Uncertainty

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Uncertainty
Knowledge can have uncertainty associated with
it - Knowledge base rule premises, rule
conclusions - User input uncertain,
unknown, partial truths Much expert system
research in inventing new methods to denote
uncertainty, and reason about it. eg. IF
dog_barks THEN dog_bites CF 80
80 means this conclusion most likely
holds with a strength of 80 eg.
Indicate intensity of pain (0-10) ___
user gives a numeric value which

will be associated with value "pain"
Three example techniques (of many) 1.
Bayesian probability 2. Fuzzy logic 3.
MYCIN certainty factors
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1. Bayesian

• Use the exact mathematics of probability theory
  to handle uncertain information
• Bayess formula determine the probability of an
  earlier event given that a later one occurred
• p(H E) P(H) P (E H)
• P(E)
• where P(H E) prob. H is true given evidence E
• P(H ) prob. H is true
• P(E H) prob of seeing E when H
  is true
• P(E) prob. E is true
• This matches with rule IF E THEN H

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1. Bayesian

• Advantages
• statistically sound formula
• Disadvantages / Requirements
• probabilities of events must be available and
  known
• often not possible for many problems
• must update probabilities in system
• sum of probabilities for a rule must equal 1
• experts often dont think this way!
• conditional independence events in Bayes
  formula are independent
• but this might not be so in many cases
• user input is not always probabilistic, but is
  arbitrary

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2. Fuzzy Logic

• Fuzzy logic a branch of logic in which there are
  degrees of membership within sets, rather than
  strictly true and false membership
• introduces uncertainty as consequence of user's
  uncertain input
• permit partial truths to be denoted
• --gt language is subjective, eg. tall, cold,
  tiny, important,...
• eg. instead of tall(bob) being True or False,
  it can be True fv 80
• fuzzy operators map language concepts let X and
  Y be fvs (between 0 and 1)
• X and Y - new_fv min(X,Y)
  intersection
• X or Y - new_fv max(X,Y)
  unton
• not X - new_fv 1 - X
  negation
• very X - new_fv X X
  concentration
• somewhat X - new_fv X0.5
  dilation
• indeed X - new_fv 2 XX (for O
  lt X lt .5)
• 1 -
  2((1 - X)2 ) for X gt .5 intensification

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2. Fuzzy

• Example IF noisy_disk fv X
• AND bad_formatting fv
  Y
• THEN defective disk fv
  0.90
• then let X.9, Y.3 then combined premise fv
  0.3 (minimum)
• confidence of rule as whole is .9 .3
  0.27
• Advantages
• gives intuitively appealing results
• can account for many ambiguous types of concepts
• Disadvantages
• doesn't consider amount of supporting evidence
  lotss of separate evidence that backs up an
  inference doesnt mean that inference is more
  likely

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Bayes vs Fuzzy

• (see Fuzzy Systems - a Tutorial by J.F. Brule
  on web)
• let t(X) X is tall, s(Y)Y is smart
• let t(bob) .90 and s(bob) .90
• Bayes t(bob) s(bob) .90.90 .81
• ie. if Bob is very tall, and Bob is very smart,
  then Bob is quite tall and smart
• Fuzzy min(t(bob), s(bob)) .90
• ie. if Bob is very tall, and Bob is very smart,
  then Bob is very tall and smart
• Hence fuzzy approach is more intuitive in this
  case Bayes combinations are always reduced in
  overall strength for all values lt 1.0

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3. MYCIN confidence factors

• Certainty theory a model of uncertain
  information used in the MYCIN
• expert system (Shortliffe and Buchanan 1975)
• uses
• - a scale for confidence factors,
  eg. -100 lt--gt 100
• - a threshold d (eg. 20)
• - premise and conclusion cf's
• - user input cf's
• - formulae for firing rules and
  combining results
• IF e1 AND e2 AND e3 THEN f1 CF w
• - combining premises minimum e min( cf(e1),
  cf(e2), cf(e3) )
• - e must be gt threshold for rule to fire if
  so...
• - inferring cf for result
  multiply w e

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3. Uncertainty

• incrementally acquired evidence combine belief
  and disbelief values of facts established by
  different rules
• cf(cf1, cf2) cf1 cf2(1-cf1) both gt 0
• ( cf1 cf2) / (1 -
  min(cf1, cf2)) one lt 0
• - cf (-cf1, -cf2) both lt 0
• net effect the more that a fact is established
  as true, the higher the overall cf will be
• likewise, if negative results for a fact are
  obtained, it will bring down the overall cf
• ---gt amount of evidence is accounted for

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MYCIN
rules
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MYCIN
premises, conclusions, and user input have
uncertainty factors (CF's) formulae are used
to combine CF's during the inference process
the intension is that resulting CF's for the
inference faithfully reflect the uncertainty
inherent in the KB and user input eg. MYCIN
Let range be -100 (totally invalid) CF
100 (totally certain)
cutoff d 20 Consider If
lives_in_water CF A and
has_gills CF B
then is_a_fish CF 75. 1.if A15 and
B30 then premise CF min(15,30) 15
lt d therefore rule not used. 2. if A
-100 and B 90 then premise CF -100 lt
d therefore rule not used 3. if A90 and
B90 then premise CF 90 and
conclusion CF 67.5 --gt assert fact(is_a_fish
CF 67.5).
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MYCIN
In addition, if a fact is determined a number
of times, then the CF's for these multiple
instances are combined to yield an overal CF eg.
Given fact(is_a_fish CF
75). () and you determine
fact(is_a_fish CF 40). Then we get 75
40 - (75x40)/100 85 ----gt replace ()
with fact(is_a_fish CF 85).
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MYCIN implementation
3.2
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Summary
the nature of expert knowledge means that
knowledge is often uncertain certainty
factors are controversial they aren't provably
correct! cf's are arbitrary values only
meaning they have is relative to one another
in an application (and that can be pretty
arbitrary also) cf's can add confusion if the
expert doesn't think in terms of them One
solution explicitly model uncertain knowledge
should it be relevant eg. values
dog_bites, dog_may_bite, dog_behavior_uncertain,..
. then the rules might be IF dog_barks

THEN dog_may_possibly_bite
IF dog_stares AND
dog_snarls
THEN dog_will_probably_bite
IF
dog_lunges
THEN dog_bites