A. Darwiche

Bayesian NetworksBayesian Networks

A. Darwiche

Reasoning SystemsReasoning Systems

• Diagnostics: Which component failed?

• Information retrieval: What document to retrieve?

• On-line help: What is he trying to do?

• E-commerce: Is he interested in books?

• CAD: Does the design meet the specification?

• Decoders: What was the original message?

• Predictive maintenance: Will the device break?

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Model-Based ReasoningModel-Based Reasoning

Formal Model (KB)

Reasoning Engine

Query

ConclusionsWhat modeling language?How to get the model?How to reason efficiently?

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Bayesian NetworkBayesian NetworkBattery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

Starter

Radio Lights Engine Turn Over

Gas Gauge

Gas

Fuel Pump Fuel Line

Distributor

Spark Plugs

Engine Start

A. Darwiche

Bayesian NetworkBayesian NetworkBattery Age Alternator Fan Belt

BatteryCharge 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

Bat

tery

Pow

er

.99 .01

.20 .800 1

.99

θ1 + θ2 = 1

A. Darwiche

Model-Based ReasoningModel-Based Reasoning

• Efficiency: Time and space resources (algorithms)

• Formalization– What modeling language? (symbolic, quantitative)

– What query semantics? (diagnosis, recommender, belief revision, causality)

• Synthesis of models– Knowledge engineering (sensitivity analysis)

– Synthesize from design information (model checking, verfication)

– Synthesize from online data (learning)

• Embeddability (compiled reasoning, anyspace reasoning)

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No children

01

62.937.1

0.37 ± 0.48

Children ages 12-17

01

82.317.7

0.18 ± 0.38

Children ages 6-11

01

80.619.4

0.19 ± 0.4

Education

Under K12K12Some collegeBachelor degreeMaster or PhD

1.3410.935.133.019.6

3.59 ± 0.97

Marital Status

Single never marriedMarriedDivorced or separatedWidowedDomestic partnership

24.460.210.21.403.79

2 ± 0.86

Income

0 to 2000020000 to 3000030000 to 4000040000 to 5000050000 to 6000060000 to 7500075000 to 1e51e5 to 1.5e51.5e5 to 2e5>= 2e5

6.358.4611.411.411.512.916.213.14.284.38

Gender

MaleFemale

48.351.7

1.52 ± 0.5

Children ages 2-5

01

85.114.9

0.15 ± 0.36

Children under age 2

01

90.29.82

0.1 ± 0.3

Children ages 18-up

01

79.420.6

0.21 ± 0.4

Age

13 to 1717 to 2424 to 3434 to 3939 to 4444 to 4949 to 5454 to 5959 to 64>= 64

0.7710.428.114.713.311.99.795.382.982.77

Demographic Bayes NetCPTs learned from 1.5M cases in the file

A. Darwiche

Diagnosis ScenarioDiagnosis ScenarioBattery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

Starter

Radio Lights Engine Turn Over

Gas Gauge

Gas

Fuel Pump Fuel Line

Distributor

Spark Plugs

Engine Start

Query

Behavior

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Diagnosis ScenarioDiagnosis ScenarioBattery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

Starter

Radio Lights Engine Turn Over

Gas Gauge

Gas

Fuel Pump Fuel Line

Distributor

Spark Plugs

Engine Start

ok on yes no

.001

ok off yes no

.090

A. Darwiche

Probabilistic ReasoningProbabilistic ReasoningBattery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

Starter

Radio Lights Engine Turn Over

Gas Gauge

Gas

Fuel Pump Fuel Line

Distributor

Spark Plugs

Engine Start

Posterior Marginals

A. Darwiche

Probabilistic ReasoningProbabilistic ReasoningBattery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

Starter

Radio Lights Engine Turn Over

Gas Gauge

Gas

Fuel Pump Fuel Line

Distributor

Spark Plugs

Engine Start

Maximum a Posteriori (MAP)

A. Darwiche

Probabilistic ReasoningProbabilistic ReasoningBattery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

Starter

Radio Lights Engine Turn Over

Gas Gauge

Gas

Fuel Pump Fuel Line

Distributor

Spark Plugs

Engine Start

Maximum a Posteriori (MAP)

A. Darwiche

Probabilistic ReasoningProbabilistic ReasoningBattery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

Starter

Radio Lights Engine Turn Over

Gas Gauge

Gas

Fuel Pump Fuel Line

Distributor

Spark Plugs

Engine Start

Maximum a Posteriori (MAP)

A. Darwiche

Probabilistic ReasoningProbabilistic ReasoningBattery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

Starter

Radio Lights Engine Turn Over

Gas Gauge

Gas

Fuel Pump Fuel Line

Distributor

Spark Plugs

Engine Start

Maximum a Posteriori (MAP)

A. Darwiche

Diagnostic SystemDiagnostic System

Battery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

Starter

Radio Lights Engine Turn Over

Gas Gauge

Gas

Fuel Pump Fuel Line

Distributor

Spark Plugs

Engine Start

Bayesian Network Inference Engine

ok on yes nook off yes no

.001

.090

. . . .

.

A. Darwiche

AgendaAgenda

• Propositional (Boolean) Logic

• Probability Calculus

• Independence & Causality

• Bayesian networks:– Markovian Assumption– Chain Rule for Bayesian Networks– d-separation

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Propositional LogicPropositional Logic

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Probablity CalculusProbablity Calculus

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Bayesian NetworksBayesian Networks

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A Bayesian NetworkA Bayesian Network

• Compact representation of a probability distribution:– Complete model– Consistent model

• Embeds many independence assumptions:– Faithful model

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A Bayesian NetworkA Bayesian Network

• Compact representation of a probability distribution:– Complete model– Consistent model

• Embeds many independence assumptions:– Faithful model

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Bayesian NetworkBayesian 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

A. Darwiche

Joint Probability DistributionJoint Probability DistributionE 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 AssumptionsIndependence Assumptionsof a Bayesian Networkof a Bayesian Network

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Chol

Test1 Test2

Causal StructureCausal Structure

I(Test1,Test2 | Chol)

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Chol

Test1 Test2

Causal StructureCausal Structure

Nurse

I(Test1,Test2 | Chol, Nurse)

I(Test1,Test2 | Chol)

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H

O1 On

Naïve BayesNaïve Bayes

O2

H: DiseaseO1, …, On: Findings (symptoms, lab tests, …)

…

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Genetic TrackingGenetic Tracking

G1 G2

G3 G4G5

G6 G7 G8 P4

Each node is independent of its non-descendants given its parents

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Genetic TrackingGenetic Tracking

G1 G2

G3 G4G5

G6 G7 G8 P4

Each node is independent of its non-descendants given its parents

A. Darwiche

Genetic TrackingGenetic Tracking

G1 G2

G3 G4G5

G6 G7 G8 P4

Each node is independent of its non-descendants given its parents

A. Darwiche

Dynamic SystemsDynamic Systems

S1

O1

S2

O2

S3

O3

S4

O4

S5

O5

Each node is independent of its non-descendants given its parents

A. Darwiche

Dynamic SystemsDynamic 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 The chain rule for Bayesian NetworksBayesian 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)

A. Darwiche

Example: Build Joint Probability Example: Build Joint Probability TableTable

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

A. Darwiche

Temperature/SensorsTemperature/Sensors

• Temperature: high (20%), low (10%), nominal (70%)

• 3 Sensors (true, false):true (90%) given high temperaturetrue (1%) given low temperaturetrue (5%) given nominal temperature

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A. Darwiche

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QueriesQueries

• Pr(Sensor1=true)?

• Pr(Temperature=high | Sensor1=true)?

• Pr(Temperature=high | Sensor1=true,Sensor2=true, Sensor3=true)?

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d-separationd-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 LinkE & C not d-separated

…Active!

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Earthquake (E) Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Chain LinkE & C are d-separated by A

…Blocked!

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Earthquake (E) Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Divergent LinkR & A not d-seperated

…Active!

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Earthquake (E) Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Divergent LinkR & A d-separated by E

…Blocked!

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Earthquake (E) Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Convergent LinkE & B d-seperated

…Blocked!

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Earthquake (E) Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Convergent LinkE & B not d-separated by A

…Active!

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Earthquake (E) Burglary (B)

Alarm (A)

Call (C)

Radio (R)

Convergent LinkE & 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 ?

ActiveBlocked

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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-separationd-separation

• Nodes X are d-separated from nodes Y by nodes Z iff every path from X to Y isblocked 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 Networkd-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

A. Darwiche

Building Building Bayesian NetworksBayesian Networks

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Three StepsThree Steps

• Identifying variables

• Catching the structure

• Defining the CPTs

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Identifying VariablesIdentifying Variables

• Hypothesis variables

• Information variables (observables)

• Others (mediate relationships)

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Car Start ProblemCar Start Problem

“In the morning, my car will not start. I can hear the starterturn, but nothing happens. There May be several reasons formy problem. I can hear the starter roll, so there must be powerin the battery. Therefore, the most probable causes are that thefuel has been stolen overnight or that the spark plugs are dirty. It may also be due to dirt in the carburetor, a leak in the ignition system, or something more serious. To find out, I first look at the fuel meter. It shows ½ full, so I decide to clean the spark plugs”

A. Darwiche

Car Start ProblemCar Start Problem

“In the morning, my car will not start. I can hear the starterturn, but nothing happens. There May be several reasons formy problem. I can hear the starter roll, so there must be powerin the battery. Therefore, the most probable causes are that thefuel has been stolen overnight or that the spark plugs are dirty. It may also be due to dirt in the carburetor, a leak in the ignition system, or something more serious. To find out, I first look at the fuel meter. It shows ½ full, so I decide to clean the spark plugs”

Fuel Meter Standing

Fuel?

Start?

Clean Spark Plugs

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Fuel Meter Standing “FM”

Fuel? “Fu”

Start? “St”

Clean Spark Plugs “Sp”

Fu = yes Fu = no

FM = full 0.39 0.001

FM = ½ 0.60 0.001

FM = empty 0.01 0.998

Fu = yes Fu = no

Sp = yes (0.99, 0.01) (0,1)

Sp = no (0.01, 0.99) (0,1)

P(FM | Fu)

P(St | Fu,Sp)

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Milk testMilk test

“Milk from a cow may be infected. To detect whether the milk is infected, you have a test, which may give either a positive or a negative test result. The test is not perfect. It may give a positive result on a clean milk as well as a negative result on infected milk”

Infected? Test

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Digital SystemsDigital Systems

. . .

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Digital SystemsDigital Systems

• Build network structure

• Specificy CPTs

• Fault modes

• What if we have two test vectors

• Single vs multiple faults

• How to pose queries

• Synthesis automatically

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Channel CodingChannel Coding

• Information bits: U1 … Uk

• Redundant bits: X1…Xm

• Code word: U1…Uk X1 … Xm

(Channel input)

• Channel output: Y1…Yk+m

Given channel output Y, restore the channel input

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Channel CodingChannel Coding

U1 U2 U4

Y1 Y2 Y3 Y4

Y5 Y6 Y7

X1 X2 X3

U3

(7,4) Hamming code

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Noisy-OrNoisy-Or

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E

C1 C2 Cn. . .

L

Noisy-ORNoisy-OR

. . .

CnC2C101 q 02 q 0nq

0lL

E

Global enabler(leak)

1

A. Darwiche

• Let v be an assignment of truth values to C1…Cn

• Let S contain all indices i such that Ci=true

Si

iqlvfalseE )|Pr(

Si

iqlvfalseE )|Pr(

Si

iqlvtrueE 1)|Pr(

E

. . .

CnC2C1 01 q 02 q 0nq

0lL

Global enabler(leak)

1

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• O(n) parameters instead of O(2n)

• CPCS network : •448 nodes• 906 links• 8254 parameters;instead of 134 million!

Noisy-ORNoisy-OR

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Noisy-OrNoisy-Or

• 5% of the mornings yields a sore throat (l = .95)

•Cold causes a sore throat with probability 0.4; (q=.6)

•Angina causes a sore throat with probability 0.7 (q=.3)

Cold?

SoreThroat?

Angina?

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Noisy-OrNoisy-Or

Cold?

SoreThroat?

Angina?

Angina? = no Angina? = yes

Cold? = no 0.95 0.95 * 0.3

Cold? = yes 0.95 * 0.6 0.95 * 0.3 * 0.6

P(SoreThroat? = no | Cold? , Angina? )

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Noisy-OrNoisy-Or

Angina? = no Angina? = yes

Cold? = no 1-0.95 1-0.95 * 0.3

Cold? = yes 1-0.95 * 0.6 1-0.95 * 0.3 * 0.6

P(SoreThroat? = yes | Cold? , Angina? )

Cold?

SoreThroat?

Angina?