1

Probability• FOL fails for a domain due to:

– Laziness: too much to list the complete set of rules, too hard to use the enormous rules that result

– Theoretical ignorance: there is no complete theory for the domain

– Practical ignorance: have not or cannot run all necessary tests

2

Probability• Probability = a degree of belief• Probability comes from:

– Frequentist: experiments and statistical assessment

– Objectivist: real aspects of the universe– Subjectivist: a way of characterizing an agent’s

beliefs• Decision theory = probability + utility theory

3

Probability• Prior probability: probability in the

absence of any other information

P(Dice = 2) = 1/6

random variable: Dicedomain = <1, 2, 3, 4, 5, 6>probability distribution: P(Dice) = <1/6, 1/6, 1/6, 1/6, 1/6,

1/6>

4

Probability• Conditional probability: probability in

the presence of some evidence

P(Dice = 2 | Dice is even) = 1/3P(Dice = 2 | Dice is odd) = 0

• P(A | B) = P(A B)/P(B)P(A B) = P(A | B).P(B)

5

Probability• Example:

S = stiff neckM = meningitisP(S | M) = 0.5P(M) = 1/50000P(S) = 1/20

P(M | S) = P(S | M).P(M)/P(S) = 1/5000

6

Probability• Joint probability distributions:

X: <x1, …, xm> Y: <y1, …, yn>

P(X = xi, Y = yj)

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Probability• Axioms:

0 P(A) 1P(true) = 1 and P(false) = 0P(A B) = P(A) + P(B) P(A B)

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Probability• Derived properties:

P(A) = 1 P(A)

P(U) = P(A1) + P(A2) + ... + P(An)U = A1 A2 ... An collectively exhaustiveAi Aj = false mutually exclusive

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Probability• Bayes’ theorem:

P(Hi | E) = P(HiE)/P(E) = P(E | Hi).P(Hi)/P(E | Hi).P(Hi)

Hi’s are collectively exhaustive & mutually exclusive

10

Probability• Problem: a full joint probability

distribution P(X1, X2, ..., Xn) is sufficient for computing any probability on Xi's, but the number of joint probabilities is exponential.

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Probability• Independence:

P(A B) = P(A).P(B) P(A) = P(A | B)

• Conditional independence:P(A B | E) = P(A | E).P(B | E)

P(A | E) = P(A | E B)

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Probability• Example:

P(Toothache | Cavity Catch) = P(Toothache | Cavity)

P(Catch | Cavity Toothache) = P(Catch | Cavity)

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Probability"In John's and Mary's house, an alarm is installed to sound in case of burglary or earthquake. When the alarm sounds, John and Mary may make a call for help or rescue."

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Probability"In John's and Mary's house, an alarm is installed to sound in case of burglary or earthquake. When the alarm sounds, John and Mary may make a call for help or rescue."Q1: If earthquake happens, how likely will John make a call?

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Probability"In John's and Mary's house, an alarm is installed to sound in case of burglary or earthquake. When the alarm sounds, John and Mary may make a call for help or rescue."Q1: If earthquake happens, how likely will John make a call?

Q2: If the alarm sounds, how likely is the house burglarized?

16

Bayesian Networks• Pearl, J. (1982). Reverend Bayes on

Inference Engines: A Distributed Hierarchical Approach, presented at the Second National Conference on Artificial Intelligence (AAAI-82), Pittsburgh, Pennsylvania

• 2000 AAAI Classic Paper Award

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Bayesian Networks

Burglary Earthquake

Alarm

JohnCalls MaryCalls

P(B).001

P(E).002

B E P(A)T TT FF TF F

.95

.94

.29

.001

A P(J)TF

.90

.05

A P(M)TF

.70

.01

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Bayesian Networks• Syntax:

– A set of random variables making up the nodes– A set of directed links connecting pairs of nodes– Each node has a conditional probability table that

quantifies the effects of its parent nodes– The graph has no directed cycles

A

B C

D

A

B C

D

yes no

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Bayesian Networks• Semantics:

Ordering the nodes: Xi is a predecessor of Xj i < j

P(X1, X2, …, Xn) = P(Xn | Xn-1, …, X1).P(Xn-1 | Xn-2, …, X1). … .P(X2 | X1).P(X1)= iP(Xi | Xi-1, …, X1) = iP(Xi | Parents(Xi))

P (Xi | Xi-1, …, X1) = P(Xi | Parents(Xi)) Parents(Xi) {Xi-1, …, X1}

Each node is conditionally independent of its predecessors given its parents

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Bayesian Networks• Example:

P(J M A B E) = P(J | A).P(M | A).P(A | B E).P(B).P(E)

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Bayesian Networks• P(Query | Evidence) = ?

– Diagnostic (from effects to causes): P(B | J)– Causal (from causes to effects): P(J | B)– Intercausal (between causes of a common

effect): P(B | A, E)– Mixed: P(A | J, E), P(B | J, E)

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P(B | A) = P(B A)/P(A)= P(B A)

Bayesian Networks

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Bayesian Networks• Why Bayesian Networks?

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General Conditional Independence

Y1 Yn

U1 Um

Z1j Znj

X

A node (X) is conditionally independent of its non-descendents (Zij's), given its parents (Ui's)

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General Conditional Independence

Y1 Yn

U1 Um

Z1j Znj

X

A node (X) is conditionally independent of all other nodes, given its parents (Ui's), children (Yi's), and children's parents (Zij's)

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General Conditional Independence

X and Y are conditionally independent given E

Z

Z

Z

X YE

27

General Conditional Independence

Example:Battery

GasIgnitionRadio

Starts

Moves

28

General Conditional Independence

Example: Battery

GasIgnitionRadio

Starts

Moves

Gas - (no evidence at all) - RadioGas Radio | StartsGas Radio | Moves

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Network Construction• Wrong ordering:

– Produces a more complicated net– Requires more probabilities to be specified– Produces slight relationships that require

difficult and unnatural probability judgments – Fails to represent all the conditional

independence relationships

a lot of unnecessary probabilities

30

Network Construction• Models:

– Diagnostic: symptoms to causes– Causal: causes to symptoms

Add "root causes" first, then the variables they influence

fewer and more comprehensible probabilities

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Dempster-Shafer Theory• Example: disease diagnostics

D = {Allergy, Flu, Cold, Pneumonia}

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Dempster-Shafer Theory• Example: disease diagnostics

D = {Allergy, Flu, Cold, Pneumonia}

• Basic probability assignment:m: 2D [0, 1]

m1: {Flu, Cold, Pneu} 0.6 D 0.4

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Dempster-Shafer Theory• Belief:

Bel: 2D [0, 1]Bel(S) = XS m(X)

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Dempster-Shafer Theory• Belief:

Bel: 2D [0, 1]Bel(S) = XS m(X)

• Plausibility:Pl(p) = 1 Bel(p)p: [Bel(p), Pl(p)]

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Dempster-Shafer Theory• Ignorance:

Bel(p) + Bel(p) 1

36

Dempster-Shafer Theory• Combination rules: m1 and m2

XY=Zm1(X).m2(Y) m(Z) =

1 XY=m1(X).m2(Y)

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Exercises• In Russell’s AIMA: Chapters 13-14.