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1990
Uncertainty reasoning and representation: A
Comparison of several alternative approaches
Barbara S. Smith
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Rochester Institute of Technology
Department of Computer Science
Uncertainty Reasoning and Representation:
A Comparison of Several Alternative Approaches
by
Barbara S. Smith
A Thesis, submitted to
The Faculty of the Department of Computer Science
in partial fulfillment of the requirements for the degree of
Master of Science in Computer Science
Approved by:
Professor John A. Biles
Date
fj/O/10
,
Professor Feredoun Kazemian
Date
_____________
IO_A~
Professor Peter G. Anderson
Date
uncertainty Reasoning and Representation:
A Comparison of Several Alternative Approaches
I, Barbara s. Smith, hereby grant permission to the Wallace
Memorial Library of RIT to reproduce my thesis in whole or
in part. Any reproduction will not be for commercial use or
profit.
Date: August 13,1990
ABSTRACT
Much
the
of
research
in Artificial
done
Intelligence
involves
investigating and developing methods of
incorporating uncertainty reasoning and representation
expert
The
situations.
on
strict
approaches
methods
have been
handling uncertainty in
for
attempted
based
Several
systems.
theories
range
probabilistic
based
on
investigates
a
Probability,
of Evidence,
Mycin
from
proposed
problem
solving
numerical
reasoning to
approaches
non-numeric
logical reasoning.
This study
of these approaches including
number
Fuzzy
into
and
Bayesian
Dempster-Shafer
Certainty Factors,
Set Theory, Possibility Theory
and
Theory
non
logic.
Each of these theories and their
formalisms
are explored by means of examples.
underlying
The discussion concentrates on a comparison of the different
approaches, noting the type of uncertainty that they best
monotonic
represent.
CR
Categories
1.2.1
1.2.4
and
Medicine
Science
and
Artificial
Formalisms
:
Uncertainty,
Shafer
Theory
Intelligence
Representation
Knowledge
Keywords
Systems
Expert
Applications
and
Methods
approximate
of
reasoning,
Evidence,
Fuzzy
set
Dempster-
theory
Table
1
2
.
.
3.
of
Contents
Introduction
The
Nature
3
of
6
Uncertainty
10
Probability Theory
3
3
3
.
.
.
1
Background:
2
Bayesian
3
3.4
3
.
5
What
are
Conditional
3.2.2
Bayes
3.2.3
Statistical
3.2.4
Ignorance
-
14
Probability Theory
3.2.1
Mycin
10
probabilities?
An
'
Probabilities
Theorem
Early
17
Independence Assumptions
Expert
Mycin
3.3.2
Combining
evidence
3.3.3
Certainty
Factors
3.3.4
Ignorance
Certainty
System
23
in Mycin
-
26
Examples
28
31
3.4.1
Representation
3.4.2
Belief
3.4.3
Combining
Evidence
3.4.4
Demster's
Rule
3.4.5
Ignorance
of
22
Factors
Dempster-Shafer Mathematical
,
19
21
3.3.1
Comparison
15
of
Theory
Uncertain
Evidence
of
Information
Commonality, Plausibility
of
Representation
3.5.2
Ignorance
3.5.3
Prior
-
from Multiple
Combination
-
An
33
Example... 38
sources
40
Examples
42
48
Probability-based
3.5.1
33
of
Probability
50
approaches
50
Belief
52
-
Problem
of
Subjectivity
....
53
3.5.4
Assumptions:
Mutual
3.5.5
4
.
Fuzzy
4
.
1
54
56
Inf erencing methods
58
Theory
Definition
of
Fuzzy
Sets
59
4
3
Possibility Theory
65
4
Ignorance
67
.
.
Logic
Comparison
Non-monotonic
63
of
Possibility Theory
and
Probability
68
74
Logic
5.1
Basic
Characteristics
75
5
Modal
Logic
76
.
2
5.3
6.
and
Exclusivity
Fuzzy
4.5
.
Independence
4.2
4
5
Set
Statistical
Doyle's
Summary
and
Truth Maintenance
Conclusion
77
78
83
REFERENCES
ANNOTATED
System
BIBLIOGRAPHY
85
Chapter One
Introduction
Every day
decisions
and
domains
These
information
to
obtained
several
questions
information
required
also
The
also
be
could
information
Expert
must
to
*
known.
not
The
known but
is uncertain,
they
are
not
the
then
answers
totally
sources
the
If
be
can
problem
the
all
multiple
inconsistent.
or
in that
available
from
obtained
The
ways.
in that
be approximate,
conflicting
systems
have been
and
The
and
goal
in
expert
system
a
domain
with
of
to
totally
attempted
or
specifically,
based
non-numeric
is
first
ability to
about
The
situations.
approaches
these theories
the
knowledge
proposed
people
support
with
conceptualize
solving
numerical
to
created
be designed
expert'
reason
problem
are
making
environments.
several
only be partial,
could
information
domains.
uncertain
combine
in
of
available
solved
uncertainty.
with
making
are
uncertain
characterized
could
answers
accurate.
only
be
the dilemma
with
in
problems
solving
can
faced
are
people
approaches
supported
for
domain.
*
these
with
design
expert'
Several
is to
ability
methods
range
from
probabilistic
reasoning
logical
reasoning
based
predicate
by
cope
handling uncertainty in
theories
strict
on
order
a
in decision
on
calculus.
different
Each
assumptions
of
and/or
interpretations
of
existing theories.
other
uncertainty best handled by
is the primary
It
investigate
by
proposed
in
making
AI
uncertain
domains.
Bayesian
Factors,
Dempster-Shafer
theory
Theory
of
each
represents
Secondly,
the
first
area
is
methods
the underlying
theory
handles
ignorance
Finally
or
we
some
assumptions
special
conflicting
will
look
information from
at
each
Chapter Two will
uncertainty
and
An
situations.
how
included.
Mycin
In
Evidence will
related
cases
they
and
How
such
uncertainty
inferences
be
how
along,
required.
will
theory
in
arise
how
of
task
can
of
problem
people
each
as
be discussed.
made
using the
Bayesian
of
solving
process
resolution
the
also
information
will
as
and
Theory
three very distinct
probability.
Zadeh's
be
Probability Theory,
the Dempster-Shafer
and
presented
interpretations
discuss Fuzzy Set
are
describe different types
Factors
be
of
evidence.
examined
information then
Chapter Three
Certainty
that
each
uncertain
be
will
how
theory.
overview
for decision making
sources
or
The
four
on
with
information
for combining
used
Set
logic.
non-monotonic
concerned
be
Certainty
Fuzzy
focus primarily
the uncertain
will
Mycin
will
information from different
with
theories that
Evidence,
decision
and
solving
theory
The
areas.
and
of
to
that have been
problem
The
Possibility Theory
discussion
this thesis
of
Probability Theory,
are:
general
for
different.
quite
approaches
researchers
reviewed
Theory,
objective
different
six
is
each
the
Therefore,
Chapter
Possibility
of
yet
Four will
Theory.
Chapter
Five will
non-numeric
Chapter
will
be
be
approach
Six the
concerned
to
strong
summarized.
with
reasoning
and
weak
non-monotonic
with
aspects
logic
uncertainty-
of
each
In
approach
as
a
Chapter Two
The
Nature
The
are
of
Uncertainty
domains
complex
very diverse
often
usually
can
be
1987].
The
first
requirements
step
alternative
to
analysis
done
In
selected.
steps
for
a
when
to
tremendous
either
information has
more
an
steps
objectives
and
necessary
solutions
are
generated.
the
all
these
impacts
last step
or
actually
and
solve
on
the
of
costs
the
most
of
a
the
appropriate
evaluation
problems,
needed
or
is
there
iteration back to
of
second
alternative
further
information is
effect
The
interpretation
an
made
amount
and
[Sage,
the decision
of
the potentially
implementation
order
process
all
of
is
situation
and
in step two is
for
alternative
the
In
alternatives.
is the formulation
analysis
evaluate
each
decisions
make
three primary
by
identified,
are
people
Though
step the desired
involves the
solutions
need
step
which
the underlying decision
characterized
this
In
acceptable
dynamic.
and
in many ways,
unique
problem.
within
is
a
earlier
new
selected
previously
alternative.
Information
block
of
this
is
without
decision
doubt the
process
understanding human reasoning
understand
the
information
most
model.
is to
used
by
important building
The
first
characterize
people
for
step
and
problem
in
In
resolution.
is done,
processing
particular
information is aquired,
the
and
represented,
The
decisions
is
solutions
are
be
can
within
however, the
nature
In
and
quickly
of
the
is
several
not
very
is
often
ill-suited
a
people
are
forced to
make
alternative
More
due to
cut
or
evidence
characterized
inaccurate, inconsistent,
positive
clear
The
make
alternative
the best
cases,
being partial,
for
forced to
are
information.
available
inferences.
make
in that
is
to how
paid
little difficulty.
with
choice
to
people
some
available
information that
used
which
be
must
information is
how the
complex
very
possible.
chosen
commonly,
the
often
attention
it then is
how
situation
information
to understand how
order
vague
otherwise
or
judgement to be
made.
decisions based
on
as
Therefore,
uncertain
information.
There
information
to the
a
or
the
the
when
the
a
number
a
could
be
partial
information
result
of
are
the high
fact that the
information has
a
reasons
of
fact that the tools
required
be
are
time
constraint
information
cannot
decision to be
The
sources
electronic
imperfect.
sensor
or
cost
our
is
placed
be
collected
on
the
person,
to
could
tools
obtain
Often,
process,
quickly enough,
partial
information,
This
adequately.
decision
a
the
proper
required
be due
could
all
obtain
obtaining the
of
on
It
available.
readily
been developed
another
Therefore,
to
technology
based
made
of
not
evidential
incomplete.
or
required
actual
not
that
all
forcing
information.
whether
are
they
be
an
inherently
information that
we
obtain
from
these
sources
solve
the majority
with
consulting
these
be
problems
multiple
that the
information
information from
or
another
an
error
tests
made
to
resolve
can
be
collected
in
an
we
rely
a
errors
effort
on
from
It
one
could
and
source
the
continue
conflicts
discrepancy,
be
must
additional
or
to outweigh the
is
the
with
has been recognized,
the
to
decision.
In order to
source.
Once
be
order
information from
support
obtained
In
actually conflicting
recognized.
can
The
to
these
solving process,
face,
we
sources.
combined
inconsistent
either
problem
the
of
is then
sources
possible
in fact be inaccurate.
could
data
more
false
information.
People
formal,
rational
the
not
this
they
perceive
in
for
it
It
expert
more
simple
not
information but
unknown.
person
evidence
all
only
also
choose
required
to
truths
a
this
its
with
of
the
world
falsities.
and
formal
state'
As
representation
it becomes less
original
meaning,
or
at
together.
important to
to
evaluate
recognize
By understanding
may
a
x
current
state'
development,
inconsistent
is lost
is
system
the
Unfortunately,
information is translated to
accurate,
times
how
expressed
in
Their decision making processes
manner.
by
decision making
approach
information describing the
easily
scheme
a
always
surrounding them.
world
perceptual
is
not
strongly biased
are
of
do
to take
what
what
this
available
information
information
specific
eliminate
the
actions
remains
to
ignorance.
remains
unknown,
obtain
the
properly describe the decision making process,
To
imperative to
belief
ignorance.
total
and
situation
where
support
decision
no
a
none
between
any
hand,
situation
not
all
certain
the
of
present
alternative
representing
between
limit
person
alternative
to
This
difference between
discussed
the
further
in the following
number
as
it
of
not
to
make
though
be
a
Based
able
and
to the
there
a
a
to
is
choice
On the
other
exists
when
completely
subset
on
to make
that
partial
the
of
this
a
She will,
alternatives
ignorance
chapters.
a
solutions.
relates
bias
as
required
words,
belief
partial
available.
will
able
other
will
partial
characterized
solutions.
required
is available,
information is
a
that
a
information
In
available.
information
information,
choice
the
decision
necessary
is
is
Ignorance
the relevant
of
evidence
supporting
a
the difference between
understand
it is
are
subset
certain
however,
be
feasible.
belief
formalisms
of
will
be
described
Chapter Three
Probability Theory
Probability theory
in AI
decision making
1987].
[Nutter,
At
and
the
theory in reasoning
subject.
theory is
it
for
dealing
however,
numbers
severe
with
are
or
often
A
subjective.
make
this
argued
theory
over
10
role
proven
The
not
generally
same
discounted
of
on
time,
assumptions
stems
1986]
from
x fuzzy'
Background
Numerical
many different
:
that the
its lack
of
formalism
to probability,
available
the
with
evidence
that
for
in
of
It
is
a
that
Theory is
order
is
to
also
probability
expressiveness,
especially
events.
What
are
probability
ways.
probabilities?
theory has been interpreted in
Each of these different
10
is
it is
Bayesian
required
inadequacy
a
requirement
the basis
contention
to be
probability
opponents
is
the
of
years
probability
mathematical
that the
At
of
continues
theory computationally feasible.
[Zadeh,
describing
for
out
that
source
in
in favor
fact
a
role
systems
uncertainty
A
active
point
strong independence
the
3.1
to
quick
an
time, the
uncertainty.
evidence
is
expert
provides
shortcoming.
available
played
same
with
widely debated
that
has
for
interpretations
* language'
theories has
or
to be
frequency
A
ratios.
Each
likelihood
of
statistical
from
the
number
of
an
the
of
outcomes
that
of
zero
represents
all
weight
defined
an
as
assigned
limited
a
(E)
event
In
one
of
to
AI,
elements
the
subjective
be noted,
studied
and
available,
In
AI,
a
for
this
with
samples
to
A weight
it is true.
The
space.
the
as
sum
An
of
of
to
(A)
is false,
hypothesis
sample
proportion
element
set.
a
the
the
as
the
a
tests,
or
in this
of
all,
the data that
no
strong
that
large
of
whereas
is
event
probability
all
the
of
weights
nature
11
of
is
in
areas
a
of
good
probability has
of
amount
not
be
data
of
available
is
available
statistical
amounts
offers
the
would
probability
approach
due to the
a
resulting from
given
is defined
that
First
however,
which
assigned
in E.
Secondly,
very
element
frequency interpretation
to derive
cases.
the
or
is defined
This weight defines
correspond
is then defined
is
space
in terms
hypotheses
space,
is to say,
given
the
of
of
fall
represents
subset
most
should
that
that the
application.
required
of
that
elements
sample
one.
(A)
element
possible
sample
That
experiment.
all
the
as
to
zero
likelihood
number
of
occurrence
sufficiently large
a
set
in the
element
ranging
have been defined
probabilities
generally known
outcomes,
weight
specific
to express probability judgement
used
Traditionally,
first.
own
1986].
[Shafer,
of
its
produced
often
foundation.
that
are
the majority
for
of
It
well
statistical
method
in
data
are
reasoning.
domains,
large
As
a
has
amounts
result
given
event,
degree
the
the
a
person
an
this
In
certain
that
chance'
will
belief
functions
the
or
is
the probability
from the
the
Bayesian
degree
hypothesis
true.
*
a
the probability
subjective
event
is
subjective
will
of
measure
of
The
occur.
directly
attached
an
to
for
Mycin,
a
subjective
measure
Dempster-Shafer
true,
not
is
In this
belief
of
Theory
in that the belief is the
believes that the
person
chance'
Dempster-Shafer Theory.
The
approach
Theory is the theory
Bayesian
whether
to
attached
as
differs
measure
evidence
proves
the
the hypothesis
itself
is
the
whether
evidence
the hypothesis.
certainty factors
Mycin
model
a
to be
The
supports
is
the probability
belief that
observation.
The
[Shortliffe,
indicate that
1976].
a
of
Confirmation
is
of
a
supported
by
a
evidence
or
to
proven,
support
the
as
x
given
factors
is
confirmation
hypothesis
a
early
diagnosis.
for certainty
of
is true
in the Dempster-Shafer Theory,
12
is
an
measure
subjective
basis
as
medical
probability known
hypothesis
indicates that there
a
in
use
hypothesis
a
theoretical
in the interpretation
developed
were
inexact reasoning for
physician's
As
a
occur
the
of
in the Bayesian Theory.
In
is
believes the
event
as
probability
approach,
evidence,
generalization
theory,
of
a
approach
event.
A
of
belief.
of
of
unavailable.
the Bayesian
limitations,
definition
the
is generally
data
statistical
these data
of
adopted
measure
of
does
not
it merely
the hypothesis.
chance'
is
attached
to
whether
All
of
the
three
present
the
3.3,
expert
Shafer
the
are
the
differences
lie
in the
Theory
will
of
of
Mycin,
attention
The
subject
to
fact that
next
section
much
they
will
considerable
features
to the
calculus
and
debate.
In
inexact reasoning implemented in
will
be discussed
13
ways.
standard
Probability in
presently
model
system,
theory
use
particular
that
the hypothesis.
approaches
in different
Bayesian
assumptions
the
Their
calculus
detail paying
section
these
of
probability-
apply this
supports
evidence
be
presented.
further
in
The
section
Dempster-
3.4.
3.2
Bayesian
The
Bayesian
Thomas
of
Bayes
(1702
represents
the
a
a
sum
space
would
All
are
be
or
range
of
expert
approach
formalism for
1984],
of
Bayes
with
will
it
occur
will
that
represented
a
by
a
A hypothesis
zero
imply
would
in
hypotheses
values
a
such
given
that their
in the following
with
provide
a
uncertainty
as
a
uncertainty
good
mathematical
[Clancy, Shortliffe,
inherent limitations
early
systems
were
mechanisms
of
dealing
Theory
several
using heurisitic
model
for
Bayesian
due to
these
Theorem,
used
in fact,
with
Unfortunately,
mathematical
of
probability
systems
does,
dealing
true Bayesian
schemes
(H)
chance'
[0,1J.
of
value
foundation for their designs
this
is
false
of
one.
Many early
and
*
The
alternative
assigned
In this
believes
person
work
is believed to be totally
one
probability
absolutely false.
a
the
on
subjective measure
evidence.
is true
probability
whereas
sample
a
1986].
[Shafer,
which
in the
value
probability
to
current
hypothesis
assigned
1761)
-
the degree
given
specific
probability is based
of
The probability that hypothesis
certainty.
true,
Theory
probability is
a
approach,
occur,
Probability Theory
but
methods.
instead
14
and
the
with
ad
A discussion
the Bayesian Theory
sections
implemented
not
will
assumptions
of
be
and
hoc
the
presented
inherent
limitations
3.2.1
this model will
of
Conditional
Probabilities
Bayesian probability
conditional
is based
probabilities,
which
otherwise
of
probability
event
a
hypothesis
(E) has already
(H)
is
event
(E)
zero
read
The
.
and
as
hypothesis
probability is
is true.
represents
fact that
we
would
An
set
P(It
the
of
always
an
is known.
are
often
they
are
set
or
the
formally,
fact
a
as:
to
a
(H)
(N)
value
given
between
belief that the
our
following
it is lightning.
rains
when
If
example.
probability,
In discussions
prior
to
to
any
15
as
to
in contrast,
hypothesis before any
referred
it
prior
evidence.
of
was
given
other
Bayesian
the
our
is lightning,
there
is raining | It is lightning)
evidence
most
on
likelihood that it is raining
unconditional
probability
More
hypothesis
of
Consider the
know that
belief that it
is incomplete
is raining | It is lightning)
P(It
This
situations
N
=
set
represents
in
probability is the
represented
the probability
which
one,
event
of
occurring based
P(H|E)
which
idea
useful
occurred.
probability is
conditional
an
A conditional
uncertain.
the
on
are
the information concerning
where
that
be highlighted.
we
one.
is the
information
or
Theory these
probabilities,
because
A prior probability
is
represented
and
P(H)
as
N,
is true before any
In
let
probabilities,
1972].
[Lukacs,
us
trials were made
and
frequency
relative
RF(X)
N
=
Nx /
the
Likewise,
a
events
(X)
and
event
(X)
is:
What we
of
are
event
the
Y
number
respect
to
space
is
which
event
X
is
a
event
outcomes
of
reduced
composed
events
represented
=
experiment
(Y)
were
observations
the
of
were
3.1
of
relative
sample
space.
hypotheses
The
to
This
we
)
3.2
frequency
to know
(Y)
reduced
with
sample
set
original
frequency
of
Y
**xny__
RF(Y|X)
=
and
RFfXOY)
RF(X)
16
3.2
we
get
(
3.3
in
given
as
from 3.1
substitution
.
)
Nx
by
(X)
event
need
event
from the
relative
The
is
observed
correspond
n
(XHY)
event
In other words,
that
all
where
(
X.
ratios
observed.
/n
is true.
(X)
RF(Y|X)
of
conditional
frequency
of
interested in is the
really
given
of
frequency
Nxny
zero
hypothesis
a
(
representing that both
=
of
n
relative
RF(XOY)
idea
statistical
the number
represents
belief that
to the
return
of
between
number
formal definition
a
Given
of
a
information is known.
other
to build
order
is
N
where
representing the degree
one
where
=
)
Given that
eguation
the
3.3
Let
P(Y|X)
called
occur ed.
P(X)
>
0
be
(.Q.,S,P)
trials
of
the
suggests
Suppose
events.
is
number
a
0
definition.
space
Now
let X
P(Y)
=
probability
Y
of
and
0.
From
that
3.4
we
see
P(XHY)
=
P(X)
P(Y|X)
P(XOY)
=
P(Y)
P(X|Y)
Pm
we
3.4
given
Y be two events
By eliminating P(Xf)Y)
Bayes
Bayes'
'
event
X has
that
suppose
obtain
PfX|Y)
(
3.5
Theorem
can
specifies
be
a
way that
calculated.
From
conditional
equation
3.5
we
see
that
P(H|E)
P(H)
=
(
PfElHl
3.6
)
P(E)
P(H)
)
Theorem
probabilities
Where
two
)
P(X)
3.2.2
be
Y
and
:
and
>
X
and
(
conditional
P(Y|X)
then
is large,
experiment
P(XQY)
P(X)
=
the
and
following
probability
>
P(X)
the
of
and
hypothesis
(H)
represents
the
P(E)
or
represent
event
(E)
conditional
17
prior
occur,
probabilites
respectively.
probability
of
the
that
P(E|H)
event
(E)
hypothesis
given
represents
the
is known.
formula
by applying
denominator.
a
set
let B be
events,
the total
an
of
of
the
eguation
probability that
Theorem
The Total
Let A be
side
Equation 3
Bayes'
formula.
left
The
.
the posterior
evidence
Bayes'
(H)
.
6
(H)
Rule
as
from this
to the
rule
probability
Probability
after
is generally known
be derived
can
states:
exclusive
mutually
occurs
and
exhaustive
3.6
we
arbitrary event,
P(B)"
^P(Aj) P(B|Aj)
=
applying this
Therefore,
Bayes'
to
equation
at
arrive
Theorem:
Assume
exhaustive
a
of
mutually
events
or
hypotheses
P(Hi)
PCElHi)
^PCHi)
P(E|Hi)
=
The
Total
all
hypotheses
This
hypothesis
in the
the
remainder
3.5
)
be
in the
means
of
our
by
decisions
made
As
are
space
given
space
can
occur.
Bayes'
,
requirement
mutually
time
any
(S)
space
As
only
a
formula
one
basis
(
for
Equation
used.
characterized
evidence.
mutually
sample
at
discussion,
Most decision making
are
in
and
introduces the
sample
that
sample
(H)
rule
Probability
exclusive.
will
exclusive
set
PfHjjE)
that
rule
new
a
single
based
environments
piece
on
a
information
18
of
are
not
evidence.
collection
of
is gathered,
generally
Instead,
information
and
beliefs must be
to
revised
to
extended
this
relect
new
accomodate
Bayes'
evidence.
multiple
pieces
of
formula
evidence
be
can
as
follows:
P(H|Ei...En)
=
P(H)
(Ei..^!!)
P(Ei...En)
Unfortunately
number
of
probabilities
above
concept
3.2.3
equation
of
events
or
has
situation
or
a
direct
a
other,
is merely
represented
on
of
an
event
(H)
are
true
to
occurrence
situations
or
has
no
of
Conditional
false.
At
a
bearing
In this
where
another
characterize
hypothesis.
situation
on
case,
the
our
event
(E)
totally independent.
is thought that
events
X
and
Y
are
it follows that the probability
equal
primarily
belief that
our
need
hypothesis
it
affect
particular
the
on
in describing
we
occurrence
concerned
probabilities.
however,
the
If
useful
based
either
where
in
events
is
extreme,
simplify
form is to apply the
workable
hypothesis
other
making this
to
One way
have been
conditional
are
the
Independence Assumptions
of
belief
increases,
also
)
independence.
probabilities
evidence
X
more
sections
probabilities
each
a
increases,
evidence
required
The preceding
other
and
to
statistical
Statistical
with
of
computationally impractical.
solution
the
the amount
as
3.7
(
to the probability
as:
19
of
Y
of
independent
Y
alone.
given
This
of
event
is
P(Y|X)
By substituting this into
events
be
describes
relation
further
to
a
to
true.
This
probabilities,
events
the world
of
be
can
can
be
&
all
new
&
E2)
form, the
to
order
based
that
the hypothesis
3.8
incorporate the idea
of
each
be
3.8
that
independent in
in
are
which
=
as
a
)
in
of
equation
P(H)
can
extended
relation
hypothesis
particular
is
follows:
P(X|H)
PfE-jH)
probabilities
case
of
is gathered,
evidence
apparent
events:
other
independent
are
Y|H)
=
in the
required
evaluated
relevant
(
P(Y|H)
Bayes'
formula
on
as
the
new
can
be
new
updated
20
(
required
one
event
are
is
From
received
easily.
3.9
)
reduced
[Charniak,
the hypothesis
evidence.
evidence
P(E2|H)
P(E2)
PfE^
As
P(Y)
as:
PfHiE!
those
that two
see
independence assumptions,
these
written
In this
we
P(Xn)
represented
P(X
By adopting
In
world.
describe
subset
that
events
to the
conditional
V(XL)...
=
P(X)
=
include
to
extended
P^O.^)
This
3.4
independent if:
are
can
P(Y)
equation
P(XOY)
This
=
must
3.9,
be
to
1984].
re
it is
the probability
of
Ignorance
3.2.4
The
Bayesian
explicitly
Probability Theory
represent
total
does
ignorance.
In
not
offer
to
order
the total
lack
is
The maximum entropy assumption assumes
used.
events
as
a
as
possible
each
result
case
probability
of
least
two
would
and
over
is
event
representing the
in the
knowledge,
independent
are
evenly
of
sets
then be
will
be
up
a
amount
further
a
of
P(B)
=
discussed
21
probability
that
all
.
As
For
example,
B the
A and
follows:
as
=
.5
that
The validity
in
1985]
value
commitment.
events
background
against.
simulate
assumption
[Cheeseman,
events
distributed
neutral
compared
all
independent
be
way to
then distributes the uncertainty
assigned
P(A)
This
the maximum entropy
a
section
any
of
3.5.3.
new
changes
these
can
assumptions
3.3
MYCIN
An
-
Early
Mycin medical
Expert System
diagnosis
the Stanford Heuristic
Shortliffe
Group
and
in collaboration
Mycin was
developed to
treatment
of
key
is
uncertainty that
is based
Mycin
during
physician,
backward
an
the
towards
critical
the
feature
diagnosis,
reasoning
express
This
One
the
of
for the
account
decision
clinical
and
making.
the
physician
condition
of
is
asked
their
a
benefit
The
questions
number
hypothesis
the
to backtrack
used
attributing to the
and
system
the
acceptance
was
Mycin
of
is
focused
are
This
1982].
[Michie,
of
A
backward chaing
of
by
asked
a
patient.
is then
method
with
a
by
physicians.
To handle
a
of
particular
a
in the diagnosis
tree to determine the disease
treatment.
groups
1982].
[Michie,
how to
of
Disease
Infectious
interactive dialogue
an
inference
and/or
appropriate
that
on
which
chaining
part
1984].
concerning the
questions
as
Edward
by
infections.
was
inherent in
[Buchanan, Shortliffe,
the
in 1974
blood
in Mycin
faced
Project
physicians
and
developed
was
with
School
aid
meningitis
challenges
through
Programming
Stanford Medical
at
system
the uncertainty
Shortliffe
model
degree
of
Buchanan
and
in Mycin
varying degrees
belief
is
22
involved in
of
which
belief
medical
developed
allows
a
as
implemented
physician
in facts
characterized
and
and
a
to
hypotheses.
probabilistic
weight
called
session
of
would
which
an
that
by entering
and
part
1984],
The
definition
factors
and
the
units
of
and
of
in
this
some
functions
Certainty
certainty
measure:
representation
of
is
as
read
h,
hypothesis,
Similarly,
the
used
cases
factors
and
belief
is
the
formal
measure
the
are
two
separate
given
and
interpretation
of
combine
based
of
=
of
[Adams,
certainty
them
be
will
evidence,
of
independent
on
independent
two
The
formal
x
increased belief
of
=
e,
is
equal
disbelief
is
x
e.
measures
confirmation
in the
to x.
increased disbelief
the evidence,
23
however,
equivalent
disbelief.
notation
measure
to
be derived from
can
is
shown,
notation
to
was
model
Factors
MD[h,e]
hypothesis, h,
10
sections.
belief
given
representing the
and
its inherent
model
MB[h,e]
which
1
of
to probability
converted
formal
and
in the following
Mycin
between
number
It has been
restrictions.
theory
Mycin's
degree
his
express
1976].
probability
3.3.1
a
interactive
an
in developing this reasoning
substantial
presented
could
Bayesian probability and
strict
a
physician
During
then be automatically
Their goal
assumptions
a
event
[Shortliffe,
values
avoid
certainty factor.
Mycin,
with
certainty
a
The
in the
requirement
is due to
theory stating
for
an
that
C[h,e]
=
1
a
-
C[not
piece
does
h,e]
of
not
[Shortliffe,
that
evidence
In
particular
1-x.
Even though
measures
are
negation
the
of
hypothesis
the
can
be
avoided
to
a
These measures
belief
of
in terms
represented
two
by using
(MB)
Bayesian
of
MB[h,e]
=
retain
MD[h,e]
=
a
independent
strong
disbelief
as
(MD)
hypothesis
prior
conditional
subjective
must
be
to any
probability
theory,
beliefs
the disbelief
3.9
represents
the
evidence
the
that
(
and
all
in the
P(h|e)
represents
3.10
)
in the hypothesis,
decrease
1984],
hypotheses
Therefore,
one.
h,
and
a
l-P(h)
equation
in disbelief
If
the
In
evidence.
alternative
to
sum
[Buchanan, Shortliffe,
24
belief
the known
proportionate
)
P(h|e)
subjective
given
3.9
(
P(h)
-
evidence
probability
assigned
represents
the
be
follows:
P(h)
represents
can
Pfh)
-
-
P(h)
and
probability
Pfh|e)
1
P(h)
degree,
in Bayesian probability theory.
foundation
where
a
inherent in
restrictions
factors
have
stating that
the hypothesis to
of
of
negation
hypothesis
a
that
states
experts
uncomfortable
of
belief, certainty
of
particular
this, many
some
probability theory
a
support
of
this
Simply,
they believe in
degree, x, they
they believe in the
the
support
that though
expressed
supports
affect
necessarily
that hypothesis.
1976].
piece
given
of
evidence
be
increases
than
greater
P(h)
representing the
if
a
piece
P(h|e)
of
our
growth
of
belief
our
On the
the
and
other
in
hand,
hypothesis,
a
of
value
will
increase
will
belief
P(h|e)
MD would
increased disbelief.
our
independent
these two
combines
into
disbelief
and
MB
of
belief.
P(h)
certainty factor
of
our
decreases
evidence
increase representing
measures
the value
and
be less than
would
The
belief in the hypothesis,
single
a
measure
as
3.9
3.10,
follows:
CF[h,e]
Substituting
and
,
into the
been
the
single
measure
values
from the
that
In Mycin,
facts
certainty
that
definition,
be
MB
were
prior
The
probability,
P(h|e),
Because
to
combined
are
it has
this,
of
factor may be
expert
and
a
more
express
reasoning
model
assumes
that the
values
of
MB
and
are
adequate
estimates
of
the
calculated
MD
if the necessary
known.
certainty
contain
and
an
1984],
expert
would
probabilities
and
CF[h,e].
intuitive term for
implemented in Mycin
equations
probablity,
of
that the
[Buchanan, Shortliffe,
received
MD[h,e]
that both the
shows
conditional
suggested
naturally
-
the probability ratios,
into this formula
P(h)
MB[h,e]
=
MD
of
[0,1].
Therefore,
of
[-1,1]
There
are
some
are
CF
a
factors
level
of
restricted
applied
of
to both
uncertainty.
to
is limited to
number
25
are
values
values
special
cases
rules
By
in the
in the
range
range
that define
the properties
1
=
MD[h,e]
and
hypothesis
0
=
and
negation
=
CF[h,e]
CF[h,e]
It
is
is
in
a
obtained
joint
out
point
is that
all
the
1984]
.
belief
MB^S-l
In
evidence
the
that
an
evidence
As
in
&
new
a
hypothesis
case
=
that
or
values
of
of
the
the
all
that
conclusion
MB
is
&
and
=
1,
by
adjusted
MD
to
belief.
our
are
and
is important to
these
functions
it is
assumed
other
[Adams,
increases
that
MB[h,S2](l
s2]
of
each
adjusted
new
-
It
is to say,
MB
on
as
incrementally
assumption
obtained
is usually
favoring
of
in Mycin.
is
+
MB
since
define how
That
MBtl^SjJ
26
MB[h,e]
Mycin,
evidence
is independent from
MDfhjSj^
in the
where
MD are
independently
combined
underlying
In
and
combination
is done
evidence
absolute
contradictory
information.
of
the
hypothesis,
s2]
or
MB[h,e]
when
certainty
situation
the
of
in Mycin
independence.
of
absolute
In the
functions
is
certain
hand,
other
MB[h,e]
where
representing
-1
or
remember
evidence
following
acquired
=
case
undefined.
effect
independent measures,
disfavoring
is
pieces
important to
The
CF[h,e]
Evidence
numerous
the
reflect
On the
there is conflicting
1,
=
Our belief
evidence
1.
=
In the
is absolutely
expert
the hypothesis.
and
on
1,
=
Combining
based
the
in the hypothesis
MD[h,e]
3.3.2
and
of
evidence
0,
=
MD[h,e]
disbelief
these measures.
of
as
-
that
our
follows:
MB^S-^)
definition MB
( 3.11)
will
0.
equal
&.s2]
MD^s-l
and
stated,
MD
be
disbelief already
Mycin,
description
hypotheses
Functions
of
evidence
is
our
a
Our
were
belief
the
acquired
also
defined to
disbelief
or
fact
particular
defined
are
values
or
for both the
belief
'and'
Boolean
defined
are
in the
as
and
min
(MB[h1,e], MB[h2,e])
MDth-L
&
h2,
e]
=
max
(MD[h1#e], MD[h2,e])
in the disjunction
or
the
by
represented
are
of
follows:
=
hypotheses
'or'
conjunction
e]
disbelief
the
multiple
h2,
two
MB
observation.
disbelief
and
in
allow
&
and
of
or
MBth-L
belief
of
MD[h,s1])
-
present.
two different hypotheses
or'
MD[h,s2](l
+
functions
given
operations.
Our
follows:
MD[h,s1]
new
disfavoring
increased proportionally to the belief
are
In
reflect
as
adjusted
=
as
to
hand,
other
MD will
evidence,
Simply
the
On
the boolean
following
equations.
MB[h1
or
h2,
e]
=
max
(MBCh^e],
MB[h2,e])
MD[hx
or
h2,
e]
=
min
(MD[hlfe],
MD[h2,e])
In viewing
these
assumptions
functions,
some
are
concerning the relationship
the
For
example,
consider
are
mutually
exclusive.
the hypotheses
either
there
would
hypothesis.
case
In
equal
The
27
where
this
zero
of
the
hypotheses.
hypothesis,
case,
the
regardless
disjunction
underlying
of
the
hi,
and
conjunction
of
our
h2
of
belief
hypotheses,
on
in
the
hand,
other
belief in
In
in
in
for
account
facts
beliefs
our
of
this,
our
belief
Shortliffe
and
Buchanan
incorporate the
the
of
evidence.
( 0,
CF[s,e]
)
MD[h,s]
=
MD'[h,s]
*
max
( 0, CF[s,e]
)
belief
MB'[h,s]
and
totally
these
the
certain
[Buchanan,
s,
are
based
Below
is
an
on
the
Factors
example
28
-
given
evidence,
rules
Shortliffe,
the
prior
that
are
1984].
describing
sections
used
actually
Certainty
of
represent
in the hypothesis,
factor
in the following
functions
MD'[h,s]
the decision
certainty
in the evidence,
and
disbelief
represent
experts
represents
3.3.3
strength
max
are
examples
functions
following
*
that
from the
actual
hypothesis,
MB'[h,s]
of
Mycin,
defined the
in the
reflected
the
our
the
h,
given
In
s.
acquired
CF[s,e]
actual
evidence,
will
belief
actual
our
=
measure
we
of
to both
applied
insure that
also
to
order
MB[h,s]
functions
In these
are
is properly
only
In
observations.
disbelief
or
not
hypothesis but
particular
To
our
1984].
[Adams,
uncertainty is
or
evidence
larger than
number
certainty factors
of
which
a
separately
a
evidence
in the supporting
measure
of
in Mycin.
rules
and
to
equal
decision making,
beliefs
our
be
hypothesis
either
our
apparent
would
e.
belief
The
illustrate how these
in Mycin.
Examples
of
an
actual
decision
rule
that
could
IF
be
The
1)
2)
3)
in the
found
strain
expert
the
of
Mycin
system
[Waterman,
1986]
.
is grampos, and
is coccus, and
organism is chains
organism
The morphology of the organism
The growth conformation of the
THEN
is
There
the
In more
suggestive
is
organism
generic
IF
X
&
Y
&
THEN
In
this
terms,
mathematical
decision
H with
factors
belief
are
are
example,
the
facts
certainty
factor
has
Consider
evidence
supporting
certainty
Our
of
is
be
represented
from experts,
based
rules
on
the
evidence.
given
absolute
In this
assigned
expert's
by
Therefore,
to this
actual
the
level
that the three
certain.
represented
prior
X
with
certainty
0.5
Y
with
certainty
0.7
Z with certainty
0.3
belief
CF'[h,x
29
y &
z]
=
0.7.
a
rule.
belief
in the
following
in the hypothesis is
&
as
certainty
relatively high
a
is true
has been
0.7
0.7.
of
factors.
experts
of
0.7
=
z]
expressed
that the
now,
y &
known to be
are
of
form
the
would
supporting
belief that the hypothesis
underlying
rule
acquired
of
expert
our
&
to these
assigned
in the truth
takes
certainty
this
notation
rules
identity
.
Z
CF[h,x
As
rule
that the
(0.7)
evidence
streptococcos
given
as
of
By applying
functions
the
incorporate the
the
calculate
actual
experts
strength
&
y &
z]
=
CF'[h,x
CF[h,x
&
y &
z]
=
0.7
Now
the
by applying
hypotheses we
*
=
0.7
*
=
0.21
&
y
&
z]
=
CF[h,x
&
y
&
z]
CF[h,x
&
y
&
z]
expert
the
supporting
certainty
expressed
downward.
It
knowledge
CF[x
&
&
y
&
y &
z])
z])
the
(
max
0,min(0.
5,
0. 7
conjunction
0.
,
of
3) )
0.3
relatively low level
a
a
zero
of
of
of
of
adjusted
expert
falsity
or
value
an
certainty
value
has been
noting that if
assign
of
consequently the
and
to the hypothesis
worth
would
can
CF[x
max(0,
describing
concerning the truth
he
evidence
is
*
z]
0,
max(
evidence
assigned
we
evidence,
to
at
0.7
CF[h,x
&
y
functions
arrive
The
*
&
of
3.3.2
section
belief in the hypothesis.
actual
CF[h,x
in
defined
a
has
no
piece
of
certainty to the
evidence.
Missing information is simply disregarded from
the
For
rule.
case
that the
is
It
example
as
evidence
several
belief
combination
consider
possible
also
of
actual
of
MBth^
&
s2]
information concerning
result
new
,
is
different
obtained
in the
all
these two
that
to
relevant
rules:
30
given
decision
hypothesis
the
a
s2
=
MBthfS-J
is
missing.
hypothesis
rules.
substantiate
must
be
rules.
in the
is the
In this
rules,
the
recalculated
For
case,
example,
as
a
RULE
1
X
IF
:
&
Y
THEN H with
EVIDENCE:
RULE
2
IF
:
certainty
X with certainty
Y with certainty
EVIDENCE:
certainty 0.8
Z with
First we must
calculate
for
based
rule
0.5
0.7
Z
THEN H with
each
0.6
the actual belief
the
on
certainty 0.5
in the hypothesis
in the supporting
belief
present
evidence.
RULE
1:
2:
RULE
The
by
CF[H,
X
&
Y]
=
0.6
CF[H,
X
&
Y]
=
0.3
CF[H,Z]
=
0.8
CF[H,Z]
=
0.4
factors
certainty
applying
equation
*
max
Y)
&
Z]
=
CF[H,(X
&
Y)
&
Z]
=
0.3
CF[H, (X
&
Y)
&
Z]
=
0.58
3.3.4
Ignorance
CF[H,X
+
&
0,min(0.
5,
0.
7) )
( 0,0.5)
rules
must
now
be
1
CF(H,X
combined
of
ignorance
the
an
expert
31
Y]
+
CF[H,Z](
-
&
Y]
0.4(0.7)
factors
representation
that
(
3.11.
&
event
max
from both
CF[H,(X
Mycin certainty
*
has
do
or
no
not
total
offer
an
unambiguous
lack
of
knowledge.
evidence
or
knowledge
In
)
a
concerning
disbelief
would
be
hypothesis
and
neither
Unfortunately,
due to
case,
the
net
belief
equal
evidence
disconfirms
0.
In
CF[h,e]=0
case,
is independent
evidence
confirms
this
or
and
disconfirms
of
the
it
1984].
[Shortliffe,
arise
to
equal
indicate that the
would
hypothesis, h, his belief
particular
a
certainty factor
non-zero
that has been
the hypothesis
of
values
zero.
32
by
of
of
MB
observed
equal
zero
and
MD.
both
amounts
may
also
In this
confirms
resulting in
and
a
3.4
Dempster-Shafer Mathematical
The
Dempster-Shafer Mathematical
first developed
further
Shafer
Mathematical
Shafer
a
on
of
This
not
the
in
the
This
between lack
of
theory
knowledge
3.4.1 Representation
with
is
of
the
with
based
an
make
First
narrow
by
the
this
explicit
all,
can
expert
set
The
set.
will
an
the
in the hypothesis
of
it
of
This
evidence.
gathered
not
affect
distinction
Information
evidence
frame
33
in
certainty.
Uncertain
of
representation
begins
subset
allows
and
of
accumulated
hypothesis
evidence
which
that
ability to
larger
a
on
support
has
single
calculus
Theory deals
reasoning.
evidence
a
probability
this
characteristics
the
on
Glenn
"A
of
of
was
Bayesian probabilities.
because
solution.
Theory
degrees
accumulation
bear
calculus
approximate
theory
evidence
Dempster-
The
applies
the
order
The
it
with
set
instead bears
actual
on
of
published
Dempster-Shafer
several
to
approach
achieved
does
but
has
and
standard
numerical
than
rather
Dempster-Shafer
hypothesis
be
and
Theory
in 1976.
although
The
way.
theory
attractive
the
Theory,
evidence
evidence
same
Evidence
of
in the 1960 's.
theory
Evidence"
of
the
uses
different
weights
the
extended
Bayesian
much
Arthur Dempster
by
Theory
Theory
the
as
Theory
of
in the Dempster-Shafer
discernment
(
e
)
[Shafer,
1976]
frame
The
.
hypotheses
or
frame
given
can
be
all
set
is
assigned
possible
subsets
possible
[Spruce,
a
the
range
of
our
exact
belief
8
0.
of
the
over
exclusive
events
2e
by
Figure
or
,
3.4.1
frame
to
in
a
and
which
belief
corresponding to
the
represents
of
possible
discernment
all
set
of
-
of
in the
belief
[0,1].
in
the
evidence,
evidence
The
a
basic probability
in
represent
the
as
theory
number
to
used
is defined
proposition
assignment.
by assigning
function
A numerical
a
Dempster-Shafer
basic
assignment
is
follows:
as
If
hypotheses
all
possibilities
mutually
represented
piece
a
probability
are
All
of
set
Pine],
indicates
defined
of
subsets
Fir,
Given
domain.
a
discernment
The
exhaustive.
in
events
of
is the
discernment
of
is
frame
a
is
m:2e->[0,l]
discernment,
of
the
function
basic probability
the
called
then
function
whenever
1)
m(^)
=
0
2)
m(0)
=
1
implies that
This
subset
will
represents
A.
This
total
a
assigned
sum
to
cannot
only the belief
total
belief
value
that
34
is
of
be
m
such
to
committed
by
all
assignment
the
the
to A
committed
to
one.
that
exactly to
subdivided
is
committed
equal
probability
committed
be
to
ought
belief
The
1.
the belief
belief
represents
The
is
0
of
numbers
one's
and
set
empty
belief
no
to
the
Each
the
m(A)
,
proposition
subsets
of
A;
alone.
a
hypothesis
is
it
[
Spruce
[ Pine,
[
Pine
The
Pine,
]
Spruce,
[ Pine,
Fir
Fir
]
(
]
]
subsets
of
the
set
of
Figure
35
coniferous
3.4.1
e
)
[ Spruce,
[
Fir
trees
(2W
Fir
]
)
]
represented
by
the belief
hypothesis
given
the hypotheses
function.
is the
(A)
that
imply
sum
of
A and
The total
the
all
the
belief in
beliefs
exact
in
in A
belief
exact
a
itself.
BEL(A)
=
^_^
m(B)
BCA
To
the total
obtain
to m(A)
the
The
belief in the hypothesis A,
quantities
for
m(B)
all
proper
basic probability assignment
from the belief
one
B
subsets
then be
can
must
of
add
A.
recovered
function:
r
=
.
m(A)
</
.
(-1)IA-BI
,-,
Bel(B)
BCA
for
It
one
follows that
for
a
all
AC0
belief
given
basic probability assignment,
the
Therefore,
belief
there
assignment
probability
function
and
is only
that
one
Corresponding to belief functions,
other
The
functions that
Doubt
evidence
function
refutes
represented
as
a
are
The
of
A
function.
by
The
the
function.
there
the degree to
hypothesis A.
three
are
which
Doubt
evidence.
the
function
is
follows:
DOU(A)
Our doubt
given
in characterizing
useful
expresses
a
conveyed
the basic probability
and
for
belief
information is being
same
is only
function there
is
equal
to
=
our
Plausibility function,
36
BEL (A)
belief
on
the
in the
other
negation
hand,
of
A.
expresses
the
extent
of
hypothesis A.
which
This
fails to doubt
one
function is
PL(A)
PL (A)
PL(A)
the
,
also
extent
terms
of
function
called
the
to which
=
1
=
1
upper
expresses
be
stated
BEL(A)
DOU(A)
-
-
probability function,
/_^
/_
-
m(B)
<C^
<
BCe
PI
function
(A)
S\
=
that
m(B)
m(B)
is
<p
useful
is defined
function
In
the plausibility
when
dealing
is the commonality function.
functions
commonality
plausible.
BCA
BOA
Another
expresses
follows:
as
=
follows:
as
the basic probability assignments,
can
the
refute
finds the hypothesis
one
P1(A)
belief
or
Q(A)
=
The
follows:
as
<C-~
with
m(B)
BC9
ACB
The
commonality Q(A)
that have A
Q(tf)
=
as
their
,
is
equal
to the
sum
of
all
subsets
It
is
interesting
to
can
be
represented
in terms
subset.
note
that
1-
The
belief
commonality Q(A)
function
now
.
Bel(A)
=
*
T (-1)
BCA
37
lBl
Q(B)
of
assignment
now
be recovered.
can
commonality functions
are
PL(A)
also
2_
=
<__
The plausibility
related
(-1)
follows:
as
|B|+1
(-1)IBI+1
and
Q(B)
BCA
it
Therefore,
correspondence
For
each
which
function
body
belief
our
and
of
the
plausibility
function
our
knowledge
a
the
BEL
(A)
and
case,
this
3.4.2
Belief,
In
an
section,
on
one
consider
to
Based
on
probability
sum
to
shown
one.
that
are
to
the
again
Bayesian
Plausibility
-
we
Spruce,
Pine
evidence,
to
in Figure 3.4.2.
3.4.2
for the belief,
38
is
of
previous
also
could
Based
take
of
represented
by
a
6,
assignments
Figure
in the
number
assign
members
all
The probability
we
special
An Example
The
Fir.
or
When
probability.
(X)
variable
belief
assign
will
assignment
the
and
precise,
in Figure 3.4.1.
scenario
is gathered,
and
In this
one.
belief
bound.
upper
certain
general
and
The
an
within
interval,
to clarify the definitions
available
calculated values
to
this
the
is
(A)
equal
reduces
values:
which
the
(A)
of
1984]
lie.
must
represents
hypothesis
Commonality
attempt
three
subsets
PL
theory
evidence
of
of
information.
same
lower bound
the
to
one
these functions define
hypothesis
a
in
are
[ Garvey , Lowrance , Wesley,
about
represents
the
represent
evidence,
interval"
"evidential
belief, plausibility,
basic probability assignment
and
commonality
one
follows that the
for
on
2e.
basic
that
such
each
shows
plausibility
set
the
and
they
are
Set
m
Bel
PI
Pine
0.2
0.2
0.8
0.8
Spruce
0.1
0.1
0.4
0.4
Fir
0.1
0.1
0.5
0.5
Pine,
Spruce
0.2
0.5
0.9
0.3
Pine,
Fir
0.3
0.6
0.9
0.4
0.0
0.2
0.8
0.1
0.1
1.0
1.0
0.1
0.0
0.0
0.0
1.0
Spruce,
Pine,
Fir
Spruce,
Fir
t
Calculations
for
a
the
set
given
of
for the belief,
body
of
evidence
coniferous
plausibility,
pertaining to the
trees.
Figure
39
and
3.4.2
commonality
subsets
of
functions, based
commonality
As
assignments.
their is
shown,
is due to the
This
probability assignments
[Spruce]
plausibility
summation
do
intersect
intersect
equal
to
the exception
it.
all
the
by
sets
the
of
that
set
[Pine]
of
Therefore, the Plausibility
value
is
0.8.
as
the
of
a
assignments
value
for Q
order
multiple
to
sources,
provides
are
combined.
body
the
of
joint effect,
and
different bodies
of
the
need
second
evidence
40
to
[Spruce, Fir, Pine]
.
is
combination
when
to be
need
two
or
is
to be
method
a
evidence.
of
evidence
combined
scenario
from
obtained
developed
where
scenarios
scenario
the
values
only
Sources
information
facility for
evidence
the
and
from Multiple
Dempster-Shafer has
first
sum
that have
sets
case,
to the
equal
is 0.1.
accomodate
generally two
The
all
[Spruce, Fir]
Evidence
Combining
for
In this
subset.
is
[Spruce, Fir]
set
resulting
given
for
with
for
There
subsets
represented
sets
those
that
is
all
,
are
In
set
In the case
with
summed
3.4.3
given
set.
the probability
The
a
that
[Spruce, Fir]
be
of
with
Commonality
of
for the basic
to the
the probability assignments
of
[Spruce, Fir]
than
greater
non-zero values
(m) pertaining
for
evidence
[Fir].
and
The
direct
no
though the belief function is
[Spruce, Fir]
zero.
these probability
on
must
more
to
when
facts
obtain
two
combined.
be
in
their
entirely
Given
a
a
&,
specified
concentrate
on
hypothesis
combining
from two
of
totally
belief
function that
frame
same
is
combined
not
commutative
=
<8
ml
m2
is the
Y
and
new
combining
are
1981].
it
and
ml
can
Let
evidence
length
in
a
new
combined
be
must
and
frame
of
of
m2
over
evidence
which
combination
Let ml
result
of
a
the
of
their orthogonal
k
=
(A-
one.
all
subsets
basic probability
In order to
rule,
evidence.
computes
order
same
ml(X)
XHY
X
a
is
rule
be basic
discernment
sum.
combining
The
ml
6.
new
and
m2
,
as:
m(A)
Where
negates
pieces
combined
the
since
the
as
probability assignment,
is defined
The
associative.
defining
,
two
on
impact
the
function to be
on
or
formal mechanism to
a
Combination
of
important
and
of
now
evidence.
represents
assignments
probability
Let m
of
discernment.
of
supports
different bodies
functions based
belief
The
have already
we
We will
that
provides
Dempster's Rule
evidence.
both
model
two belief
evidence,
of
evidence,
evidence
different bodies
Given
the
body
given
Dempster-Shafer
combine
is
a
how to combine
elementary facts.
shown
The
and
be
=
m2(Y))
A
whose
intersection is A.
assignment
as
a
understand
this
result
m(A)
of
m2.
intuitively
represented
in
a
the basic probability
be depicted
Now
*
41
a
unit
model
assignments
portions
as
Consider
geometric
of
square
a
combination
line
with
for
[Barnett,
body
each
segment
of
two sides,
one
side
Figure
3.4.3
vertical
in
mass
for
committed
a
is
0
between
square.
The
one
with
intersection
m2
strip
the
(A)
in the
we
of
results
of
combination
therefore,
subset;
.
m2
and
rectangle
to
between
us
look
square
sum
must
due
ml
Rule
at
and
in the
impossibility
certain
impossibility
of
of
some
Ml
hypothesis
to the empty
a
of
as
shows
of
c.
a
of
The
m2
where
K
is
the
sum
The
weight
log(K)
Dempster's
large
on
of
.
rule
are
two
amount
the
a
and
other
hypothesis
result
of
interesting
There
hypothesis
M2,
e
Example
An
particularly
3.4.4).
a.
-
ml
as
,
are
values
set.
is defined
possibility
hypothesis
by K,
<f
states,
setting
k is defined
of
examples
(Figure
42
by
=
aob
therefore,
are
new
all
value
other
Combination
consider
m2.
values
and
=0,
sources
possibility
in the
of
value
Dempster
set.
accomplished
each
two
Zadeh
to
evidence,
certainty
is
intersection,
no
assigned
(^)
assigned
First
combination.
example
The
values
Dempster's
Let
m2
This
l/(l-k).
conflict
3.4.4
The
0.
=
is
to the empty
multiplying
non-zero
very
made
1.
and
and
=0
equal
of
ml
horizontal
a
there
where
the ml
so
normalized
of
a
m2
representing
segments
associated
particular
that m($)
however,
all
and
m2(Aj)
rectangles
committment
(6)
(A)
from
side
values.
For
a
ml
*
ml(Aj)
other
There may be more than
AOB.
all
to
combined
strip
the
and
the two line
shows
orthogonally
a
ml
representing
c
shows
sources
of
also
in the
hand,
and
is
the
hypothesis
ml
(m2)
m2
1
probability mass
(ml(Aj) * m2(Aj))
m2
(Aj )
Figure
43
3.4.3
Normalized
Orthogonal
orthogonal
sum
Set
m2
+
ml
sum
m2
(ml + m2)/(l-k)
a
0.9
0.0
0.0
0.0
b
0.1
0.01
1.0
c
0.0
0.1
0.9
0.0
0.0
a
(0.9)
ml
(0.0)
b
(m2)
(0.1)
c(0.9)
a
(ml)
k
=
*
ml(X)
<
m2(Y)
=
(1-k))
=
0.99
xny=log(K)
=
log(l
=
2
/
Figure
44
3.4.4
weight
of
conflict
b
the
as
possible
only
it to be
considered
answer
highly improbable
assignment
probability
0.1.
of
for this
source
has distributed the entire
a,b,and
of
discrepency between
the beliefs
If
questionable.
the
reliable,
the
not
interesting
to
note
assignments
do
not
reflect
probability
of
the
sources.
is
combination
sources
the
of
same
ml
evidence
This
shows
example
narrow
in
on
frame
of
be
of
seen
sources
a
as
particular
3.4.6
45
have
b
a,
rule
and
for
are
committed
c
sources
hypothesis.
each
is
c,
have
The
reinforced.
reinforces
the
two
the
conflicts.
is collected,
we
This
can
hypothesis.
illustrates
discernment
evidence
fully
is
there
in that the two
for
to be
in the
seen
sources
discards
and
more
It
Again
combination
of
is slightly
Dempster's
Both
.
it
probability
conflict
assignments
rule
implies that
Figure
the
the
between
agreements
m2
large
considered
that the dominant hypothesis
that
shows
and
can
identical probability
result
not
belief to hypotheses
of
This
respectively.
source
resultant
example
has any
source
unexpected.
the
is
the two sources,
in Figure 3.4.5.
shown
amount
so
that the
interesting
Another
are
an
Each
source
Due to the
each
sources
is
result
of
of
the
among the
weight
neither
m2
by
shown
provided
implies that the
ignorance.
or
that the reliability
seems
unit
In other words,
uncertainty
and
counter-intuitive result.
which
c,
reliable.
entirely
degree
rather
ml
is
which
Shafer has
answer
hypothesis
though both
even
a
combination
[Spruce,
Pine,
of
Fir].
beliefs
This
over
Normalized
Orthogonal
orthogonal
sum
Set
m2
+
ml
sum
m2
(ml
+
m2)/(l-k)
a
0.3
0.3
0.09
b
0.3
0.3
0.09
0.26
0.26
c
0.4
0.4
0.16
0.48
a
(0.3)
ml
(0.3)
(m2)
(0.3)
b
c(0.4)
a
0.09
a
0.09
f
0.12
<f
(ml)
(0.3)
b
0.09
<ji
0.09
b
0.12
<f
(0.4)
c
0.12
<f
0.12
^
0.16
c
k
=
<_
*
ml(X)
m2(Y)
=
(1-k))
=
0.66
*ny=^
log(K)
=
log(l
=
47
/
Figure
46
3.4.5
weight
of
conflict
Normalized
Orthogonal
orthogonal
sum
Set
Pine
(P)
(S)
(F)
ml
m2
0.2
0.0
0.3
Spruce
0.1
Fir
0.1
Pine, Spruce (PS)
Pine, Fir (PF)
Spruce, Fir (SF)
Pine,
Spruce,
Fir
0.2
0.1
0.3
0.3
0.0
0.0
0.2
0.1
0.1
+
ml
sum
(ml + m2)/(l-k)
m2
0.17
0.24
0.22
0.14
0.315
0.20
0.11
0.16
0.04
0.03
0.03
0.02
0.01
0.015
(PSF)
(m2)
PS
(.2)
P
0
.06
t
(.1)
S
0
.03
s
(.1)
F
0
.03
(
.
2
)
PS
0
.06
(
.
3
)
PF
0
.09
(.0)SF
0
0
(.1)PSF
0
.03
k
=
r
S
f
^
log(K)
SF
PSF
.02
9I
.06
P
0
.04
*S
.02
P
.01
<f>
.03
S
0
.02
S
.01
S
.01
F
.03
(fl
0
.02
F
.01
F
.06
PS
0
.04
S
.02
PS
.09
P
0
.06
F
.03
PF
0
0
.02
.03
p
F
0
0
s
PF
.01
ml(X)
=
log(l
=
0.154
F
.03
*
m2(Y)
=
/ (1-k))
=
Figure
47
PS
3.4.6
0
.02
0
SF
.01
0.3
weight
of
conflict
PSF
completes
3.4.5
discussion
our
Ignorance
The
Dempster-Shafer theory
distinction between
explicit
ignorance
and
high
a
ignorant concerning
the hypothesis
level
its
case
set
of
discernment
all
lack
no
consists
[Male,
=
possible
BEL
of
knowledge
or
a
person
belief
In terms
-
an
When
he has
PL (A)
=
for
allows
of
disbelief.
of
determining
of
9
The
total
a
Theory ignorance is defined
Consider the
of
evidence
negation.
Ignorance
frame
of
hypothesis,
a
in
or
Dempster-Shafer
The
the coniferous tree example.
of
either
in
the
of
as
is
follows:
(A)
the
sex
of
male
and
female:
a
person.
Female]
is
subsets
2"
[0,^, Male, Female]
=
Bel
(Male)
male
likewise
Bel
is female.
In
and
person
was
available
the
person
would
Bel
be
evidence
called
be
all
zero
ignorance.
would
was
and
vacuous
of
the case,
male
It
of
be very
Bel
belief
Generally
(9)
48
person
belief
the
supported
the belief that
female,
follow
In the
low.
Bel
the
(Male)
be
would
that
one.
Bel
and
Bel
This
m(Female)
(Male)
case
represents
vacuous
evidence
and
m(Male)
is
that the
little
function,
stated,
that the
that very
would
available,
the
belief
represents
specifically
either
also
at
(Female)
very low.
set
the
that
was
(Female)
would
the degree
represents
that
and
no
(Female)
situation,
total
belief
function
is
obtained
by
setting:
m(0)
=
1, m(A)
Bel(O)
=
1,
Bel
49
=
(A)
0,
=
for
0,
A
f
0
all
A
4
all
for
0
3.5
Comparison
In
three
an
attempt
theory.
x
A
functions
theory.
is
decision
people
to
truly
of
a
make
psychological
therefore maintaining
The
these
Dempster-Shafer
intuitive way to
restrictive.
3.5.1
Representation
hypotheses
to
one.
Bayesian
the
are
The
varies
theory,
a
define
An
concept
that
of
functional
approach
to
describes
uncertainty
offering
a
the
requirements
problem
restriction
of
uncertainty,
actual'
require
among the three
by
x
manner.
each
approach
always
situations
level
probability
is
not
the
of
approaches.
more
appears
imposed
naturally
to
be
the
Belief
of
theory,
probability
assigned
expert
specific
actual
express
least
In
all
decision that
a
these theories
of
rational
a
the
requirements
in
decisions
to their
However,
do
rational
three
as
that
structures
however,
truly
a
All
.
adhere
solving.
provide
limitations
and
data
and
People,
in
more
1971]
[Carnap,
by
decision making,
is defined
decision
requirements
decision making
expert
to describe rational
rational'
the
by
underlying
how
Dempster-Shafer Approaches
formalism primarily based in probability
mathematical
specific
and
these approaches to uncertainty
of
abides
Bayesian, Mycin
of
forced
50
all
values
so
alternative
that
to divide his belief
they
among
sum
individual hypothesis in the
each
relatively easy task
of
of
the propositions.
belief
our
in
very difficult
belief.
This
often
In
However,
limit
forced to
expert
In
the
of
discernment
possible
we
is
so
a
a
In
this
restricted
to
our
doubtful
of
probability
approach.
actual
belief.
by independently
avoided
the
case,
to
all
expert
is
alternative
in his belief.
confidence
point
assigning
that the
so
upper
one.
where
theory
our
offers
Dempster-Shafer
of
that the total
more
expert
a
good
longer forced to
to
commit
therefore providing
advantage
51
a
a
of
alternative
space
belief will
approach,
the
assigned
the
or
to
sum
to
ways
frame
one.
express
supporting only
the
belief that he
better
incomplete,
theory, belief is
alternative
In this
is
knowledge
hypothesis
the
individual hypotheses.
This
is
to
given
his
of
all
it
find
number
a
in
a
confident
might
using the Bayesian
longer restricting him to
no
not
either
probabilities
his
of
subsets
the
provides
when
assign
is
sum
the
In
all
beliefs.
are
to assign
required
overcommittment
situation
to
in,
is
individual hypothesis
to
their
beliefs,
we
assign
expert
disbelief.
regardless
the
an
beliefs
our
however,
restriction
and
Dempster-Shafer
This
an
this
of
approach.
his
still
leads to
probabilities
the
that
of
that
case
proposition,
event
measuring belief
hypotheses
sure
uncomfortable to
is
he
Mycin,
longer
are
m the
to the hypothesis
value
we
is
This
space.
in fact ignorant
concerning
or
hypothesis,
a
or
In the
his belief
no
when
sample
expert
is
representation
Dempster-Shafer
not
of
is
no
confident
his
true
theory is
most
in the
evident
3.5.2
a
no
is
both
In this
described in
ignorant concerning
a
belief to the
frame
of
entire
set
possible
assign
his belief to any
of
a
allowing
and
is
known
Though this
established
by
in
found
it does
not
offer
express
his
lack
assign
a
point
the
of
in
is
his
the
forced to
argues
that this
events
all
a
therefore
express
his
Even
are
amount
a
clear
and
probability
to
person
amount
more
of
than
is
independence
and
is further
theoretically,
least
A
by
independent.
approaches
the
knowledge.
least
assumes
though
that
neutral
statistical
three
all
expert
52
not
representing the
argue
represents
entropy
all
way to
assumptions
3.5.4.
section
maximum
natural
is justified,
argument
is
person
represents
which
He
a
assign
Cheeseman
assuming that
assumption
discussed
entropy
Opponents
when
overcommitting his belief.
Bayesian probability,
without
way to
a
individual hypothesis,
In
committment.
an
hypotheses.
support
Dempster-Shafer
can
(9)
to
surpasses
offering
3.4.5,
discernment
knowledge
background
by
there
where
available
In the
hypothesis he
of
maximum
situation
Dempster-Shafer
section
comfortable
more
applying the
truly
the
ignorance.
represent
as
approach,
lack
a
knowledge
or
case,
as
Bayesian probability and Mycin
explicitly
to
total ignorance.
characterized
information
relevant
decision.
is
of
Ignorance
Ignorance
is
case
of
committment,
unambiguous
is
individual
still
way to
required
hypotheses,
therefore
forcing
In
person
has
be naturally
provides
a
more
in
maximum
neither
3.5.3
approach
Prior
All
theory
beliefs
Prior
than
because
to any
each
tremendously.
a
a
value
of
probability
approach
ignorance than
and
ignorance,
simulate
Problem
-
of
experts.
also
requires
as
of
explicit
the
as
evidence
Mycin,
53
on
rely
The
integrity
of
They differ in that
prior
reason
to be
being
probabilities
highly
are
more
often
subjective
known the
may be
the
and
to be
or
that prior
quite
probabilities
though
the
considered
are
expert
prior
Subjectivity
probabilities
valid.
stated
In
to
he
ambiguous
very
approaches
individual
therefore their
non-zero
a
-
considered
are
of
a
assigning
representation
conditional
being
as
prior
a
probabilities
probabilities
world
his
Given that
hypothesis,
a
express
still
methods
these
Bayesian probability
disregarded
to
was
it is
offers
of
subjective
subjective
hypothesis.
Even though this
entropy.
Probability
three
beliefs.
to
contrast
reasonable
Dempster-Shafer
the
exceeds
Although both Bayesian probability
representation.
offer
a
comfortable
intuitive
more
Bayesian probability
Mycin
to
zero
information concerning
no
representing
to
equal
to his belief
zero
belief that
of
ignorance is represented
by assigning
Mycin,
certainty factor
would
statement
belief.
actual
a
a
*
is
state'
different
of
the
and
may vary
theoretical
basis
of
MB
included
MD
and
experts
practice,
which
conditional
probabilities
are
both
combine
are
probability theory.
in
ways
estimates
as
Many
concept
that
new
still
of
not
problem
3.5.4
Statistical
that
discussed here.
well
sufficient
systems
and
prior
to
reason
can
refine
be designed
our
[Nutter,
strict
must
more
be
initial
1987].
statistical
idea
Independence
can
In
a
to
the
cases
is simply
approach
subjective
support
or
decision
be
most
found
actually known to be
on
which
the
Exclusivity
mutual
exclusivity
in
all
situations,
these
outcome
assumptions
is actually known
The
invalid.
of
part
are
three theories
involving
is due in
54
& Mutual
and
information than
assumptions
this
of
applied
of
solving.
especially those
documented
support
the
independence
Statistical
these
of
quality
and
These
value.
in the majority
Therefore,
and
are
one
gathered
that the
data to
probability
many cases,
or
Expert
is
However,
statistical
assumptions
In
probability is the only legitimate
making
predicate
into
improve
can
maintain
available.
personal
they
evidence
interpretation.
required
not
(P(H))
probability
in themselves
not
abandon
such
prior
(P(H|E))
probability
probabilities
are
to estimate the
certainty
asked
concerning the validity
questions
experts
to estimate their prior beliefs.
asked
specifically
factors,
prior
medical
and
diagnosis,
repercussions
these
approaches
to the
in
limited
of
is
not
number
of
implementations.
actual
A
number
verify the
was
of
integrity
designed to
of
compare
known probability data
certainty factor
known
and
One
factor
P(H|E)
Mycin to
on
results.
the certainty
( P(H)
and
conducted
the program
computed
for MB
values
have been
studies
such
from
computed
known
are
)
study
and
using the combining functions
MD.
The outcome
of
this
that the largest descrepancies between the two CF
values
were
impact
their
overall
to be
rather
short
reasoning
A
small
number
accomodate
independent
of
piece
note
due to the
predominant
use
evidence
or
be
a
rules.
There
techniques
help
the
by increasing
also
A
Caduceus.
good
together
It
the
a
the
solve
actual
have been
proposed
exclusivity.
system
makes
suggests
rule.
single
to
offered
independence
grouped
that though this may
unmanageable
have been
of
of
that
into
mutual
all
a
non-
single
is important to
problem
implementation
much
more
difficulty in obtaining
number
of
heuristic
the
restrictions
example
can
be
seen
[Charniak,McDermott,
55
relatively
help
and
avoid
to
appears
1984].
[Adams,
solutions
such
evidence
However,
in Mycin
restrictions
theoretically it
1984].
assumptions
solutions
One
exclusivity.
longer
and
these
of
chains.
of
the
evidence
[Buchanan, Shortliffe,
chains
reasoning
interrelated
to
and
study
showed
due
the
of
in the
1984].
mutual
expert
3.5.5
Methods
Inferencing
A
function
key
decisions
information.
falls
not
In the
making.
offering any
are
through the combination
The
represented
BEL and
the bounds
defining
statement
There
is presently
based
on
these bounds.
could
be
used
these values
The
should
ignorance in the
on
The
.
reasoning.
reasoning
cases,
prior
will
in
are
a
make
number
inductive-style
of
People
and
Inductive
reasoning usually
perception
56
of
the
loss
of
1981].
our
of
on
in
of
a
implemented
style
of
to deductive
In
Mycin.
how
most
people
incorporate
their
-state'
in making
[Thompson,
were
represent
tend to
results
which
masks
used
deductive
success
reasoning based
intuition
be
can
drawbacks
not
PI
or
The probabilities
in Mycin
a
of
have limited the
decisions.
[Barnett,
expected
with
Bel
However,
in that it
used
methods
for
making.
mechanism
the
each
for decision making
value
probabilites.
minimize
and
interval.
evidential
unknown
deductive reasoning does
actually
the
differs
conjunction
There
which
is
used
approach
reasoning
basis
this
be
be calculated for
mechanism
Possibly
the updating
by
decisions that
1985]
accepted
can
for decision
solely
Bayesian
calculated
no
and
[Barnett,
process
the
of
ignorance
in belief functions
directly
PL functions
for decision
means
Dempster-Shafer approach,
are
1981]].
make
available
effective
uncertainty
carried
is the ability to
systems
is here that the Dempster-Shafer
Theory
It
by
short
expert
inferences from the
draw
and
of
a
more
personal
their
solution
situation.
that best
represents
Deductive
or
is
most
reasoning,
shortcomings
with
Typicality-based
individual
its kind
has
also
[Nutter,
the data
has been
uncertainty,
a
of
shown
default reasoning
such
generalizations
typical
as
"Birds
which
Fly",
or
57
evidence.
to have
severe
typicality.
is involved in
centers
property that is typical
1987].
or
of
on
whether
other
an
things
of
Chapter Four
Fuzzy
Set
The
Zadeh
Theory
theory
in 1965,
complex
[Kandel,
fuzzy
of
Possibility Theory
a
development
approaches,
sets
due to his
systems
development
fuzzy
of
Since
has
theory
Possibility Theory
Fuzzy
set
language
as
tall,
is
a
applications
young
short,
generalization
presented
basis
in
fuzzy
of
sections
Possibility Theory
model
for
ignorance
be
included.
be
A
sets
4.1
be
will
and
comparison
Probability Theory
apply to
and
will
58
of
as
of
hypotheses
or
exist.
for
natural
variables
Fuzzy
theory
set
in that
such
the
all
fuzzy
sets
also
logic
will
be
hold
1982].
fuzzy
4.2,
respectively.
in
examined
represented
suited
theory,
set
method
not
linguistic
section
4.3
A discussion
uncertainty.
representing
can
that
[Kandel,
sets
well
do
prevalent.
are
abstract
of
theorems
and
for non-fuzzy
The
etc.
concepts
definitions
vague
where
the
significantly.
formal
a
vague
theory is particularly
definitions
true
or
of
probability-based
offers
representing uncertainty due to
distinct boundaries
that time,
grown
Unlike
sets.
where
analysis
introduced by Zadeh in 1978
later
was
fuzzy
of
strong interest in the
1982].
set
introduced by Lotfi
first
was
in possiblity theory
both
Possibility
be discussed
in the
as
of
a
how
also
will
and
final
section.
Definition
4.1
Since
it
is
Fuzzy
of
fuzzy
a
Sets
is
set
a
generalization
first necessary to understand the
In
a
crisp set,
assigned
a
value
of
either
within
or
is strictly
set.
contained
set.
In
describing
universal
set
are
A
non-members.
a
element
is defined by
of
example
temperature
set
value
1
seen
of
(X)
crisp set,
of
a
the universal
indicating
all
a
concept
outside
crisp set,
a
of
0
or
the
of
elements
discrimination
set
in
specified
of
the
members
a
is
it is
and
delineates
boundary
Formally,
crisp
whether
into two groups,
non-members.
crisp
the
set
each
function:
1
if
and
only
if XA
0
if
and
only
if XC A
all
integer
=
crisp
is the
set
in the
values
universal
a
a
1
sharp distinctive
from the
An
element
separated
members
uA
each
of
set
of
[50F,
range
X
=
(0,30,60,90,120),
or
0
indicting it
each
membership
in figure 4.1.
(T)
Elements
0
0
30
0
60
1
90
1
120
0
Figure
59
100F]
4
.
l
.
Given
element
in the
is
set
the
assigned
(T)
,
as
It
can
be determined with
certainty that
either
contained
the
set.
and
non-members.
A
In
is
set
a
In
fuzzy
a
assigned
the membership
fuzzy
a
a
a
each
delineates
of
members
is
boundary
sharp
element
specified
grade
of
each
This
is
formally
set.
outside
boundary.
set,
in
value
this
set,
is
element
is strictly
or
boundary
* fuzzy'
gradual
defining
specified
set
sharp distinctive
by
replaced
the
within
each
element
in
of
range
the
universal
representing
respect
by
represented
to the
the
following membership function:
Let
X
X
:
uA
in
As
a
to
0
a
1.
of
the
from
a
values
in the
position
within
represents
the
uncertainty
the
elements
elements
of
or
of
a
0
to
an
the
an
the
case
vagueness
60
specified
set
it is
of
allows
Clearly,
a
exists
set
is
0.
between
of
specified
universal
universal
A
fuzzy
the
fuzzy
being
a
concerning
differs
set
of
elements
either
crisp
set
assigned
assignment
an
set
outside
strictly
representing
set.
numbers
the
value
1
real
of
element
also
of
1988].
element
boundary
special
in the
the
of
vague
outside
or
if
in that it
range
in the
if
within
assigned
set
crisp
interval
the
[Klir,Folger,
Likewise,
it is
set
1
crisp set,
contained
of
[0,1]
->
is clearly
value
set,
[0,1] denotes
Where
from
Universal
=
when
wholly
set
no
the membership
set.
In
this
would
be
assigned
case,
of
all
values
of
0
or
1,
and
To
no
elements
help illustrate
the
consider
universal
membership
Elements
X
set,
a
assigned
HOT
and
=
fuzzy
(set
of
of
sets:
hot
-fuzzy'
fuzzy
a
temperatures)
in the
range
particular
(Temperatures,
fuzzy
to
0
of
1
of
cold
element
representing its
4.2).
HOT
COLD
F)
the
Given
.
each
(Figure
set
boundary.
set,
(set
COLD
(0,30,40,50,60,70,80,100),
value
in the
in the
exist
the concepts
two
following
temperatures)
is
would
10
0
10
30
40
0.8
0.1
50
0.5
0.3
60
0.3
0.4
70
0.1
0.7
80
0
1
100
0
1
4.2
FIGURE
In
the
fuzzy
boundary
set
vagueness
uncertainty
or
fuzzy
This
inherent
or
assigned
representing the
element
The
all
the
belongs
support
elements
It
to
of
a
indicated
is
that
degree
in the
of
concept
element
each
a
are
of
fuzzy
that have
set
by
is
non-zero
The
the
the
that
believes
equal
amount
size
subjective
fuzzy
in the
vagueness
hot.
note
person
particular
61
level
some
is due to the
important to
is
is
there
concerning their membership
uncertainty
vagueness
boundary-
grades
that
in the
are
(40,50,60,70)
elements
in the linguistic
uncertainty
fuzzy
the
indicating
area,
set.
HOT,
of
of
the
membership
in nature,
that
an
set.
to
values.
the
crisp
From
set
Figure
of
the
4.2,
support
the
of
SUPP(HOT)
A
fuzzy
element
the
membership
set
(40,50,60,70,80,100)
fuzzy sets,
height
grade
HOT
of
fuzzy
a
to any
assigned
and
COLD,
are
at
least
one
possible
is
set
of
its
normalized
to the
equal
elements.
with
height
a
l.
of
Fuzzy
can
be
modifiers
or
from
fuzzy
subsets.
as
illustration,
HOT,
a
Elements
very,
really,
are
more
modifiers
if the
the
subset,
support
of
*very'
is
1982].
the
applied
to
to
created.
subset,
VERYHOT,
HOT
(Temperatures, F)
resulting
less
in the
As
is
VERYHOT,-
fuzzy
therefore,
referred
[Kandel,
etc.,
new
subsets
and,
often
usually,
create
the
concise
are
modifier
fuzzy
to
variables
linguistic hedges
new
in Figure 4.3,
as
Generally speaking,
Fuzzy
literature
such
linguistic
modifiers
uncertain.
set,
to
applied
sets
an
the
fuzzy
As
shown
is half
VERYHOT
0
0
30
0
0
40
0.1
0
50
0.3
0
60
0.4
0
70
0.7
4
80
1
8
100
1
1
FIGURE
of
is
HOT,
is assigned the highest
The
grade.
highest membership
Both
=
set,
is considered to be normalized if
set
of
fuzzy
the
support
for the
subset,
62
0
4.3
HOT,
indicating
that the
new
is
fuzzy
set
found
in the
only two
less
4.2
a
more
fuzzy boundary
uncertain
or
It
a
value
goal
approximate
true
either
is
1988].
with
In
in the
value
important to note,
fuzzy
a
as
and
is
=
1
fuzzy
of
-
Intersection
uAnB(X)
=
a
fuzzy
two
no
X
subsets
min[uA(X),uB(X)]
63
false.
propositions
is
is
It
uncertainty
to two-valued
subset
fuzzy
and
to do
false.
or
uA(X)
of
clear
representing the
that
of
other
hypothesis
have been defined
subsets
In
.
method
each
true
either
reduced
a
fuzzy
or
[0,1]
case
(0)
completely
provide
is
hypothesis
fuzzy logic, the functions
B be
Complement
uA(X)
2)
of
is
complement
and
or
follows:
Let A
1)
for
be
can
true
fuzzy logic,
in the
that
logic
basis
intersection
theory
include
subset
to be very
imprecise
range
false
or
considered
degree that the hypothesis
As
fuzzy
new
logic, every
(1)
logic to
fuzzy
of
reasoning
[Klir,Folger,
a
of
being completely
either
is the
exists,
that the
two-valued
or
hypothesis
each
assigned
decreased to
also
vague.
In classical
precise,
has
elements
of
number
Logic
Fuzzy
words,
The
set.
indicating
elements
assigned
concise
of
logic.
union,
in fuzzy
set
3)
Union
two
of
UAUB<X)
The
=
fuzzy
max[uA(X),uB(X) ]
intersection
complement,
illustrated in Figure 4.4,
subsets,
TEMP
HOT
HOT
subsets
and
as
functions
union
they
pertain
are
the
to
fuzzy
VERYHOT.
and
VERYHOT
HOT U VERYHOT
HOT
VERYHOT
HOT D
0
0
0
0
0
30
0
0
0
0
1
40
0.1
0
0.1
0
0.9
50
0.3
0
0.3
0
0.7
60
0.4
0
0.4
0
0.6
70
0.7
0.4
0.7
0.4
0.3
80
1
0.8
1
0.8
0
FIGURE
In
fuzzy logic,
consist
(very,
of
fuzzy
the
fuzzy
of
value
the
propostion
and
Consider
for
depends
the
values
was
on
a
yesterday
the
of
cold
day is very
x
a
cold
day is fairly
of
these
claims
will
64
result
The
the
grade
of
fuzzy
truth
two
truth
being
claims:
true'
a
was
consider
day.
in the
following
Yesterday was
Yesterday
(very true,
example,
cold
can
modifiers
the membership
strength
example
proposition
fuzzy
,
For
1988].
recorded
on
also
cold)
truth
^Yesterday
temperature
COLD,
claimed.
Each
this
fuzzy
[Klir,Folger,
proposition
actual
subet,
also
and
fuzzy
or
(hot,
predicates
usually)
fairly false)
4.4
imprecise
an
1
true'
in different truth
values
based
on
claim
expressed.
the
Fuzzy
fuzzy
of
truth
subsets
real
can
fuzzy
be
as
numbers
of
absolutely true
in the range,
by these
and
is
false.
unit
Given
[0,1],
represented
0.
can
be
applied
subsets
as
shown
now
U
True
Fuzzy
to
modifiers
these
fuzzy
for
as
value
false
would
create
False
Very
1
true
of
or
very
of
0.
of
grade
to
sets
True
Fairly
new
fairly
fuzzy
False
0
0
0
1
1
0.2
0.2
0.3
0.8
0.8
0.4
0.4
0.5
0.6
0.5
0.6
0.6
0.7
0.4
0.2
0.8
0.8
1
0.2
0
1
1
1
0
0
4
,.5
Possibility Theory
Possibility Theory
1978
that
in Figure 4.5.
Figure
4.3
such
U,
linguistic
value
the membership
grade
set,
varying
example
the
by
own
assuming that truth
to the
for
truth
universal
numerically by the
represented
as
a
by their
values,
assigned
increases towards 1 the membership
decrease toward
and
Consider
interval,
appropriate
represented
numerical
be
can
absolutely false is
Within this
be
can
described below.
membership
true
variables,
and
values
represented
degrees
representing the
set
as
behind
a
development
was
of
Possibility Theory
65
initially
fuzzy
was
set
to
proposed
theory.
provide
a
The
good
by
Zadeh
main
in
purpose
formalism to
deal
information
theory
inherent fuzziness found in
the
with
is
is
that
centered
to
used
make
the
around
much
decisions.
concept
of
a
the
of
Possibility
possibility
distribution.
A possibility
as
X.
fuzzy
a
constraint
Zadeh defines
fuzzy
distribution,
restriction
F
be
a
fuzzy
X be
a
variable
Then
with
distribution
=
the
In
an
example,
day'.
In
that
values
formal
order
U be
and
taking
values
the Universal set.
in U, and let F act
"X
is
F"
translates
a
as
into
F.
this proposition is
is equal to R(X)
=
=
a
possibility
R(X)
uF
consider
this
can
terms, the
example
be
to
possibility
relate
the
above
proposition
the
assigned
R (Temperature
In
of
possiblity distribution function of X is defined
numerically equal to the membership function of F.
TTX
cold
in terms
which
TTX
As
subset
the proposition
Associated
to be
to
assigned
restriction.
R(X)
This
that may be
values
is defined
TTX,
follows:
as
Let
fuzzy
the
as
the possibility distribution
Let
a
a
on
denoted
fuzzy
to
this
distribution,
66
idea
of
a
consider
COLD,
yesterday's
propostion
(Yesterday) )
set,
'Yesterday
=
restricts
temperature.
written
as
restriction
to
Cold
fuzzy
the
be
can
was
temperature,
40F,
a
degree
whose
The
degree
of
of
combining this
day'
yesterday
we
concept
state
The
in the
values
for the
0.5/50
fuzzy
+
set,
0.3/60
term
is
read
cold
is
equal
as
to
of
The
is
TTX
=
1/0
0/80
+
0/90
0.1/70
+
+
Although
possibility distribution may
probability distribution,
4
section
.
4
a
a
possibility that
was
4 OF with
value
that temperature
is numerically
,
equal
assigned
lies
1/30
+
a
+
0.8/40
+
0/100
is
40
given
where
given
set,
to be very
a
number
each
that X
to
will
Possibility Theory,
total
ignorance
possibility distribution consisting
=
Generally speaking,
(1,1,1
the
67
less
of
is
all
represented
l's.
1}
specific
the
a
fundamental
investigated in
be
which
is
a
similar
of
+
4.5.
TTx(u)
a
the
Ignorance
In
by
for
appear
there
differences between the two
the
was
possibility distribution
the possibility that X
0.8.
of
by
Yesterday
therefore may be
and
[0,1].
COLD,
degree
is true.
function
range
of
of
distribution, TTx(u)
the membership
x
a
Now
cold.
the
of
becomes the possibility
cold
of
as
0.8.
fact that yesterday
0.8,
compatibility,
The possibility
to
the
given
that the proposition
given
concept
that the degree
is 0.8
40F
the
with
is
COLD,
interpreted
the proposition that
with
can
was
day.
cold
4 OF
of
is
0.8,
membership,
compatibility
cold
in the fuzzy set,
membership
evidence
or
information,
the
representing
the high degree
ignorance is
represented
by
the
x largest'
consisting
of
all
l's
distribution,
the
other
and
consequently
larger the possibility distribution
the
hand,
of
situation
no
uncertainty.
where
uncertainty,
Hence, total
possibility
[Klir,Folger,
1988].
have perfect
we
evidence
be represented
would
On
by
the
following possibility distribution.
TTx(u)
It
relevant
evidence,
Unfortunately,
address
all
propositions
Comparison
Both
to
of
to
problem
is
quite
represent
and
solving
and
best
possible.
are
still
as
we
of
forced to
were
in
Probability
found
decision
given
with
or
methods
situations
However,
making.
Probabilities
the
in
our
evidence
This
being
and
are
well
belief
the
faced
with
On
a
information
many
the
suited
of
in the
uncertainty lies
solutions.
represent
68
in
offer
is best handled by these two
available.
hypotheses
possibilities
are
Possibility Theory
different.
associated
alternative
and
the uncertainty inherent
is currently
ambiguity
we
singletons
Possibility
hypothesis,
particular
that
of
uncertainty that
theories
as
uncertainty that is
represent
type
propositions
absence
-
Probability
involving
all
in the
that
state
in possibility theory
probability theory
4.5
(1,0,0,0...0)
naturally intuitive to
seems
any
=
well
other
the uncertainty that
defined
hand,
results
from the
use
of
vague
indistinct
or
concepts
linguistic
or
terms.
As
must
clear
type
and
which
most
of
available
a
the alternative
in diagnostic
prevalent
This
choose.
and
environments,
is best handled by
For
suppose
instance,
to determine
this
as
example,
the
have
a
set
of
person's
person
two
problem
be
all
people
viewed
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age
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sets
is
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old.
our
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is limited to the
person
drives
is currently attending highschool.
information
with
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well
years
observations;
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is true.
solutions
probabilistic
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person
hypotheses,
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age.
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concerning this
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hypothesis
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support
of
solutions
69
is
guess'
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and
are
decision.
very
clear
we
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and
is
inherent in
is
uncertainty
information that
our
consequently
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the
other
certain.
hand,
If we
were
given
perfect
the person's
definitions do
on
the
actual
measure
is
given
concept
of
subjective
The
young.
Zadeh
for
eggs
1977].
breakfast"
can
also
eat
be
u
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of
possibility
and
look
as
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taking
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a
young.
and
possibilites
example
given
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1111
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of
person
vague
values
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eggs.
uncertainty
to the
statement
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associated
probability
may
probabilities
distribution, P(u)
by interpreting P(u)
can
relates
concept
Consider the
with
of
the problem
example,
illustrated by the following
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Hans
age
the
of
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than the
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interpretation
be best
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person's
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concentrates
rather
determining
of
certainty
distinct
language applications.
age.
to determine how this
information
1987].
the task
their
where
with
the uncertainty
Possibility theory
the
of
natural
consider
is young,
exist.
[Klir,Folger,
in
prevalent
example,
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predict
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meaning
it
of
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understanding
possibility
can
be
probability
and
the possibility
measure
based
on
represents
certain
represents
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probabilities
whereas
value
of
measure
of
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concepts
and
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occur,
measure
is
feasible.
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by
sum
the
of
in the distribution space,
are
represented
fact,
special
of
Specifically,
may be
71
so
an
by
the
maximum
A possibility
of
measure
it has been
subset
of
the
proposed
that
plausibility
1988].
introduction
approach
does
words,
hypothesis
distribution.
In
Theory.
event
the
measure.
represented
debate concerning the validity
probabilistic
other
to the plausibility
[Klir,Folger,
applicability.
In
hypothesis
a
measures
measures
With the
an
A possibility
events
the possibility
Dempster-Shafer
measures
the
possibility
that
are
measures
of
between probability
that
chance
evidence.
the degree
of
by comparing
gained
the
is based
improbable.
also
nor
-possibility/probability
event.
the differences
of
low degree
a
This principle
the
of
probability,
probabilities
the possibility
as
the probability
event
the
called
consistency
does
imply
of
Zadeh did propose that there
heuristic
possibilities
high degree
a
high degree
a
probability
However,
weak
that
observe
can
imply
not
low degree
a
we
example,
possibility
theory
began
the
theory
and
of
new
Cheeseman
used
claims
that
to describe the
a
its
a
uncertainty due to
eliminating the
Cheeseman
-fuzzy'
that
argues
the probability that
instead
with
our
In
event
space
support
of
his
the underlying
consider
fuzzy
does
in the later
in
it
the
case
as
following
=
"event X
represented
One
possible
assumptions
severe
proposition
values
for the
of
X,
1986].
identifies
further
rules
max/min
For
shortcomings.
intersection
for the
rule
problem
[Bhatnagar,Kanal,
Cheeseman
as
occurs"
X
event
occurs".
all
it does
example
of
two
and
instead be
suffers
also
B
are
defends
from
as
the
minimum
certain
the underlying
with
assumptions.
contends
uncertain
In
that the
equal
particular
the
defense
vagueness
of
Although
verification
72
fact,
assumptions
to
make
forced to
Possibility Theory,
about
fuzzy
found
is
one
surrounding
to the uncertainty
context.
empirical
information
order
a
theory
value
lacks
make
Zadeh
particular
its truth
set
and
Probability
In
assumptions.
independence
rule
the two
of
However,
zero.
as
stated
this
mutually exclusive,
in probability by stating that in
decisions
not
min[uA(X),uB(X)]
their possibility
should
found
that A
case
Cheeseman
the
that
possible
include
not
dependency
fuzzy logic
state
Theory
is
is
"it
be
can
1986].
sets,
the
will
theory [Cheeseman,
possibility
arguments,
uAnB(X)
In
new
interpretation is that in the first
this
found
a
a
the proposition
of
whereas
for
need
therefore
and
concepts,
concept
in
a
much
in probability theory,
it
of
offers
that
is
a
good
method
inherent in
to reason
natural
73
with
a
language.
type
of
uncertainty
Chapter
5
Non-Monotonic Logic
The preceding
to
approaches
based
set
by
his
on
of
unique
mathematical
as
a
be
should
approaches
for modeling
methods
Non-monotonic
as
formalisms based
One
specific
many have
such
further
offers
numerical
approaches.
systems
logics
logic
offer
and
[Shoham,
unique
consideration.
include
logic
default
Some
McDermott
1987].
the
non-monotonic
been
people
into hard
more
realistic
each
purposes
logic
74
developed
of
approach,
represents
by
represented
the
the
better known
circumscription,
and
Each
in the following
discussions
of
McCarthy's
approaches
For the
the
has
non-numeric
it best
uncertainty that
of
previous
monotonic
This
non-numeric
good
a
from the uncertainty
non-monotonic
as
shared
express
must
that
own
uncertainty.
logic
the type
were
their
fact that
investigated
different
Reiter's
on
numbers.
argued
numeric
requirement
expert
quite
is
various
approaches
shortcoming due to the
Consequently,
numbers.
these
is that the
of
with
it difficult to translate their beliefs
find
often
severe
of
assumptions.
uncertainty in terms
viewed
and
and
these methods,
of
Each
uncertainty-
requirements
each
have dealt
sections
of
Doyle's
these
worthy
this
sections
non
logical
of
thesis, however,
will
my McDermott
be
and
limited to
Doyle.
5.1
Basic Characteristics
The
revise
information is
current
that
ability to
are
or
changing
essential
uncertain.
the underlying uncertainty
contradictory information
Classical
traditional
or
does
therefore,
not
conclusions
as
traditional
logic
problem
new
solving
beliefs based
our
of
the
environments
In
truly
order
be
must
to
revision
of
in
and
nature
and,
previous
obtained.
be effectively
Consequently,
to dynamic
applied
having incomplete
situations
reflect
considered.
monotonic
information is
cannot
most
in modeling
logic is
allow
the
hypothesis, incomplete
a
also
on
or
uncertain
information.
Non-monotonic
inconsistencies that
monotonic
and
logic
Consequently,
value.
either
contradictory
front to
resolve
It
assigned.
there
each
is presently
method.
In
previous
chapters,
In
the
case
of
variable
in the
or
no
be
noted
way to
a
that
could
logic,
Non
calculus
to
a
single
is
evidence
assumptions
and
previous
on
predicate
that the
single
must
value
due to this
approaches
non-montonic
based
restricted
represent
hypotheses
75
order
is
allows
information.
new
case
missing,
numerical
revised
first
any conflicts,
should
the
on
hand,
other
or
from
arise
is based
that
requires
the
on
invalidated
to be
conclusions
logic,
made
must
up
be
restriction
ignorance
discussed
be
with
this
in the
be partially believed.
each
hypothesis
is
either
believed
or
that
applied.
are
disbelieved based
suppositions
The
made
not
process.
in
decision strongly depends
underlying
5.2
Modal
The
Infer
modal
this
In
1985]
be
the
as
confidence
nature
inability
as
"p is
consistent
drawn based
As
conclusions
operators
the
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to
infer
representing
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with
may be
~
BIRD
2)
PENGUIN
allow
a
3
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4
)
theory
M( CAN-FLY)
we
can
BIRD
prove
CAN-FLY
76
and
idea
of
extended
As
an
->
A
NOT
(NOT p)
"consistency"
contingent
fact that there
withdrawn.
->
is
theory"
is
no
information is received,
new
1)
In this
these
the
represent
the
CAN-FLY
(CAN-FLY)
and
[Maselli,
conclusion
evidence
example
following.
)
of
have
we
follows:
operator
modal
to
classical
modal
The
contrary-
logic,
the
read
.
of
introduced by McDermott
operators
from the
(Mp)
Mp is
be
logic
non-monotonic
by adding modality, M,
can
level
on
in the
Logic
"consistency".
Here
masked
assumptions.
incorporates
Doyle
the
Therefore,
assumptions
through the decision
carried
making
our
is
uncertainty
is
and
initial
the
on
to
the
previous
consider
to
the
the premises
information is
5)
and
be
must
previous
Doyle's
on
the
The
obtained.
tree
each
structure
each
to
the
revises
of
user
monotonic
the
necessary
to
the
logic
with
for truth
each
changes
the
its
child
In this
The
in the
decision
reflect
contain
may
system
receives
or
TMS
also
systems
77
new
theorem,
in the tree in
order
inform
will
made.
the
chains,
amount
In
may degrade
the
a
non
by TMS,
reached
assumptions.
inferencing
the
and
justifications
the hypothesis
were
the
structure.
node
a
information
new
maintains
various
consistency.
that
as
hypothesis
a
assumptions
explicitly
maintenance
tree
provides
to effectively demonstrate
able
longer
a
If the
with
changes
system.
not
models
represents
turn,
underlying uncertainty
systems,
in
(TMS)
that
program
represent
system
all
does
however,
node
In
has been
TMS
a
inconsistent
the
restore
operator
System
justifications.
and
input that is
TMS
inconsistent
now
situation.
revising
reasoner
node
assumptions.
hypotheses
a
and
is
TMS
of
of
the
of
Truth Maintenance
knowledge base
and
new
Truth Maintenance System
framework for updating
children
The modal
context
is
(4)
conclusion
re-evalutated.
Doyle's
is
following
received
our
meaning based
5.3
if the
Now
3.
and
PENGUIN
find that
we
1,2
of
of
large
system
expert
response
significantly.
Chapter
Summary
Conclusion
and
Probability theory
has played
modeling uncertainty in
This
years.
many
Dempster-Shafer
theories,
of
but
theories
problem
be
can
Probability Theory
subsequent
in
seen
also
such
Theory
as
of
probabilities
belief,
6
as
not
in
role
for
environments
only in the Bayesian
the basis
for
Mycin certainty
defined
to the
contrast
solving
Evidence.
are
important
an
In
as
number
factors
three
all
a
a
traditional
frequency
the
and
these
of
subjective
of
measure
ratio
approach.
that
In
Bayesian probability,
an
event
[0,1],
range
large
volume
computational
Its
systems.
lack
model
data
limited
the
that
when
in the
first
incorporates
weight
limited
the
due
a
in
representation
uncertainty.
to
of
to
its
partial
of
events.
breakthrough
certainty
due
expert
part
'fuzzy'
In
general,
resulting
is
and
In
success
in large
also
in the
value
event.
implementations
called
78
a
the
and
tremendous
workable
by
'chance'
or
applied
ignorance
a
represented
probabilistic
required
success
for
to
experienced
complexity
especially
Mycin
of
of
represented
directly
attached
expressiveness
of
beliefs,
one
is
occur,
probability has
Bayesian
the
will
the probability
a
Mycin,
factor
by offering
reasoning
a
is
used
to
represent
degree
one's
belief.
of
certainty factors lies in
Consequently,
and
probability
model
consequently it
such
has
as
chains
have
mutual
exclusivity
The
on
theory is
bear
subsets
that the
on
of
a
explicit
effective
restrictions.
theory,
the
Much
same
of
It's
part
this
of
and
assumptions,
Mycin's
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in
model,
inferencing
short
independence
the
of
the
is
One
based
to
and
accumulated
Theory
of
ignorance.
method
79
and
and
is based
In
of
other
this
expert
all
on
a
also
Drawbacks
does
possible
truer
allows
of
include the lack
difficulties
it
evidence
the
provides
beliefs
evidence
by
instead
This
set.
evidence.
the
same
however,
advantage
notable
expert's
of
on
the
approach
whether
but
uses
theory,
This
way.
hypothesis
the
Evidence
Bayesian
attached
evidence
inferencing
of
support
representation
Dempster-Shafer
and
simplified
Theory
the hypothesis
of
more
as
of
single
representation
developing
substantial
of
effects
the hypothesis.
supports
not
'chance'
the
some
different
much
degrees
numerical
words,
a
evidence
assumptions.
probability
in
it
applies
the
minimized
of
a
probability-
Dempster-Shafer
calculus
that
from
its
to
in
goals
assumptions
independence.
to Bayesian
comparison
the
of
from probability
suffers
attributed
certainty factor
a
to avoid strict Bayesian
however,
statistical
been
One
was
be derived
can
theory.
belief that the available
its inherent
has been shown,
to
assigned
of
model
reasoning
theoretical basis for
confirmation
the hypothesis.
supports
It
value
the degree
represents
this
the
The
an
the
of
an
in defining
frames
discernment
of
in
Probability-based theories
partial
in
beliefs
domains.
complex
evidence
are
hypotheses
and
for expressing
effective
uncertainty lies
in the
approaches
be best applied to domains that
would
defined
and
realize
that
reasonable
and
assumptions
set
offer
formal
vague
concepts
natural
the
methods
of
or
fuzzy
and
feasibility
of
the
actual
of
the
assumptions
such
empirical
from
max/min
testing
Non-monotonic
a
its
rules.
since
found
logic
80
information.
used
turn,
to
relate
whereas
a
hypothesis
set
own
of
hypotheses
offers
probability
the
measure
occur.
will
and
restrictions
dependency
it presently lacks
with
reasoning
represents
Many have been
this
In
fuzzy logic
measure
the underlying
as
Possibility Theory
that
for
suited
imprecise
or
approximate
A possibility
chance
of
In
concepts.
hypothesis,
well
very
unclear
are
methodology to do
suffers
in the
restrictions
Possibility theory
meaning
membership
Possibility
found
are
applications.
a
give
representing uncertainty due to
consequently
propositons.
represents
of
'fuzzy'
functional
to
expected
the underlying
of
well
is important to
It
be
are
Possibility theories, developed by Zadeh,
and
degrees
vague
all
can
the uncertainty due to
definitions
to
model
These
followed.
are
language
represents
theory,
unless
the hypotheses.
of
available.
mathematical
results
Fuzzy
is
the data
where
no
context
the
where
restrictions
skeptical
much
of
of
the
in probability theory.
offers
a
non-numeric
approach
to
handling
translations
Doyle's
as
beliefs
of
"consistency"
in the
conclusions
logic
Non-monotonic
in
reasoning
absence
is
are
with
to
used
to the
and
revised
modal
represent
to
us
allowing
evidence
jump
the
to
contrary.
in modeling default
effective
environments
to be
conclusions
thereby
of
McDermott
by incorporating
operators
in the model,
the difficult
avoids
numbers.
allows
received
These modal
operators.
into hard
logic
non-monotonic
information is
new
therefore,
uncertainty and,
incomplete
or
missing
information.
In
conclusion,
provide
a
panacea
Probability-based
our
hypothesis.
most
The
expressive
ignorance.
In
order
systems,
we
uncertainty
domains
types
need
set
is
to
human
into
one
must
that
make
the
of
is
these
allow
when
the
our
information.
modes
in
expert
of
In the majority
of
all
evidential
In
order
misrepresented,
experts
based
reasoning.
by incorporating
partial
not
a
to be
completely
generalizations.
information
where
generalizations
system.
decisions
and
to be developed that
used
all
combine
including
81
be
reasoning
reasoning
vagueness
our
to
of
appears
for default
method
can
in describing
in
applicable
situations
context
Theory
vagueness
that
theory
uncertainty.
in
best
are
should
a
need
of
especially
allowing
model
global
evidential
theory
uncertainties
information,
ensure
in the
approach,
would
people
of
approaches
logic
typicality
one
situations
is due to the
Non-monotonic
on
all
not
Dempster-Shafer
Fuzzy
uncertainty
for
lies
uncertainty
is
there
to
represent
to
systems
all
their different types
appropriate
of
The
method.
combining the
inferences to be
rules
allowing
good
these
types
systems
The best
developed.
theory for the
best
limitations
It
alternative
is
with
that
the
Many
the
difficulty in acquiring
the
full
Consequently,
usually
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have
global
are
All
in
key
82
to
not
currently
is to
the
choose
aware
of
the
the
made.
all
not
the
the
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of
the
of
that handle
systems
information
limitations
information
world
and
into
Unfortunately,
present
the
all
most
methods
made.
base that
is extemely diverse
the
in successfully
that
of
all
expert's
knowledge
incomplete.
reasoning
will
methods.
the
lies
representation
inferencing
of
are
reasoning
and
state
at
remember
difficulty in developing
uncertainty
the
domain, being
particular
trade-offs
and
ideal
though
important to
is
difficulty
with
from these different
numbers
of
uncertainty
or
we
due to
describing
domain.
begin
realm
complex,
answer.
are
of
with
is
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