Transcript Chapters 13 and 14

Abduction, Uncertainty,
and
Probabilistic Reasoning
Chapters 13, 14, and more
1
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
Others, e.g., Probabilistic logic, Markov Logic Network
2
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 meets the
given chemical formula, and would most likely produce the
given mass spectrum if subjected to electron beam
bombardment
3
– 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.
4
• 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
( 1
2)
( 1|
2) ( 2)
11
• Independence assumption
– Two events E1 and E2 are said to be independent of each other if
( 1 | 2)
( 1) (given E2 does not change the likelihood of
E1)
– Computation can be simplified with independent events
( 1
2)
( 1
2)
( 1|
2) ( 2)
( 1)
( 2)
( 1)
( 2)
1 (1
( 1)(1
( 1) ( 2)
( 1
2)
( 1) ( 2)
( 2))
• Mutually exclusive (ME) and collectively exhaustive (CE) set
of events
– ME:
– CE:
( (
1
...
)
( (
1
0), ,
...
1,.., ,
) 1)
12
Bayes’ Theorem
• In the setting of diagnostic/evidential reasoning
(
(
|
hypotheses
)
)
evidence/m anifestati ons
1
– 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 1 ,...,
is more plausible given , then we can ignore the denominator
and rank them use
(
|
)
(
|
) (
)
13
•
) can be computed from ( | ) and ( ) , if we
assume all hypotheses 1 ,...,
are ME and CE
(
P( E j )
P( E j
( H1 ... H n ) ) (by CE)
n
P( E j
Hi )
(by ME)
i 1
n
P( E j | H i ) P( H i )
i 1
• Then we have another version of Bayes’ theorem:
(
|
(
)
|
(
1
(
where
|
) (
|
) (
)
(
)
|
(
)
|
)
1
) (
) , the sum of relative likelihood of all
1
n hypotheses, is a normalization factor
14
Probabilistic Inference for simple diagnostic problems
• Knowledge base:
,...,
:
evidence/m anifestati on
:
hypotheses /disorders
1 ,...,
E j and H i are binary and hypotheses form a ME & CE set
1
(
),
1,...
(
|
),
prior probabilit ies
1,... ,
1 ,...,
1,...
conditional probabilit ies
• Case input:
• Find the hypothesis with the highest posterior
probability ( | 1 ,..., )
( 1 ,... | ) (
• By Bayes’ theorem ( | 1 ,..., )
(
1
,...
)
)
• Assume all pieces of evidence are conditionally
independent, given any hypothesis
(
1
,...
|
)
1
(
|
)
15
• The relative likelihood
(
|
1
,...,
)
(
1
,...,
|
) (
)
(
)
(
1
|
)
• The absolute posterior probability
(
|
1
,...,
(
)
|
(
1
|
,...,
1
(
)
,...,
)
)
(
1
(
1
)
1
|
)
(
|
)
1
• Evidence accumulation (when new evidence discovered)
If El+1 present
(
|
1
,...,
,
1
)
(
1
|
)
(
|
1
,...,
|
1
)
If ~El+1 present
(
|
1
,...,
,~
1
) (1
(
1
|
))
(
,...,
)
16
Assessing the Assumptions
• Assumption 1: hypotheses are mutually exclusive and
collectively 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
(
1
2
)
(
1
) (
2
) 0 if both (
1
) and (
2
) 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
17
Limitations of the simple Bayesian system
• Cannot handle well hypotheses of multiple disorders
– Suppose 1 ,...,
are independent of each other
– Consider a composite hypothesis 1 ^ 2
– How to compute the posterior probability (or relative likelihood)
( 1 ^ 2 | 1 ,...,
– Using Bayes’ theorem
(
1^
(
1
(
^
2 |
2
^ 2 ) ( 1^ 2 )
( 1 ,... )
( 1 ) ( 2 ) because they are independen t
1 ,...,
)
,... |
assuming
1
)?
1
^
)
(
1
,...
|
1
)
| 1^
1 (
are independen t, given
2
)
1^
2
2
How to compute P( E j | H1 ^ H 2 ) ?
18
– Assuming H ,..., H are independent, given E ,..., E ?
1
n
1
l
( 1 ^ 2 | 1 ,..., )
( 1 | 1 ,..., ) (
but this is a very unreasonable assumption
B: burglar
E: earth quake
A: alarm set off
• Cannot handle causal chaining
2
|
1
,...,
)
E and B are independent
But when A is given, they
are (adversely) dependent
because they become
competitors to explain A
P(B|A,E) <<P(B|A)
– 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
19
Bayesian 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)
(
|
) where
is the set of all parent nodes of
– Root nodes are a special case – no parents, so just use priors
, so ( | )
( )
in CPD:
20
Example BN
P(a0) = 0.001
A
P(c0|a0) = 0.2
C P(c0|a1) = 0.005
P(b0|a0) = 0.3
B
P(b0|a1) = 0.001
D
P(d0|…)
b0
b1
c0
0.1
0.01
c1
0.01
0.00001
P(d0|b0, c0) = 0.1
P(d0|b0, c1) = 0.01
P(d0|b1, c0) = 0.01
P(d0|b1, c1) = 0.00001
E
P(e0|c0) = 0.4
P(e0|c1) = 0.002
Uppercase: variables (A, B, …)
Lowercase: values/states of variables (A has two states a0 and a1)
Note that we only specify P(a0) etc., not P(a1), since they have
to add to one. On the other hand, we need all combinations of
conditionals
21
Netica
• An commercial BN package by Norsys
• Down load limited version for free from
http://www.norsys.com/
• May also down load APIs
22
Conditional independence and
chaining
• Conditional independence assumption
–
( | , )
( | )
where is any set of variables
(nodes) other than
and its successors
–
blocks influence of other nodes on
and its successors (q influences
only
through 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 CPTs:
( 1 ,...,
)
1
(
|
)
23
Chaining: Example
A
Computing the joint probability for all
B
C
variables is easy:
The joint distribution of all variables
D
E
P(A, B, C, D, E)
= P(E | A, B, , D) P(A, B, C, D) by Bayes’ theorem
= P(E | C) P(A, B, C, D)
by cond. indep. assumption
= P(E | C) P(D | A,
) P(A, B, C)
= P(E | C) P(D | B, C) P(C | B) P(A, B)
= P(E | C) P(D | B, C) P(C | A) P(B | A) P(A)
For a particular state:
P(a0, b0, c1, d1, e0) = P(a0)P(b0|a0)P(c1|a0)P(d1|b0, c1)P(e0| c1)
= 0.001*0.3*0.8*0.99*0.002 = 4.752*10^(-7)
24
P(E) = 0.002
P(B) = 0.01
E: earth quake
B: burglar
A: alarm set off
P(A|…)
P(E|A) = 0.167; P(B|A) = 0.835
B
~B
E
0.9
0.8
~E
0.8
0.0
P(E|A, E) = 1.0; P(B|A, E) = 0.0112
P(B|A, E) = P(B,A,E)/P(A,E)
= P(B,A,E)/(P(B,A,E) + P(~B,A,E))
= 0.01*0.002*0.9/(0.01*0.002*0.9 + 0.99*0.002*0.8)
= 0.000018/(0.000018 + 0.001548)
= 0.000018/0.001566
= 0.01123
25
Topological semantics
• A node is conditionally independent of its nondescendants 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
A
C
B
C
Chaining: A and C
are independent,
given B
B
C
Diverging: B and
C are independent,
given A
A
Converging: B and
C are independent,
NOT given A
26
Inference tasks
• Simple queries: Compute posterior probability P(Xi | E=e)
– E.g., P(NoGas | Gauge=empty, Lights=on, Starts=false)
– Posteriors for ALL nonevidence nodes
– Priors for 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;
– inference is 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?
27
• MAP problems (explanation)
– Let X denote the set of all variables in a BN, V
X the set
of instantiat ed variables, U X V the set of all un - instantiat ed
varialbes. Then the MAP (maximum aposterior i probabilit y) problem
is to find the most probable instantiat ion of U , given V , i.e.,
max u ( P(U | V ))
– The solution provides a good explanation for your action
– This is an optimization problem
28
Approaches to inference
• Exact inference
– Enumeration
– Variable elimination
– Belief propagation in polytrees (singly connected BNs)
– Clustering / junction tree algorithms
• Approximate inference
– Stochastic simulation / sampling methods
• Markov chain Monte Carlo (MCMC) methods
– Loopy propagation
– Others
• Mean field theory
• Neural networks
29
Inference by enumeration
• To compute P(X|E=e), where X is a single variable and E is
evidence (instantiation of a set of variables)
• Add all of the terms (atomic event probabilities) from the full
joint distribution that are consistent with E
• If 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
A
Example: Enumeration
B
C
D
• P(xi) = Σ πi P(xi | πi) P(πi)
• Suppose we want P(D), and only the value of E is given as true
E
• P (D|e) = ΣA,B,CP(a, b, c, d, e)
= ΣA,B,CP(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))
31
Variable elimination
• Basically just enumeration, but with caching of local
calculations
• Linear for polytrees
• Potentially exponential for multiply connected BNs
Exact inference in Bayesian networks is NP-hard!
32
Variable elimination
General idea:
• Write query in the form
P( X n , e )
L
xk
P( xi | pai )
x3
x2
i
• Iteratively
– Move all irrelevant terms outside of innermost sum
– Perform innermost sum, getting a new term
– Insert the new term into the product
33
Variable elimination
Example:
8 x 4 = 32 multiplications
ΣAΣBΣCP(a) P(b|a) P(c|a) P(d|b,c) P(e|c)
= ΣAΣBP(a)P(b|a)ΣCP(c|a) P(d|b,c) P(e|c)
= ΣAP(a)ΣBP(b|a)ΣCP(c|a) P(d|b,c) P(e|c)
8 x 2 + 4 + 2 = 22
multiplications
for each state of A = a
Variable C is summed out
for each state of B = b
compute fC(a, b) = ΣCP(c|a) P(d|b,c) P(e|c)
compute fB(a) = ΣBP(b)fC(a, b)
variable B is summed out
Compute result = ΣAP(a)fB(a)
Here fC(a, b), fB(a) are called factors, which are vectors or matrices
34
Belief Propagation
• 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
• The instantiated variable influences
its ancestors and descendents differently
(using CPT along opposite directions)
• Computation is linear to the diameter of
B
the network (the longest undirected
path)
• Update belief (posterior) of every non- D
evidence node in one pass
A
C
E=e
F
– For multi-connected net: conditioning
35
Conditioning
A
B
C
D
E
• 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|))
36
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 potential (similar to CPT), constructed
from CPT of variables in the original BN
37
Junction Tree
• Example
A
A
B
B
C
D
E
C
D
Triangulated graph
E
A
A simple BN
B
A,B,C
(B, C)
B,C,D
C
(C, D)
C,D.E
D
E
Junction tree of 3 nodes
Moralized graph
38
Junction Tree
• Reasoning
– Since it is now a tree, polytree algorithm can be applied,
but now two cliques exchange P(S), the distribution over
S, their shared variables.
– 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.
39
Some comments on BN reasoning
– Let X ( X1 ,K , X n ) be the set of all variables in a BN. Any BN
reasoning task can be expressed in the form of calculating
P(U | V ) where U ,V
X
– This can be done by marginalization of the joint distribution P(X)
over Y = X \ U \ V:
Y P(U , V , Y )
where each entry P(x) = P(u,v,y) can be calculated by chain rule
from CPTs
– Computation can be done more efficiently using, say Junction tree,
by utilizing variable interdependencies
– Computational complexity of BN reasoning is proved to be NPhard by reducing 3SAT problems to BN reasoning (Cooper 1990)
40
Approximate inference: Direct sampling
• Suppose you are given values for some subset of the
variables, E, and want to infer distributions for unknown
variables, Z
• Randomly generate a very large number of instantiations
from the BN according to the distribution
– 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 and inefficient
41
Likelihood weighting
• Idea: Don’t generate samples that need to be rejected in the
first place!
• Sample only from the unknown variables Z and Y=X\Z\E
(E are fixed)
• 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 E1 = e1-0) is selected for
weighting, its parents are already instantiated (say parents A
and B are assigned state a and b)
– Modify w = w * P(e1-0 | a, b) based on E1’s CPT
– Repeat for the other evidence nodes
42
Markov chain Monte Carlo algorithm
• So called because
– Markov chain – each instance generated in the sampling 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, randomly select a non-evidence variable x, randomly
sample its value by
P( x | mb ( x))
P( x | parent( x))
Π
Y child ( X )
P( y | parents(Y )
• Given enough samples, MCMC gives an accurate estimate of the
true distribution of values
43
Loopy Propagation
• Belief propagation
– Works only for polytrees (exact solution)
– Each evidence propagates once throughout the network
• Loopy propagation
– Let propagation 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 of approximation are
under intense investigation
44
Noisy-Or BN
• A special BN of binary variables (Peng & Reggia, Cooper)
– Each link xi
x j is associated with a probabilit y value called
causal strengthcij that measures the strength of xi alone may
cause x j , i.e., cij P( xi | x j is true and all others in i are false)
– 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
Complete joint prob. distributi on : 2 n
n
| i|
General BN :
2
i 1
Noisy - Or BN :
– Computation is less expensive
n
i 1|
i
|
• Disadvantage: less expressive (less general)
45
Learning BN (from case data)
• Needs for learning
– Difficult to construct BN by humans (esp. CPT)
– 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
46
BN Learning Approaches
• 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)
47
BN Learning Approaches
• 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 using a pre-determined partial order
of the variables
– Compute CPTs after the DAG is determined
– Extensions to handle some form of missing values
48
BN Learning Approaches
• Minimum description length (MDL) (Lam, etc.)
– Sacrifices accuracy for simpler (less dense) structure
• Case data not always accurate
• Outliers are hard to model (needs more links)
• Fewer links imply smaller CPD tables and less expensive
inference
–
where
•
: the length of the encoding of DAG (smaller for simpler
DAG)
•
: the length of the encoding of the difference between DAG
and DB (smaller for better match of DAG with DB)
• Smaller
implies less accurate DAG, and thus larger
– Find DAG by heuristic best-first search that Minimizes
49
BN Learning Approaches
• Neural network approach (Neal, Peng)
– For noisy-or BN
~
P (V V r ) where
Maximizing L ln
Vr D
D : case database;
V r : case in D;
~
V : state vector of the learned network
L measures the similarity of the two distributi ons : one in D,
another in the learned network
– Change inter-node link strength locally, following gradient
descent approach to maximize L.
50
• Compare Neural network approach with Cooper’s K2
• Network: Alarm (37 nodes)
# 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
51
Current research in BN
• 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 (FOL); Semantic web
• Reasoning with uncertain evidence
– Virtual evidence
– Soft evidence
52
Other formalisms for Uncertainty
Fuzzy sets and fuzzy logic
• Ordinary set theory
1 if x A
0 otherwise
f A ( x) is called the characteristic or membership function of set A
– f A ( x)
Predicate A( x)
1 if x A
0 otherwise
When it is uncertain if
, use probabilit y (
)
– There are sets that are described by vague linguistic terms (sets
without hard, clearly defined boundaries), e.g., tall-person, fastcar
• Continuous
• Subjective (context dependent)
• Hard to define a clear-cut 0/1 membership function
53
• Fuzzy set theory
– Relax
( ) from binary {0, 1} to continuous [0, 1]
stands for the degree is thought t o belong to set
height(john) = 6’5”
height(harry) = 5’8”
height(joe) = 5’1”
Tall(john) = 0.9
Tall(harry) = 0.5
Tall(joe) = 0.1
– Examples of membership functions
1-
Set of teenagers
0
12
19
1-
Set of young people
0
12
19
1-
20
35
50
65
80
Set of mid-age
people
54
• Fuzzy logic: many-value logic
– Fuzzy predicates (degree of truth) FA ( x) y if f A ( x) y
– Connectors/Operators
negation : F A ( x) 1 FA ( x)
conjunction : FA B ( x) min{FA ( x) , FB ( x)}
disjunctio n : FA B ( x) max{FA ( x) , FB ( x)}
• 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 born
unsure after grown up (height = 5’6”)
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– 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) = 0.1
Tall(john) ^ ~Tall(john) = min{0.1, 0.9) = 0.1
john’s degree of membership in the fuzzy set of “medianheight people” (both Tall and not-Tall)
• In prob. theory: P(john Tall ^ john Tall) = 0
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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 certainty
Ex: flu => sore throat with high probability
• Combining symbolic rules with numeric uncertainty: Mycin’s
Certainty 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).
57
– CF propagation:
• Within a rule: each Ci has CFi, then the certainty of Action is
{
}
• When more than one rules can apply to the current WM for the
same Conclusion with different CFs, the
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.
58
Dempster-Shafer theory
• A variation of Bayes’ theorem to represent ignorance
• Uncertainty and ignorance
– Suppose two events A and B are ME and CE, given an evidence
E
A: 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 E
I may have some belief in B, given E
I 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
59
• Representing ignorance
{h1 ,..., hn }, a set of ME and CE
– Frame of discernment:
hypotheses. The power set 2 is organized as a lattice of super/subset
relations. Each node S is a subset of hypotheses (S
)
– Ex: = {A,B,C}
Each node is associated with a
{A,B,C} 0.15
basic probabilit y assignment ( )
0
( ) 1;
{A,B} 0.1 {A,C} 0.1 {B,C}0.05
( ) 0;
{A} 0.1
{B} 0.2
{C}0.3
1
S
{} 0
• Belief function (lower bound of belief)
( )
( ' );
( ) 0;
( ) 1
'
({ , })
({ , })
({ })
({ })
( )
0.1 0.1 0.2 0 0.4
({ , } )
({ }) 0.3
60
– Plausibility (upper bound of belief of a node)
All belief not committed to S C may be commited to S
Pls ( S ) 1 Bel ( S C )
Pls ({ A, B}) 1 Bel ({C}) 1 0.3 0.7
Pls ({ A}) 1 Bel ({B, C}) 1 0.55 0.45
[ Bel ( S ),
Lower
bound
(known
belief)
Pls( S )] belief interval
Upper
bound
(maximally
possible)
{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
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• Evidence combination (how to use D-S theory)
– Each piece of evidence has its own m(.) function for the same
{ , } : A : having cancer; B : not having cancer
{A,B} 0.3
{A} 0.2
{B} 0.5
{} 0
( )
1 : smoking
1
{A,B} 0.1
{A} 0.7
{B} 0.2
{} 0
( )
2 : living in high radiation area
2
– Belief based on combined evidence can be computed from
) 2( )
1(
( )
1( )
2( )
1
) 2( )
1(
normalization factor
incompatible combination
62
{A,B} 0.3
{A} 0.2
{A,B} 0.1
{B} 0.5
{A} 0.7
{B} 0.2
{} 0
{} 0
E1
E2
({ })
2
({ })
2
{A} 0.607
E1 ^ E2
({ })
1 ({ }) 2 ({ , })
({ })
1 [ 1 ({ }) 2 ({ })
1 ({ })
0.5 0.2 0.5 0.1 0.3 0.2 0.21
0.344
1 [0.2 0.2 0.5 0.7]
0.61
1
({ , })
1
({ , }) 2 ({ , })
0.61
{B} 0.344
{} 0
({ })
1 ({ }) 2 ({ , })
1 [ 1 ({ }) 2 ({ })
1 ({ })
0.2 0.7 0.2 0.1 0.3 0.7 0.37
0.607
1 [0.2 0.2 0.5 0.7]
0.61
1
({ })
{A,B} 0.049
0.03
0.61
({ , })
2 ({ })]
1
({ , })
2 ({ })]
1
2
({ })
2
({ })
0.049
63
{A,B} 0.3
{A} 0.2
{B} 0.5
{A,B} 0.1
{A} 0.7
{B} 0.2
{} 0
{} 0
E1
E2
{A,B} 0.049
{A} 0.607
{B} 0.344
{} 0
E1 ^ E2
– 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 CE
– How to obtain m(.) for each piece of evidence is not clear, except
subjectively
64