e5 to 2e5
>= 2e5
6.35
8.46
11.4
11.4
11.5
12.9
16.2
13.1
4.28
4.38
Gender
Male
Female
48.3
51.7
1.52 ± 0.5
Children ages 2-5
0
1
85.1
14.9
0.15 ± 0.36
Children under age 2
0
1
90.2
9.82
0.1 ± 0.3
Children ages 18-up
0
1
79.4
20.6
0.21 ± 0.4
Age
13 to 17
17 to 24
24 to 34
34 to 39
39 to 44
44 to 49
49 to 54
54 to 59
59 to 64
>= 64
0.77
10.4
28.1
14.7
13.3
11.9
9.79
5.38
2.98
2.77
Demographic Bayes Net
CPTs learned from 1.5M cases in the file
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A Bayesian Network


Compact representation of a probability

distribution:


Complete model


Consistent model


Embeds many independence assumptions:


Faithful model

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A. Darwiche

A Bayesian Network


Compact representation of a probability

distribution:


Complete model


Consistent model


Embeds many independence assumptions:


Faithful model

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Bayesian Network

Earthquake (E)

Burglary (B)

Alarm (A)

Pr(E=true)

Pr(E=false)

.1

.9

Pr(B=true)

Pr(B=false)

.2

.8

Pr(A=true)

Pr(A=false)

E=true, B=true

.95

.05

E=false, B=true

.9

.1

E=true, B=false

.7

.3

E=false, B=false

.01

.99

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Joint Probability Distribution

E

B

A

Pr(.)

True

True

True

.019

True

True

False

.001

True

False

True

.056

True

False

False

.024

False

True

True

.162

False

True

False

.018

False

False

True

.0072

False

False

False

.7128

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Independence Assumptions

of a Bayesian Network

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Chol

Test1

Test2

Causal Structure

I(Test1,Test2 | Chol)

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Chol

Test1

Test2

Causal Structure

Nurse

I(Test1,Test2 | Chol, Nurse)

I(Test1,Test2 | Chol)

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H

O1

On

Naïve Bayes

O2

H: Disease

O1, …, On: Findings (symptoms, lab tests, …)



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Genetic Tracking

G1

G2

G3

G4

G5

G6

G7

G8

P4

Each

node

is independent of its

non
-
descendants

given its

parents

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Genetic Tracking

G1

G2

G3

G4

G5

G6

G7

G8

P4

Each

node

is independent of its

non
-
descendants

given its

parents

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Genetic Tracking

G1

G2

G3

G4

G5

G6

G7

G8

P4

Each

node

is independent of its

non
-
descendants

given its

parents

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Dynamic Systems

S1

O1

S2

O2

S3

O3

S4

O4

S5

O5

Each

node

is independent of its

non
-
descendants

given its

parents

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Dynamic Systems

S1

O1

S2

O2

S3

O3

S4

O4

S5

O5

Each

node

is independent of its

non
-
descendants

given its

parents

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The chain rule for

Bayesian Networks

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Earthquake (E)

Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Pr(c|a)

Pr(craeb)=

Pr(c|raeb)Pr(r|aeb)Pr(a|eb)Pr(e|b)Pr(b)



Pr(r|e)

Pr(a|eb)

Pr(e)

Pr(b)

Pr(e)

Pr(b)

Pr(a|eb)

Pr(r|e)

Pr(c|a)

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Example: Build Joint Probability Table

Earthquake (E)

Burglary (B)

Alarm (A)

Pr(E=true)

Pr(E=false)

.1

.9

Pr(B=true)

Pr(B=false)

.2

.8

Pr(A=true)

Pr(A=false)

E=true, B=true

.95

.05

E=false, B=true

.9

.1

E=true, B=false

.7

.3

E=false, B=false

.01

.99

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Temperature/Sensors


Temperature:

high (20%), low (10%), nominal (70%)


3 Sensors (true, false):

true (90%) given high temperature

true (1%) given low temperature

true (5%) given nominal temperature

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Queries


Pr(Sensor1=true)?



Pr(Temperature=high | Sensor1=true)?



Pr(Temperature=high | Sensor1=true,

Sensor2=true, Sensor3=true)?

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d
-
separation

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Earthquake (E)

Burglary (B)

Alarm (A)

Call (C)

Radio (R)

… (F)

Is A Independent of R given E?

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Earthquake (E)

Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Chain Link

E & C not d
-
separated

…Active!

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Earthquake (E)

Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Chain Link

E & C are d
-
separated by A

…Blocked!

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Earthquake (E)

Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Divergent Link

R & A not d
-
seperated

…Active!

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Earthquake (E)

Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Divergent Link

R & A d
-
separated by E

…Blocked!

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Earthquake (E)

Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Convergent Link

E & B d
-
seperated

…Blocked!

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Earthquake (E)

Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Convergent Link

E & B not d
-
separated by A

…Active!

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Earthquake (E)

Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Convergent Link

E & B not d
-
separated by C

…Active!

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Earthquake (E)

Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Are B & R d
-
separated by E & C ?

Active

Blocked

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Earthquake (E)

Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Active

Active

Are C & R d
-
separated ?

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blocked

blocked

active

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d
-
separation


Nodes X are
d
-
separated

from nodes Y

by nodes Z iff
every path

from X to Y is

blocked by Z.



A
path

is
blocked

by Z if some link on the

path is blocked:


For some

X


or

X

,

X in Z


For some

X

,

neither X nor one of its

descendents in Z

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d
-
separation in Asia Network


Visit to Asia / Smoker:


No evidence: No


Given TB
-
or
-
Cancer: Yes


Given +ve X
-
Ray: Yes


Visit to Asia / +ve X
-
ray:


No evidence: Yes


Given TB: No


Given TB
-
or
-
Cancer: No


Bronchitis / Lung Cancer:


No evidence: Yes


Given Smoker: No


Given Smoker and Dysnpnoea: Yes