Download CF(R1), CF(R2)

Kunstig intelligens (IT-2702) - høst 2006.
Forelesning 7
Emner:
Usikkerhetsbehandling
- Utvidelse av standard logikk
- Den klassiske ”certainty factor” metoden
- Fuzzy mengder
- Statistikk-baserte metoder
- og spesielt Bayesianske nett
- Kunnskapsbaserte metoder
Den reelle verden er usikker
It is the mark of an instructed mind to rest satisfied with that degree of precision
which the nature of the subject admits, and not to seek exactness where only
an approximation of the truth is possible.
(Aristotle)
All traditional logic habitually assumes that precise symbols are being employed.
It is therefore not applicable to this terrestrial life but only to an imagined
celestial existence.
(Bertrand Russell)
So far as the laws of mathematics refer ro reality they are not certain. And so far
as they are certain they do not refer to reality.
(Albert Einstein)
Inferens-metoder
• Deduksjon
- sannhetsbevarende slutning
- basis er slutningsregelen modus ponens
(P(x) -> Q(x)) & P(a) -> Q(a)
Klassisk eksempel:
&
->
(Isa-man(x) -> Is-mortal(x))
Isa-man(Socrates)
Is-mortal(Socrates)
• Abduksjon
- ikke sannhetsbevarende
- "inference to the best explanation"
(P(x) -> Q(x)) & Q(a) ~> P(a)
Eksempel:
(Has-appendicitis(x) -> Has-abdominal-pain(x))
& Has-abdominal-pain(Socrates)
~> Has-appendicitis(Socrates)
Ikke-monotone systemer
Forutsetninger for 1. ordens predikatlogikk:
- komplett domenebeskrivelse
- konsistent domenebeskrivelse
- monotont voksende kunnskapsbase
I ikke-monotone systemer er en eller flere av disse forutsetningene
ikke oppfylt.
Logikk-tilnærminger:
Modal-operatorer
Truth Maintenance systemer
unless, is-consistent-with, …
JTMS, ATMS, . . .
Andre
CWA, Circumscription, . . .
Modal-operatorer:
p(X) unless q(X) => r(X)
good-student(X) ^ M study-hard(X) => graduates(X)
is-consistent-with
Set cover approach:
En abduktiv forklaring på et sett av fakta (S2) er et annet sett av fakta
(S1) som er tilstrekkelig for å forårsake S2.
En optimal forklaring er det minimale sett S1.
Logikkbasert approach:
En abduktiv forklaring på et sett av observasjoner (O) er det minimale sett av
hypoteser (H) som er konsistent med den aktuelle bakgrunnskunnskap (K).
O kan ikke være utledbar fra K alene.
Usikkerhetsbehandling - Certainy Factors
- usikkerhetsanslag i regel-baserte systemer
- benyttet i MYCIN og avledede ES-skall
- basert på anslag av
- degree of belief
- degree of disbelief
MB(H/E)
MD(H/E)
- som kobineres i en Certainty Factor
CF(H/E) = MB(H/E) - MD(H/E)
Usikkerhetsbehandling - CF (forts.)
- eksempel, Mycin-type regel:
IF
(P1 and P2) or P3
THEN R1 (0.7) and R2 (0.3)
- kombinasjon av to regler som peker på samme konklusjon:
CF(R1)+CF(R2) - (CF(R1)xCF(R2))
|
CF(R1), CF(R2) pos.
CF(R1)+CF(R2) + (CF(R1)xCF(R2)) |
CF(R1), CF(R2) neg.
CF(R1)+CF(R2)
|
1 - min((abs(CF(R1), abs(CF(R2))
ellers
Usikkerhetsbehandling - Fuzzy Sets
- et fuzzy set (fose mendge?) er en mendge der elementene
i større eller mindre grad kan sies å være medlem av
mendgen
- en medlemsskapsfunksjon definerer i hvilken grad
(mellom 0 og 1) et element er medlem av mengden
- øvelse: tegn medlemsskapsfunksjonene [0,1] for
ung og gammel i mendgen av aldre [1,100].
1
0.5
0
1
10
20
30
40
50
60
70
80
90
100
Fig 9.6 the fuzzy set representation for “small integers.”
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Fig 9.7 A fuzzy set representation for the sets short, medium, and tall
males.
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Fig 9.8 The inverted pendulum and the angle θ and dθ/dt input values.
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Fig 9.9 The fuzzy regions for the input values θ (a) and dθ/dt (b).
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Fig 9.10 The fuzzy regions of the output value u, indicating the movement
of the pendulum base.
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Fig 9.11 The fuzzificzation of the input measures X1 = 1, X2 = -4
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Fig 9.10 The Fuzzy Associative Matrix (FAM) for the pendulum problem.
The input values are on the left and top.
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Fig 9.13 The fuzzy consequents (a) and their union (b). The centroid of the
union (-2) is the crisp output.
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Usikkerhetsbehandling - Statistiske metoder
Basisbegreper:
Prior probability
(a priori sanns., ubetinget sanns.)
Sannsynligheten, P, for en hendelse, A, uten at
noe informasjon er gitt:
P(A)
Posterior probability (a posteriori sanns., betinget sanns.)
Sannsynligheten, P, for en hendelse, A, gitt
informsjonen E:
P(A/E)
Kombinasjon av uavhengige (ubetingede) sanns.
P(A & B) = P(A) x P(B)
Probability theory, the general form of Bayes’ theorem
The application of Bayes’ rule to the car purchase problem:
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Naïve Bayes, or the Bayes classifier, that uses the partition
assumption, even when it is not justified:
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Fig 5.4 The Bayesian representation of the traffic problem with potential
explanations.
Table 5.4 The joint probability distribution for the traffic and construction
variables of Fig 5.3.
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Fig 9.14 The graphical model for the traffic problem, first introduced in
Section 5.3.
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= Belief Networks
Defining the d-separation of nodes in a belief network (after Pearl 1988)
Another way to express the same thing:
Nodes are independent of their non-descendants given their parents.
(d-separated)
E
R
(d-connected)
If Known
A
B
A
C
If Known
E
B
If Known
A
If Known
C
OR
Fig 9.15 a is a serial connection of nodes where influence runs between A
and B unless V is instantiated. 9.15b is a diverging connection,
where influence runs between V’s children, unless V is
instantiated. In 9.15c, a converging connection, if nothing is
known about V the its parents are independent, otherwise
correlations exist between its parents.
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Rød kule: Verdien er kjent (gjelder for de tre siste eksempeluttrykkene)
Fig 9.16 An example of a Bayesian probabilistic network, where the
probability dependencies are located next to each node. This
example is from Pearl (1988).
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Table 9.4 The probability distribution for p(WS), a function of p(W) and
p(R) given the effect of S. We calculate the effect for x, where
R = t and W = t.
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A junction tree algorithm.
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Fig 9.17 A junction tree (a) for the Bayesian probabilistic network of (b).
Note that we started to construct the transition table for the
rectangle R, W.
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Fig 9.18 A Markov state machine or Markov chain with four states, s1, ..., s4
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Fig 9.19 A hidden Markov model of two states designed for the coin
flipping problem. The aij values are determined by the elements
of the 2 x 2 transition matrix.
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Fig 9.20 A hidden Markov model for the coin flipping problem. Each coin
will have its own individual bias.
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Fig 9.21 A PFSM representing a set of phonemically related English words.
The probability of each word occurring is below that word.
Adapted from Jurasky and Martin (2000).
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Fig 9.22 A trace of the Viterbi algorithm on the paths of Fig 9.21. Rows
report the maximum value for Viterbi on each word for each input
value (top row). Adapted from Jurafsky and Martin (2000).
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Usikkerhetsbehandling - Forklaringsbasert
- modell-basert tilnærming
- kausale relasjoner oftest benyttet, men også andre relasjoner
- relasjonene i modellen antas usikre, og kan ha ”degree of belief” anslag
- usikkerhet begrenses ved multiple forklaringer,
dvs. en hypotese støttes i større eller mindre
grad av forklaringene som genereres i modellen
eks. ABEL (Stanford), HeartFailureModel (MIT),
”Endorsement Theory” (UMass), CREEK (NTNU-IDI)