A. Darwiche Bayesian Networks. A. Darwiche Reasoning Systems Diagnostics: Which component failed?...
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Transcript of A. Darwiche Bayesian Networks. A. Darwiche Reasoning Systems Diagnostics: Which component failed?...
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A. Darwiche
Bayesian NetworksBayesian Networks
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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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A. Darwiche
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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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
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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
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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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A. Darwiche
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A. Darwiche
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A. Darwiche
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A. Darwiche
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
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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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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
ok on yes no
.001
ok off yes no
.090
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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
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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)
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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)
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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)
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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)
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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
. . . .
.
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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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A. Darwiche
Propositional LogicPropositional Logic
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A. Darwiche
Probablity CalculusProbablity Calculus
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A. Darwiche
Bayesian NetworksBayesian Networks
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A. Darwiche
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. Darwiche
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A. Darwiche
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A. Darwiche
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A. Darwiche
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A. Darwiche
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. Darwiche
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
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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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A. Darwiche
Independence AssumptionsIndependence Assumptionsof a Bayesian Networkof a Bayesian Network
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A. Darwiche
Chol
Test1 Test2
Causal StructureCausal Structure
I(Test1,Test2 | Chol)
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A. Darwiche
Chol
Test1 Test2
Causal StructureCausal Structure
Nurse
I(Test1,Test2 | Chol, Nurse)
I(Test1,Test2 | Chol)
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A. Darwiche
H
O1 On
Naïve BayesNaïve Bayes
O2
H: DiseaseO1, …, On: Findings (symptoms, lab tests, …)
…
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A. Darwiche
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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A. Darwiche
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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A. Darwiche
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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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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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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A. Darwiche
The chain rule for The chain rule for Bayesian NetworksBayesian Networks
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A. Darwiche
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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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
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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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A. Darwiche
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A. Darwiche
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A. Darwiche
QueriesQueries
• Pr(Sensor1=true)?
• Pr(Temperature=high | Sensor1=true)?
• Pr(Temperature=high | Sensor1=true,Sensor2=true, Sensor3=true)?
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A. Darwiche
d-separationd-separation
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A. Darwiche
Earthquake (E) Burglary (B)
Alarm (A)
Call (C)
Radio (R)
… (F)
Is A Independent of R given E?
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A. Darwiche
Earthquake (E) Burglary (B)
Alarm (A)
Call (C)
Radio (R)
Chain LinkE & C not d-separated
…Active!
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A. Darwiche
Earthquake (E) Burglary (B)
Alarm (A)
Call (C)
Radio (R)
Chain LinkE & C are d-separated by A
…Blocked!
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A. Darwiche
Earthquake (E) Burglary (B)
Alarm (A)
Call (C)
Radio (R)
Divergent LinkR & A not d-seperated
…Active!
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A. Darwiche
Earthquake (E) Burglary (B)
Alarm (A)
Call (C)
Radio (R)
Divergent LinkR & A d-separated by E
…Blocked!
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A. Darwiche
Earthquake (E) Burglary (B)
Alarm (A)
Call (C)
Radio (R)
Convergent LinkE & B d-seperated
…Blocked!
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A. Darwiche
Earthquake (E) Burglary (B)
Alarm (A)
Call (C)
Radio (R)
Convergent LinkE & B not d-separated by A
…Active!
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A. Darwiche
Earthquake (E) Burglary (B)
Alarm (A)
Call (C)
Radio (R)
Convergent LinkE & B not d-separated by C
…Active!
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A. Darwiche
Earthquake (E) Burglary (B)
Alarm (A)
Call (C)
Radio (R)
Are B & R d-separated by E & C ?
ActiveBlocked
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A. Darwiche
Earthquake (E) Burglary (B)
Alarm (A)
Call (C)
Radio (R)
Active
Active
Are C & R d-separated ?
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A. Darwiche
blocked
blocked
active
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A. Darwiche
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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A. Darwiche
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
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A. Darwiche
Building Building Bayesian NetworksBayesian Networks
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A. Darwiche
Three StepsThree Steps
• Identifying variables
• Catching the structure
• Defining the CPTs
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A. Darwiche
Identifying VariablesIdentifying Variables
• Hypothesis variables
• Information variables (observables)
• Others (mediate relationships)
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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”
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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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A. Darwiche
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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A. Darwiche
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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A. Darwiche
Digital SystemsDigital Systems
. . .
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A. Darwiche
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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A. Darwiche
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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A. Darwiche
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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A. Darwiche
Noisy-OrNoisy-Or
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A. Darwiche
E
C1 C2 Cn. . .
L
Noisy-ORNoisy-OR
. . .
CnC2C101 q 02 q 0nq
0lL
E
Global enabler(leak)
1
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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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A. Darwiche
• 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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A. Darwiche
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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A. Darwiche
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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A. Darwiche
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?