abduction, uncertainty, and probabilistic reasoning

1

Abduction, Uncertainty, Abduction, Uncertainty, and and

Probabilistic ReasoningProbabilistic ReasoningChapters 13, 14, and more

2

Introduction

• Abduction is a reasoning process that tries to form plausible explanations for abnormal observations– Abduction is distinct different from deduction and induction– Abduction is inherently uncertain

• Uncertainty becomes an important issue in AI research• Some major formalisms for representing and reasoning about

uncertainty– Mycin’s certainty factor (an early representative)– Probability theory (esp. Bayesian networks)– Dempster-Shafer theory– Fuzzy logic– Truth maintenance systems

3

Abduction

• Definition (Encyclopedia Britannica): reasoning that derives an explanatory hypothesis from a given set of facts– The inference result is a hypothesis, which if true, could

explain the occurrence of the given facts• Examples

– Dendral, an expert system to construct 3D structure of chemical compounds • Fact: mass spectrometer data of the compound and the

chemical formula of the compound• KB: chemistry, esp. strength of different types of bounds• Reasoning: form a hypothetical 3D structure which meet the

given chemical formula, and would most likely produce the given mass spectrum if subjected to electron beam bombardment

4

– Medical diagnosis• Facts: symptoms, lab test results, and other observed findings

(called manifestations)

• KB: causal associations between diseases and manifestations

• Reasoning: one or more diseases whose presence would causally explain the occurrence of the given manifestations

– Many other reasoning processes (e.g., word sense disambiguation in natural language process, image understanding, detective’s work, etc.) can also been seen as abductive reasoning.

5

Comparing abduction, deduction and induction

Deduction: major premise: All balls in the box are black minor premise: These balls are from the box conclusion: These balls are blackAbduction: rule: All balls in the box are black observation: These balls are black explanation: These balls are from the boxInduction: case: These balls are from the box observation: These balls are black hypothesized rule: All ball in the box are black

A => B A ---------BA => B B-------------Possibly A

Whenever A then B but not vice versa-------------Possibly A => BInduction: from specific cases to general rules

Abduction and deduction: both from part of a specific case to other part of

the case using general rules (in different ways)

6

Characteristics of abduction reasoning

1. Reasoning results are hypotheses, not theorems (may be false even if rules and facts are true),

– e.g., misdiagnosis in medicine2. There may be multiple plausible hypotheses

– When given rules A => B and C => B, and fact B both A and C are plausible hypotheses

– Abduction is inherently uncertain– Hypotheses can be ranked by their plausibility if that can be

determined 3. Reasoning is often a Hypothesize- and-test cycle

– hypothesize phase: postulate possible hypotheses, each of which could explain the given facts (or explain most of the important facts)

– test phase: test the plausibility of all or some of these hypotheses

7

– One way to test a hypothesis H is to query if some thing that is currently unknown but can be predicted from H is actually true.• If we also know A => D and C => E, then ask if D and E are

true.• If it turns out D is true and E is false, then hypothesis A

becomes more plausible (support for A increased, support for C decreased)

• Alternative hypotheses compete with each other (Okam’s razor)

4. Reasoning is non-monotonic – Plausibility of hypotheses can increase/decrease as new

facts are collected (deductive inference determines if a sentence is true but would never change its truth value)

– Some hypotheses may be discarded/defeated, and new ones may be formed when new observations are made

8

Source of Uncertainty

• Uncertain data (noise)• Uncertain knowledge (e.g, causal relations)

– A disorder may cause any and all POSSIBLE manifestations in a specific case

– A manifestation can be caused by more than one POSSIBLE disorders

• Uncertain reasoning results– Abduction and induction are inherently uncertain– Default reasoning, even in deductive fashion, is uncertain– Incomplete deductive inference may be uncertain

9

Probabilistic Inference

• Based on probability theory (especially Bayes’ theorem)– Well established discipline about uncertain outcomes– Empirical science like physics/chemistry, can be verified by

experiments• Probability theory is too rigid to apply directly in many

knowledge-based applications– Some assumptions have to be made to simplify the reality– Different formalisms have been developed in which some aspects

of the probability theory are changed/modified.• We will briefly review the basics of probability theory before

discussing different approaches to uncertainty

• The presentation uses diagnostic process (an abductive and evidential reasoning process) as an example

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Probability of Events

• Sample space and events– Sample space S: (e.g., all people in an area)– Events E1 S: (e.g., all people having cough)

E2 S: (e.g., all people having cold)• Prior (marginal) probabilities of events

– P(E) = |E| / |S| (frequency interpretation)– P(E) = 0.1 (subjective probability)– 0 <= P(E) <= 1 for all events – Two special events: and S: P() = 0 and P(S) = 1.0

• Boolean operators between events (to form compound events)– Conjunctive (intersection): E1 ^ E2 ( E1 E2)– Disjunctive (union): E1 v E2 ( E1 E2) – Negation (complement): ~E (E = S – E)C

11

• Probabilities of compound events– P(~E) = 1 – P(E) because P(~E) + P(E) =1– P(E1 v E2) = P(E1) + P(E2) – P(E1 ^ E2)– But how to compute the joint probability P(E1 ^ E2)?

• Conditional probability (of E1, given E2)– How likely E1 occurs in the subspace of E2

E

~E

E2E1

E1 ^ E2

)2(

)21(

||/|2|

||/|21|

|2|

|21|)2|1(

EP

EEP

SE

SEE

E

EEEEP

)2()2|1()21( EPEEPEEP

12

• Independence assumption– Two events E1 and E2 are said to be independent of each other if

(given E2 does not change the likelihood of E1)

– Computation can be simplified with independent events

• Mutually exclusive (ME) and exhaustive (EXH) set of events– ME:

– EXH:

)1()2|1( EPEEP

)2()1()2()2|1()21( EPEPEPEEPEEP

))2(1)(1(1(1 )2()1()2()1(

)21()2()1()21(

EPEPEPEPEPEP

EEPEPEPEEP

jinjiEEPEE jiji ,,..,1, ),0)((

)1)...(( ... 11 nn EEPSEE

13

Bayes’ Theorem

• In the setting of diagnostic/evidential reasoning

– Know prior probability of hypothesis

conditional probability – Want to compute the posterior probability

• Bayes’ theorem (formula 1):• If the purpose is to find which of the n hypotheses

is more plausible given , then we can ignore the denominator

and rank them use relative likelihood

onsanifestatievidence/m

hypotheses

1 mj

i

EEE

H

)(/)|()()|( jijiji EPHEPHPEHP

nHH ,...,1

)()|()|( iijji HPHEPEHrel

)( iHP

)|( ij HEP

)|( ij HEP

)|( ji EHP

)( iHP

jE

14

• can be computed from and , if we assume all hypotheses are ME and EXH

• Then we have another version of Bayes’ theorem:

where , the sum of relative likelihood of all n

hypotheses, is a normalization factor

nHH ,...,1

n

iiij

n

iij

njj

HPHEP

HEP

HHEPEP

1

1

1

)()|(

ME) (by )(

EXH) (by ) )...(()(

n

kjk

ji

n

kkkj

iijji

EHrel

EHrel

HPHEP

HPHEPEHP

11

)|(

)|(

)()|(

)()|()|(

n

kkkj HPHEP

1

)()|(

)( jEP )|( ij HEP )( iHP

15

Probabilistic Inference for simple diagnostic problems

• Knowledge base:

• Case input: • Find the hypothesis with the highest posterior

probability

• By Bayes’ theorem

• Assume all pieces of evidence are conditionally independent, given any hypothesis

onanifestatievidence/m :,...,1 mEE/disordershypotheses :,...,1 nHH

set EXH & ME a form hypotheses andbinary are and ij HE

iesprobabilit lconditiona,...,,... ),|( m jniHEP ij 11

iHlEE ,...,1

),...,|( 1 li EEHP

),...(

)()|,...(),...,|(

1

11

l

iilli EEP

HPHEEPEEHP

)|()|,...( 11 ijljil HEPHEEP

iesprobabilitprior ,... ),( n iHP i 1

16

• The relative likelihood

• The absolute posterior probability

• Evidence accumulation (when new evidence discovered)

n

kkj

ljk

ijlji

n

klk

lili

HEPHP

HEPHP

EEHrel

EEHrelEEHP

11

1

11

11

)|()(

)|()(

),...,|(

),...,|(),...,|(

)|()()()|,...,(),...,|( 111 ijljiiilli HEPHPHPHEEPEEHrel

),...,|()|(),,...,|( 1111 liillli EEHrelHEPEEEHrel

),...,|())|(1()~,,...,|( 1111 liillli EEHrelHEPEEEHrel

17

Assessment of Assumptions

• Assumption 1: hypotheses are mutually exclusive and exhaustive – Single fault assumption (one and only hypothesis must true)– Multi-faults do exist in individual cases– Can be viewed as an approximation of situations where

hypotheses are independent of each other and their prior probabilities are very small

• Assumption 2: pieces of evidence are conditionally independent of each other, given any hypothesis– Manifestations themselves are not independent of each other, they

are correlated by their common causes– Reasonable under single fault assumption– Not so when multi-faults are to be considered

small very are )( and )(both if 0)()()( 212121 HPHPHPHPHHP

18

Limitations of the simple Bayesian system

• Cannot handle well hypotheses of multiple disorders– Suppose are independent of each other– Consider a composite hypothesis– How to compute the posterior probability (or relative likelihood)

– Using Bayes’ theorem

),...(

)^()^|,...(),...,|^(

1

21211121

l

ll EEP

HHPHHEEPEEHHP

? ),...,|^( 121 lEEHHP

tindependen are they because )()()^( 2121 HPHPHHP

21

211211

^given t,independen are assuming )^|()^|,...(

HHEHHEPHHEEP

j

jljl

?)^|( compute toHow 21 HHEP j

nHH ,...,1

21^ HH

19

–

but this is a very unreasonable assumption

• Cannot handle causal chaining– Ex. A: weather of the year

B: cotton production of the year C: cotton price of next year

– Observed: A influences C– The influence is not direct (A -> B -> C)

P(C|B, A) = P(C|B): instantiation of B blocks influence of A on C• Need a better representation and a better assumption

),...,|( ),...,|( ),...,|^( 1211121 lll EEHPEEHPEEHHP

? ),...,given t,independen are ,..., Assuming 11 ln EEHH

E: earth quake B: burglar

A: alarm set off

E and B are independentBut when A is given, they are (adversely) dependent because they become competitors to explain A

P(B|A, E) <<P(B|A)

20

Bayesian Belief Networks (BNs)

• Definition: BN = (DAG, CPD) – DAG: directed acyclic graph (BN’s structure)

• Nodes: random variables (typically binary or discrete, but methods also exist to handle continuous variables)

• Arcs: indicate probabilistic dependencies between nodes (lack of link signifies conditional independence)

– CPD: conditional probability distribution (BN’s parameters)• Conditional probabilities at each node, usually stored as a

table (conditional probability table, or CPT)

– Root nodes are a special case – no parents, so just use priors in CPD:

iiii xxP of nodesparent all ofset theis where)|(

)()|( so , iiii xPxP

22

• Conditional independence assumption–

where q is any set of variables (nodes) other than and its successors

– blocks influence of other nodes on and its successors (q influences onlythrough variables in )

– With this assumption, the complete joint probability distribution of all variables in the network can be represented by (recovered from) local CPDs by chaining these CPDs:

ix

)|(),...,( 11 iinin xPxxP

)|(),|( iiii xPqxP

ix i ix

i

q

ix i

Conditional independence and chaining

24

Topological semantics• A node is conditionally independent of its non-

descendants given its parents• A node is conditionally independent of all other nodes in

the network given its parents, children, and children’s parents (also known as its Markov blanket)

• The method called d-separation can be applied to decide whether a set of nodes X is independent of another set Y, given a third set Z

A

B

CA

B CA

B C

Chain: A and C are independent, given B

Diverging: B and C are independent, given A

Converging: B and C are independent, NOT given A

25

Inference tasks

• Simple queries: Computer posterior marginal P(Xi | E=e)– E.g., P(NoGas | Gauge=empty, Lights=on, Starts=false)

– Posteriors for ALL nonevidence nodes

– Priors for and/all nodes (E = )

• Conjunctive queries: – P(Xi, Xj | E=e) = P(Xi | E=e) P(Xj | Xi, E=e)

• Optimal decisions: Decision networks or influence diagrams include utility information and actions; probabilistic inference is required to find P(outcome | action, evidence)

• Value of information: Which evidence should we seek next?

• Sensitivity analysis: Which probability values are most critical?

• Explanation: Why do I need a new starter motor?

26

• MAP problems (explanation)–

– The solution provides a good explanation for your action

– This is an optimization problem

))|((max i.e., ,given , ofion instantiat probablemost thefind tois

problem y)probabilit iaposterior (maximum MAP Then the varialbes. edinstantiat-un all ofset the , variablesedinstantiat of

set the BN, ain variablesall ofset thedenote Let

VUPVU

VXUXVX

u

27

Approaches to inference• Exact inference

– Enumeration– Variable elimination– Belief propagation in polytrees (singly connected BNs)– Clustering / join tree algorithms

• Approximate inference– Stochastic simulation / sampling methods– Markov chain Monte Carlo methods– Loopy propagation– Mean field theory– Simulated annealing– Genetic algorithms– Neural networks

28

Direct inference with BNs

• Instead of computing the joint, suppose we just want the probability for one variable

• Exact methods of computation:– Enumeration

– Variable elimination

• Belief propagation (for polytree only)

• Join/junction trees: clustering closely associated nodes into a big node in JT

29

Inference by enumeration

• Add all of the terms (atomic event probabilities) from the full joint distribution

• If E are the evidence (observed) variables and Y are the other (unobserved) variables, excluding X, then the posterior distribution

P(X|E=e) = α P(X, e) = α ∑yP(X, e, Y)

• Sum is over all possible instantiations of variables in Y

• Each P(X, e, Y) term can be computed using the chain rule

• Computationally expensive!

30

Example: Enumeration

• P(xi) = Σ πi P(xi | πi) P(πi)

• Suppose we want P(D), and only the value of E is given as true

• P (d|e) = ΣABCP(a, b, c, d, e)

= ΣABCP(a) P(b|a) P(c|a) P(d|b,c) P(e|c)

• With simple iteration to compute this expression, there’s going to be a lot of repetition (e.g., P(e|c) has to be recomputed every time we iterate over C for all possible assignments of A and B))

A

B C

D E

31

Exercise: Enumeration

smart study

prepared fair

pass

p(smart)=.8 p(study)=.6

p(fair)=.9

p(prep|…) smart smart

study .9 .7

study .5 .1

p(pass|…)smart smart

prep prep prep prep

fair .9 .7 .7 .2

fair .1 .1 .1 .1

Query: What is the probability that a student studied, given that they pass the exam?

32

Variable elimination

• Basically just enumeration, but with caching of local calculations

• Linear for polytrees

• Potentially exponential for multiply connected BNsExact inference in Bayesian networks is NP-hard!

33


General idea:• Write query in the form

• Iteratively– Move all irrelevant terms outside of innermost sum– Perform innermost sum, getting a new term– Insert the new term into the product

kx x x i

iin paxPXP3 2

)|(),( e

36

Computing factorsR S C P(R|C) P(S|C) P(C) P(R|C) P(S|C) P(C)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

R S f1(R,S) = ∑c P(R|S) P(S|C) P(C)

T T

T F

F T

F F

37

Variable elimination algorithm

• Let X1,…, Xm be an ordering on the non-query variables

• For i = m, …, 1

– Leave in the summation for Xi only factors mentioning Xi

– Multiply the factors, getting a factor that contains a number for each value of the variables mentioned, including X i

– Sum out Xi, getting a factor f that contains a number for each value of the variables mentioned, not including Xi

– Replace the multiplied factor in the summation

j

jjX XX

))X(Parents|X(P...1 m2

38

x

kxkx yyxfyyf ),,,('),,( 11

m

ilikx i

yyxfyyxf1

,1,1,11 ),,(),,,('

Complexity of variable eliminationSuppose in one elimination step we compute

This requires

multiplications (for each value for x, y1, …, yk, we do m multiplications) and

additions (for each value of y1, …, yk , we do |Val(X)| additions)

►Complexity is exponential in the number of variables in the intermediate factors

►Finding an optimal ordering is NP-hard

i

iYXm )Val()Val(

i

iYX )Val()Val(

39

Exercise: Variable elimination

smart study

prepared fair

pass


p(fair)=.9


study .9 .7

study .5 .1


prep prep prep prep

fair .9 .7 .7 .2

fair .1 .1 .1 .1

Query: What is the probability that a student is smart, given that they pass the exam?

40

• Singly connected network, SCN (also known as polytree) – there is at most one undirected path between any two nodes

(i.e., the network is a tree if the direction of arcs are ignored)– The influence of the instantiated variable (evidence) spreads

to the rest of the network along the arcs

A

B C

D E = e F

Belief Propagation

• The instantiated variable influences its predecessors and successors differently (using CPT along opposite directions)

• Computation is linear to the diameter ofthe network (the longest undirected path)

• Update belief (posterior) of every non-evidence node in one pass

– For multi-connected net: conditioning

41

Conditioning

• Conditioning: Find the network’s smallest cutset S (a set of nodes whose removal renders the network singly connected)

– In this network, S = {A} or {B} or {C} or {D}

• For each instantiation of S, compute the belief update with the belief propagation algorithm

• Combine the results from all instantiations of S (each is weighted by P(S = s))

• Computationally expensive (finding the smallest cutset is in general NP-hard, and the total number of possible instantiations of S is O(2 |

S|))

A

B C

D E

42

Junction Tree

• Convert a BN to a junction tree– Moralization: add undirected edge between every pair of

parents, then drop directions of all arc: Moralized Graph

– Triangulation: add an edge to any cycle of length > 3: Triangulated Graph

– A junction tree is a tree of cliques of the triangulated graph

– Cliques are connected by links• A link stands for the set of all variables S shared by these

two cliques

• Each clique has a CPT, constructed from CPT of variables in the original BN

43

Junction Tree

• Reasoning– Since it is now a tree, polytree algorithm can be applied,

but now two cliques exchange P(S), the distribution of S

– Complexity: • O(n) steps, where n is the number of cliques

• Each step is expensive if cliques are large (CPT exponential to clique size)

• Construction of CPT of JT is expensive as well, but it needs to compute only once.

44

Approximate inference: Direct sampling

• Suppose you are given values for some subset of the variables, E, and want to infer values for unknown variables, Z

• Randomly generate a very large number of instantiations from the BN– Generate instantiations for all variables – start at root variables and

work your way “forward” in topological order

• Rejection sampling: Only keep those instantiations that are consistent with the values for E

• Use the frequency of values for Z to get estimated probabilities

• Accuracy of the results depends on the size of the sample (asymptotically approaches exact results)

• Very expensive

45

Exercise: Direct sampling

smart study

prepared fair

pass


p(fair)=.9


study .9 .7

study .5 .1


prep prep prep prep

fair .9 .7 .7 .2

fair .1 .1 .1 .1

Topological order = …?Random number generator: .35, .76, .51, .44, .08, .28, .03, .92, .02, .42

46

Likelihood weighting

• Idea: Don’t generate samples that need to be rejected in the first place!

• Sample only from the unknown variables Z

• Weight each sample according to the likelihood that it would occur, given the evidence E– A weight w is associated with each sample (w initialized to 1)

– When a evidence node (say E = e) is selected for sampling, its parents are already sampled (say parents A and B are assigned state a and b)

– Modify w = w * P(e | a, b) based on E’s CPT

47

Markov chain Monte Carlo algorithm• So called because

– Markov chain – each instance generated in the sample is dependent on the previous instance

– Monte Carlo – statistical sampling method

• Perform a random walk through variable assignment space, collecting statistics as you go– Start with a random instantiation, consistent with evidence variables

– At each step, for some nonevidence variable x, randomly sample its value by

• Given enough samples, MCMC gives an accurate estimate of the true distribution of values (no need for importance sampling because of Markov blanket)

)(|(Π))(|())(|()(

YparentsyPxparentxPxmbxPXchildY

48

Exercise: MCMC sampling

smart study

prepared fair

pass


p(fair)=.9


study .9 .7

study .5 .1


prep prep prep prep

fair .9 .7 .7 .2

fair .1 .1 .1 .1

Topological order = …?Random number generator: .35, .76, .51, .44, .08, .28, .03, .92, .02, .42

49

Loopy Propagation

• Belief propagation– Works only for polytrees (exact solution)

– Each evidence propagates once throughout the network

• Loopy propagation– Let propagate continue until the network stabilize (hope)

• Experiments show– Many BN stabilize with loopy propagation

– If it stabilizes, often yielding exact or very good approximate solutions

• Analysis– Conditions for convergence and quality approximation are

under intense investigation

50

• A special BN of binary variables (Peng & Reggia, Cooper)–

– Causation independence: parent nodes influence a child independently

• Advantages:– One-to-one correspondence between causal links and causal

strengths – Easy for humans to understand (acquire and evaluate KB)– Fewer # of probabilities needed in KB

– Computation is less expensive• Disadvantage: less expressive (less general)

false) are in others all and trueis |( i.e., , causemay ofstrength themeasures that

called y valueprobabilit a with associated is link Each

ijiijj

iij

ji

xxPcxalonexc strengthcausal

xx

|| :BNOr -Noisy

2 :BN General

2 :ondistributi prob.joint Complete

1

1||

n

ni i

ni

i

Noisy-Or BN

51

Learning BN (from case data)

• Need for learning– Experts’ opinions are often biased, inaccurate, and incomplete– Large databases of cases become available

• What to learn– Parameter learning: learning CPT when DAG is known (easy)– Structural learning: learning DAG (hard)

• Difficulties in learning DAG from case data– There are too many possible DAG when # of variables is large

(more than exponential) n # of possible DAG 3 25 10 4*10^18

– Missing values in database– Noisy data

52

• Early effort: Based on variable dependencies (Pearl)– Find all pairs of variables that are dependent of each

other (applying standard statistical method on the database)

– Eliminate (as much as possible) indirect dependencies– Determine directions of dependencies– Learning results are often incomplete (learned BN

contains indirect dependencies and undirected links)

BN Learning Approaches

53

• Bayesian approach (Cooper)– Find the most probable DAG, given database DB, i.e.,

max(P(DAG|DB)) or max(P(DAG, DB))– Based on some assumptions, a formula is developed to

compute P(DAG, DB) for a given pair of DAG and DB– A hill-climbing algorithm (K2) is developed to search a

(sub)optimal DAG– Extensions to handle some form of missing values


54

• Minimum description length (MDL) (Lam, etc.)– Sacrifices accuracy for simpler (less dense) structure

• Case data not always accurate• Fewer links imply smaller CPD tables and less expensive

inference– L = L1 + L2 where

• L1: the length of the encoding of DAG (smaller for simpler DAG)

• L2: the length of the encoding of the difference between DAG and DB (smaller for better match of DAG with DB)

• Smaller L2 implies more accurate (and more complex) DAG, and thus larger L1

– Find DAG by heuristic best-first search, that Minimizes L


55

• Neural network approach (Neal, Peng)– For noisy-or BN

network learned in theanother ,in one :onsdistributi two theof similarity themeasures

network learned theof vector state :~

;in case : database; case :

DLV

DVD

r

where)~

(ln Maximizing r

DV

VVPLr


56

• Compare Neural network approach with K2

# cases missing links extra links time

500 2/0 2/6 63.76/5.91

1000 0/0 1/1 69.62/6.04

2000 0/0 0/0 77.45/5.86

10000 0/0 0/0 161.97/5.83

57

• Missing data– Missing value: EM (expectation maximization)– Missing (hidden) variables are harder to handle

• BN with time– Dynamic BN: assuming temporal relation obey Markov

chain• Cyclic relations

– Often found in social-economic analysis– Using dynamic BN?

• Continuous variable– Some work on variables obeying Gaussian distribution

• Connecting to other fields– Databases– Statistics– Symbolic AI

Current research in BN

58

• Ordinary set theory

–

– There are sets that are described by vague linguistic terms (sets without hard, clearly defined boundaries), e.g., tall-person, fast-car• Continuous• Subjective (context dependent)• Hard to define a clear-cut 0/1 membership function

otherwise 0

if 1)( Predicate Ax

xA

) (y probabilit use , ifuncertain isit When AxPAx

Axf

Axxf

A

A

set offunction membershipor sticcharacteri thecalled is )(

otherwise 0

if 1)(

Other formalisms for Uncertainty Fuzzy sets and fuzzy logic

59

• Fuzzy set theory–

height(john) = 6’5” Tall(john) = 0.9height(harry) = 5’8” Tall(harry) = 0.5height(joe) = 5’1” Tall(joe) = 0.1

– Examples of membership functions

Axxf A

set tobelong o thought tis degree for the stands1] [0, continuous to1} {0,binary from )(Relax

Set of teenagers

0 12 19

1 -

Set of young people

0 12 19

1 -

Set of mid-age people

20 35 50 65 80

1 -

60

• Fuzzy logic: many-value logic– Fuzzy predicates (degree of truth)– Connectors/Operators

• Compare with probability theory– Prob. Uncertainty of outcome,

• Based on large # of repetitions or instances• For each experiment (instance), the outcome is either true or false

(without uncertainty or ambiguity)unsure before it happens but sure after it happens

Fuzzy: vagueness of conceptual/linguistic characteristics• Unsure even after it happens

whether a child of tall mother and short father is tall unsure before the child is bornunsure after grown up (height = 5’6”)

y )( ify )( xfxF AA

)}(, )(max{ )( )( :ndisjunctio )}(, )(min{ )( )( :nconjunctio

)(1 )( :negation

xFxFxFxFxFxFxFxF

xFx~F

BABA

BABA

AA

61

– Empirical vs subjective (testable vs agreeable)– Fuzzy set connectors may lead to unreasonable results

• Consider two events A and B with P(A) < P(B)• If A => B (or A B) then

P(A ^ B) = P(A) = min{P(A), P(B)} P(A v B) = P(B) = max{P(A), P(B)}

• Not the case in general P(A ^ B) = P(A)P(B|A) P(A) P(A v B) = P(A) + P(B) – P(A ^ B) P(B)

(equality holds only if P(B|A) = 1, i.e., A => B)– Something prob. theory cannot represent

• Tall(john) = 0.9, ~Tall(john) = 0.1Tall(john) ^ ~Tall(john) = min{0.1, 0.9) = 0.1john’s degree of membership in the fuzzy set of “median-height people” (both Tall and not-Tall)

• In prob. theory: P(john Tall ^ john Tall) = 0

62

Uncertainty in rule-based systems

• Elements in Working Memory (WM) may be uncertain because– Case input (initial elements in WM) may be uncertain

Ex: the CD-Drive does not work 70% of the time– Decision from a rule application may be uncertain even if the

rule’s conditions are met by WM with certaintyEx: flu => sore throat with high probability

• Combining symbolic rules with numeric uncertainty: Mycin’s Uncertainty Factor (CF)– An early attempt to incorporate uncertainty into KB systems– CF [-1, 1]– Each element in WM is associated with a CF: certainty of that

assertion– Each rule C1,...,Cn => Conclusion is associated with a CF:

certainty of the association (between C1,...Cn and Conclusion).

63

– CF propagation: • Within a rule: each Ci has CFi, then the certainty of Action is

min{CF1,...CFn} * CF-of-the-rule• When more than one rules can apply to the current WM for the

same Conclusion with different CFs, the largest of these CFs will be assigned as the CF for Conclusion

• Similar to fuzzy rule for conjunctions and disjunctions– Good things of Mycin’s CF method

• Easy to use• CF operations are reasonable in many applications• Probably the only method for uncertainty used in real-world

rule-base systems– Limitations

• It is in essence an ad hoc method (it can be viewed as a probabilistic inference system with some strong, sometimes unreasonable assumptions)

• May produce counter-intuitive results.

64

Dempster-Shafer theory

• A variation of Bayes’ theorem to represent ignorance• Uncertainty and ignorance

– Suppose two events A and B are ME and EXH, given an evidence EA: having cancer B: not having cancer E: smoking

– By Bayes’ theorem: our beliefs on A and B, given E, are measured by P(A|E) and P(B|E), and P(A|E) + P(B|E) = 1

– In reality, I may have some belief in A, given EI may have some belief in B, given EI may have some belief not committed to either one,

– The uncommitted belief (ignorance) should not be given to either A or B, even though I know one of the two must be true, but rather it should be given to “A or B”, denoted {A, B}

– Uncommitted belief may be given to A and B when new evidence is discovered

65

• Representing ignorance–

– Ex: = {A,B,C}

• Belief function

)( hypotheses ofsubset a is nodeEach relations.et super/subs of lattice a as organized is 2set power The .hypotheses

EXH and ME ofset a },,...,{ :tdiscernmen of Frame 1

SS

hh n

{A,B,C} 0.15

{A,B} 0.1 {A,C} 0.1 {B,C}0.05

{A} 0.1 {B} 0.2 {C}0.3

{} 0

S1

;0)( ;1)(0

)( assignmenty probabilit basic a with associated is nodeEach

m(S)m

SmSm

S

3.0})({)},({4.002.01.01.0

)(})({})({}),({}),({1)( ;0)( ;)'()(

'

CBelBABel

mBmAmBAmBABelBelBelSmSBel

C

SS

66

– Plausibility (upper bound of belief of a node)

{A,B,C} 0.15

{A,B} 0.1 {A,C} 0.1 {B,C}0.05

{A} 0.1 {B} 0.2 {C}0.3

{} 0

interval belief )]( ),([ 7.03.01})({1}),({

)(1)( tocommited bemay tocommittednot belief All

SPlsSBelCBelBAPls

SBelSPlsSS

C

C

Lower bound (known belief)

Upper bound (maximally possible)

67

• Evidence combination (how to use D-S theory)– Each piece of evidence has its own m(.) function for the same

– Belief based on combined evidence can be computed from

{A,B} 0.3

{A} 0.2 {B} 0.5

{} 0

{A,B} 0.1

{A} 0.7 {B} 0.2

{} 0

cancer havingnot :B cancer; having :A :},{ BA

arearadiation high in living : smoking :)( )(

21

21

EESmSm

YX

SYX

YmXm

YmXmSmSmSm

)()(1

)()()()()(

21

2121

normalization factor incompatible combination

68

607.061.0

37.0

]7.05.02.02.0[1

7.03.01.02.07.02.0

})]({})({})({})({[1

})({}),({}),({})({})({})({})({

2121

212121

AmBmBmAm

AmBAmBAmAmAmAmAm

344.061.0

21.0

]7.05.02.02.0[1

2.03.01.05.02.05.0

})]({})({})({})({[1

})({}),({}),({})({})({})({})({

2121

212121

AmBmBmAm

BmBAmBAmBmBmBmBm

049.061.0

03.0

61.0

}),({}),({}),({ 21

BAmBAmBAm

{A,B} 0.3

{A} 0.2 {B} 0.5

{} 0

{A,B} 0.1

{A} 0.7 {B} 0.2

{} 0

{A,B} 0.049

{A} 0.607 {B} 0.344

{} 0

E1 E2 E1 ^ E2

69

– Ignorance is reduced from m1({A,B}) = 0.3 to m({A,B}) = 0.049)

– Belief interval is narrowed A: from [0.2, 0.5] to [0.607, 0.656] B: from [0.5, 0.8] to [0.344, 0.393]

• Advantage:– The only formal theory about ignorance– Disciplined way to handle evidence combination

• Disadvantages– Computationally very expensive (lattice size 2^– Assuming hypotheses are ME and EXH– How to obtain m(.) for each piece of evidence is not clear,

except subjectively

70

Appendix: A more complex examplefor variable elimination in BN

Visit to Asia

Smoking

Lung CancerTuberculosis

Abnormalityin Chest

Bronchitis

X-Ray Dyspnea

• “Asia” network:

71

V S

LT

A B

X D

),|()|(),|()|()|()|()()( badPaxPltaPsbPslPvtPsPvP

• We want to compute P(d)• Need to eliminate: v,s,x,t,l,a,b

Initial factors

72

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: v,s,x,t,l,a,b

Initial factors

Eliminate: v

Note: fv(t) = P(t)In general, result of elimination is not necessarily a probability term

Compute: v

v vtPvPtf )|()()(

),|()|(),|()|()|()()( badPaxPltaPsbPslPsPtfv

73

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: s,x,t,l,a,b

• Initial factors

Eliminate: s

Summing on s results in a factor with two arguments fs(b,l)In general, result of elimination may be a function of several variables

Compute: s

s slPsbPsPlbf )|()|()(),(


),|()|(),|(),()( badPaxPltaPlbftf sv

74

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: x,t,l,a,b

• Initial factors

Eliminate: x

Note: fx(a) = 1 for all values of a !!

Compute: x

x axPaf )|()(



),|(),|()(),()( badPltaPaflbftf xsv

75

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: t,l,a,b

• Initial factors

Eliminate: t

Compute: t

vt ltaPtflaf ),|()(),(




),|(),()(),( badPlafaflbf txs

76

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: l,a,b

• Initial factors

Eliminate: l

Compute: l

tsl laflbfbaf ),(),(),(




),|(),()(),( badPlafaflbf txs

),|()(),( badPafbaf xl

77

V S

LT

A B

X D


• We want to compute P(d)• Need to eliminate: b

• Initial factors

Eliminate: a,bCompute:

b

aba

xla dbfdfbadpafbafdbf ),()(),|()(),(),(




),|()(),( badPafbaf xl),|(),()(),( badPlafaflbf txs

)(),( dfdbf ba

78

Dealing with evidence

• How do we deal with evidence?

• Suppose we are give evidence V = t, S = f, D = t• We want to compute P(L, V = t, S = f, D = t)

V S

LT

A B

X D

79


• We start by writing the factors:

• Since we know that V = t, we don’t need to eliminate V• Instead, we can replace the factors P(V) and P(T|V) with

• These “select” the appropriate parts of the original factors given the evidence• Note that fp(V) is a constant, and thus does not appear in elimination of other variables


V S

LT

A B

X D

)|()()( )|()( tVTPTftVPf VTpVP

80


• Given evidence V = t, S = f, D = t• Compute P(L, V = t, S = f, D = t )• Initial factors, after setting evidence:

),()|(),|()()()( ),|()|()|()|()()( bafaxPltaPbflftfff badPsbPslPvtPsPvP

V S

LT

A B

X D

81


• Eliminating x, we get


V S

LT

A B

X D

),()(),|()()()( ),|()|()|()|()()( bafafltaPbflftfff badPxsbPslPvtPsPvP


82




• Eliminating t, we get


V S

LT

A B

X D


),()(),()()( ),|()|()|()()( bafaflafbflfff badPxtsbPslPsPvP

83





• Eliminating a, we get


V S

LT

A B

X D



),()()( )|()|()()( lbfbflfff asbPslPsPvP

84





• Eliminating a, we get

• Eliminating b, we get


V S

LT

A B

X D



),()()( )|()|()()( lbfbflfff asbPslPsPvP

)()()|()()( lflfff bslPsPvP

abduction, uncertainty, and probabilistic reasoning

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