-
7/31/2019 Chapter14-15
1/20
1
Reasoning Under UncertaintyPart (1) Certainty Factors
Kostas Kontogiannis
E&CE 457
Objectives
This unit aims to investigate techniques that allow for
analgorithmic process to deduce new facts from a knowledgebase with
a level of confidence or a measure of belief.
These techniques are of particular importance when:1. The rules
in the knowledge base do not produce a conclusion that is
certain even though the rule premises are known to be certain
and/or
2. The premises of the rules are not known to be certain
The three parts in this unit deal with:1. Techniques related to
certainty factors and their application in rule-
based systems2. Techniques related to the Measures of Belief,
their relationship to
probabilistic reasoning, and finally their application in
rule-basedsystems
3. The Demster Shafer model for reasoning under uncertainty in
rule-based systems
-
7/31/2019 Chapter14-15
2/20
2
Uncertainty and Evidential Support
In its simplest case, a Knowledge Base contains rules of
the form :
A & B & C => D
where facts A, B, C are considered to be True (that is these
facts hold with probability 1), and D is asserted in the
Knowledge Base as being True (also with probability 1)
However for realistic cases, domain knowledge has to be
modeled in way that accommodates uncertainty. In other
words we would like to encode domain knowledge using
rules of the form:A & B & C => D (CF:x1)
where A, B, C are not necessarily certain (i.e. CF = 1)
Issues in Rule-Based Reasoning
Under Uncertainty Many rules support the same conclusion with
various
degrees of Certainty
A1 & A2 & A3 => H (CF=0.5)
B1 & B2 & B3 => H (CF=0.6)
(If we assume all A1, A2, A3, B1, B3, B3 hold then H is
supported with CF(H) = CFcombine(0.5, 0.6))
The premises of a rule to be applied do not hold with
absolute certainty (CF, or probability associated with apremise
not equal to 1)
Rule: A1 => H (CF=0.5)
However if during a consultation, A1 holds with CF(A1) =
0.3 the H holds with CF(H) = 0.5*0.3 = 0.15
-
7/31/2019 Chapter14-15
3/20
3
The Certainty Factor Model
The potential for a single piece of negative evidence
should not overwhelm several pieces of positive evidence
and vice versa
the computational expense of storing MBs and MDs
should be avoided and instead maintain a cumulative CF
value
Simple model:
CF = MB - MD
Cfcombine(X, Y) = X + Y*(1-X)
The problem is that a single negative evidence overwhelmsseveral
pieces of positive evidence
The Revised CF ModelMB - MD
1 - min(MB, MD)CF =
{X + Y(1 - X) X, Y > 0
X + Y1 - min(|X|, |Y|)
One of X, Y < 0
- CFcombine(-X, -Y) X, Y < 0
CFcombine(X,Y) =
-
7/31/2019 Chapter14-15
4/20
4
Additional Use of CFs
Provide methods for search termination
A B C D E
In the case of branching in the inference sequencing paths
should be kept distinct
0.8 0.4 0.7 0.7
R1 R2 R3 R4
Cutoff in Complex Inferences
A B C
D E
0.8 0.4F0.9
R30.7
0.7
R4
R5
R1 R2
We should maintain to paths for cutoff (0.2), one being
(E, D, C, B, A) and the other (F, C, B, A). If we had one
path
then E, D, C would drop to 0.19 and make C unusable later in
path F, C, B, A.
-
7/31/2019 Chapter14-15
5/20
5
Reasoning Under UncertaintyPart (2) Measures of Belief
Kostas Kontogiannis
E&CE 457
Terminology
The units of belief follow the same as in probability theory
If the sum of all evidence is represented by e and dis the
diagnosis (hypothesis) under consideration, then the
probability
P(d|e)
is interpreted as the probabilistic measure of belief or
strength that the hypothesis dholds given the evidence e.
In this context: P(d) : a-priori probability (the probability
hypothesis d occurs
P(e|d) : the probability that the evidence represented by e
are
present given that the hypothesis (i.e. disease) holds
-
7/31/2019 Chapter14-15
6/20
6
Analyzing and Using Sequential
Evidence
Let e1 be a set of observations to date, and s1 be some new
piece of data. Furthermore, let e be the new set of
observations once s1 has been added to e1. Then
P(di | e) = P(s1 | di & e1) P(di | e1)
Sum (P(s1 | dj & e1) P(dj | e1))
P(d|e) = x is interpreted:
IF you observe symptom e
THEN conclude hypothesis d with probability x
Requirements
It is practically impossible to obtain measurements
for P(sk|dj) for each or the pieces of data sk, in e,
and for the inter-relationships of the sk within
each possible hypothesis dj
Instead, we would like to obtain a measurement of
P(di | e) in terms of P(di | sk), where e is the
composite of all the observed sk
-
7/31/2019 Chapter14-15
7/20
7
Advantages of Using Rules in
Uncertainty Reasoning
The use of general knowledge and abstractions in the
problem domain
The use of judgmental knowledge
Ease of modification and fine-tuning
Facilitated search for potential inconsistencies and
contradictions in the knowledge base
Straightforward mechanisms for explaining decisions
An augmented instructional capability
Measuring Uncertainty Probability theory
Confirmation
Classificatory: The evidence e confirms the hypothesis h
Comparative: e1 confirms h more strongly than e2 confirms h
or
e confirms h1 more strongly than e confirms h2
Quantitative: e confirms h with strength x usually denoted
as
C[h,e]. In this context C[h,e] is not equal to 1-C[~h,e]
Fuzzy sets
-
7/31/2019 Chapter14-15
8/20
8
Model of Evidential Strength
Quantification scheme for modeling inexact reasoning
The concepts ofbeliefand disbeliefas units of measurement
The terminology is based on:
MB[h,e] = x the measure of increased belief in the hypothesis
h,
based on the evidence e, is x
MD[h,e] = y the measure of increased disbelief in the hypothesis
h,
based on the evidence e, is y
The evidence e need not be an observed event, but may be a
hypothesis
subject to confirmation
For example, MB[h,e] = 0.7 reflects the extend to which the
experts
belief that h is true is increased by the knowledge that e is
true
In this sense MB[h,e] = 0 means that the expert has no reason to
increase
his/her belief in h on the basis of e
Probability and Evidential Model
In accordance with subjective probability theory, P(h)reflects
experts belief in h at any given time. Thus 1 - P(h)reflects
experts disbelief regarding the truth of h
If P(h|e) > P(h), then it means that the observation of
eincreases the experts belief in h, while decreasing
his/herdisbelief regarding the truth of h
In fact, the proportionate decrease in disbelief is given bythe
following ratio
P(h|e) - P(h)
1 - P(h)
The ratio is called the measure of increased belief in
hresulting from the observation of e (i.e. MB[h,e])
-
7/31/2019 Chapter14-15
9/20
-
7/31/2019 Chapter14-15
10/20
10
Characteristics of Belief Measures
Range of degrees:
0
-
7/31/2019 Chapter14-15
11/20
11
More Characteristics of Belief Measures
CF[h,e] + CF[~h,e] =/= 1
MB[h,e] = MD[~h,e]
The Belief Measure Model as anApproximation
Suppose e = s1 & s2 and that evidence e confirms d. Then
CF[d, e] = MB[d,e] - 0 = P(d|e) - P(d) =
1 - P(d)
= P(d| s1&s2) - P(d)
1 - P(d)
which means we still need to keep probability measurementsand
moreover, we need to keep MBs and MDs
-
7/31/2019 Chapter14-15
12/20
12
Defining Criteria for Approximation
MB[h, e+] increases toward 1 as confirming evidence is
found, equaling 1 if and only f a piece of evidence
logically
implies h with certainty
MD[h, e-] increases toward 1 as disconfirming evidence is
found, equaling 1 if and only if a piece of evidence
logically
implies ~h with certainty
CF[h, e-]
-
7/31/2019 Chapter14-15
13/20
13
Combining Functions
{MB[h, s1&s2] =0 If MD[h, s1&s2] = 1
MB[h, s1] + MB[h, s2](1 - MB[h, s1]) otherwise
MD[h, s1&s2] = {0 If MB[h, s1&s2] = 1
MD[h, s1] + MD[h, s2](1 - MD[h, s1]) otherwise
MB[h1 or h2, e] = max(MB[h1, e], MB[h2, e])MD[h1 or h2, e] =
min(MD[h1, e], MD[h2, e])
MB[h, s1] = MB[h, s1] * max(0, CF[s1, e])MD[h,s1] = MD[h, s1] *
max(0, CF[s1, e])
Probabilistic Reasoning and Certainty Factors(Revisited)
Of methods for utilizing evidence to select diagnoses
ordecisions, probability theory has the firmest appeal
The usefulness of Bayes theorem is limited by
practicaldifficulties, related to the volume of data required
tocompute the a-priori probabilities used in the theorem.
On the other hand CFs and MBs, MDs offer an intuitive,yet
informal, way of dealing with reasoning underuncertainty.
The MYCIN model tries to combine these two areas(probabilistic,
CFs) by providing a semi-formal bridge(theory) between the two
areas
-
7/31/2019 Chapter14-15
14/20
14
A Simple Probability Model
(The MYCIN Model Prelude)
Consider a finite population ofn members. Members of the
population may possess one or more of several properties
that define subpopulations, or sets.
Properties of interest might be e1 or e2, which may be
evidence for or against a diagnosis h.
The number of individuals with a certain property say e,
will be denoted as n(e), and the number of two properties
e1 and e2 will be denoted as n(e1&e2).
Probabilities can be computed as ratios
A Simple Probability Model (Cont.) From the above we observe
that:
n(e1 & h) * n = n(e & h) * n
n(e) * n(h) = n(h) * n(e)
So a convenient form of Bayes theorem is:
P(h|e) = P(e|h)
P(h) P(e)
If we consider that two pieces of evidence e1 and e2 bear on a
hypothesis h,and that if we assume e1 and e2 are independent then
the following ratios hold
n(e1 & e2) = n(e1) * n(e2)
n n n
andn(e1 & e2 & h) = n(e1 & h) * n(e2 & h)
n(h) n(h) n(h)
-
7/31/2019 Chapter14-15
15/20
15
Simple Probability Model
With the above the right-hand side of the Bayes Theorem
becomes:
P(e1 & e2 | h) = P(e1 | h) * P(e2 | h)
P(e1 & e2) P(e1) P(e2)
The idea is to ask the experts to estimate the ratios
P(ei|h)/P(h)
and P(h), and from these compute P(h | e1 & e2 & &
en)
The ratios P(ei|h)/P(h) should be in the range [0,1/P(h)]
In this context MB[h,e] = 1 when all individuals with e have
disease h, and MD[h,e] = 1 when no individual with e has h
Adding New Evidence Serially adjusting the probability of a
hypothesis with new
evidence against the hypothesis:
P(h | e) = P(ei | h) * P(h | e)
P(ei)
or new evidence favoring the hypothesis:
P(h | e) = 1 - P(ei | ~h) * [ 1 - P(h | e)]P(ei)
-
7/31/2019 Chapter14-15
16/20
16
Measure of Beliefs and Probabilities
We can define then the MB and MD as:
MB[h,e] = 1 - P(e | ~h]
P(e)
and
MD[h,e] = 1 - P(e | h)
P(e)
The MYCIN Model
MB[h1 & h2, e] = min(MB[h1,e], MB[h2,e])
MD[h1 & h2, e] = max(MD[h1,e], MD[h2,e])
MB[h1 or h2, e) = max(MB[h1,e], MB[h2,e])
MD[h1 or h2, e) = min(MD[h1,e], MDh2,e])
1 - MD[h, e1 & e2] = (1 - MD[h,e1])*(1-MD[h,e2])
1- MB[h, e1 & e2] = (1 - MB[h,e1])*(1-MB[h, e2])
CF(h, ef& ea) = MB[h, ef] - MD[h,ea]
-
7/31/2019 Chapter14-15
17/20
17
Reasoning Under UncertaintyPart (3) Demster-Shafer Model
Kostas Kontogiannis
E&CE 457
The Demster-Shafer Model
So far we have described techniques, all of which consideran
individual hypothesis (proposition) and and assign toeach of them a
point estimate in terms of a CF
An alternative technique is to consider sets of propositionsand
assign to them an interval of the form
[Belief, Plausibility] that is
[Bel(p) , 1-Bel(~p)]
-
7/31/2019 Chapter14-15
18/20
18
Belief and Plausibility
Belief (denoted asBel) measures the strength of the
evidence in favor of a set of hypotheses. It ranges from 0
(indicating no support) to 1 (indicating certainty).
Plausibility (denoted as Pl) is defined as
Pl(s) = 1 - Bel(~s)
Plausibility also ranges from 0 to 1, and measures the
extent to which evidence in favor of ~s leaves room for
belief in s. In particular, if we have certain evidence infavor
of ~s, then the Bel(~s) = 1, and the Pl(s) = 0. This
tells us that the only possible value for Bel(s) = 0
Objectives for Belief and Plausibility
To define more formally Belief and Plausibility we need to
start with an exhaustive universe of mutually exclusive
hypotheses in our diagnostic domain. We call this set
frame of discernment and we denote it as Theta
Our goal is to attach a some measure of belief to elements
ofTheta. In addition, since the elements ofTheta are
mutually exclusive, evidence in favor of some may have an
effect on our belief in the others.
The key function we use to measure the belief of elementsofTheta
is a probability density function, which we denote
as m
-
7/31/2019 Chapter14-15
19/20
19
The Probability Density Function in
Demster-Shafer Model The probability density function m used in
the Demster-
Shafer model, is defined not just for the elements ofThetabut
for all subsets of it.
The quantity m(p) measures the amount of belief that iscurrently
assigned to exactly the setp of hypotheses
IfTheta contains n elements there are 2n subsets ofTheta
We must assign m so that the sum of all the m valuesassigned to
subsets ofTheta is equal to 1
Although dealing with 2n hypotheses may appear
intractable, it usually turns out that many of the subsetswill
never need to be considered because they have nosignificance in a
particular consultation and so their mvalue is 0
Defining Belief in Terms of
Function m Having defined m we can now defineBel(p) for a set p,
as
the sum of the values of m forp and for all its subsets.
ThusBel(p) is our overall belief, that the correct answer
lies somewhere in the setp
In order to be able to use m, and thus Bel and Pl in
reasoning programs, we need to define functions that
enable us to combine ms that arise from multiple sources
of evidence
The combination of belief functions m1 and m2 issupported by the
Dempster-Shafer model and results to a
new belief function m3
-
7/31/2019 Chapter14-15
20/20
20
Combining Belief Functions
To combine the belief functions m1 and m2 on sets X and Y we
use
the following formula
SumY intersect Y = Z m1(X) * m2(Y)
m3(Z) =
1 - SumX intersect Y = empty m1(X) * m2(Y)
If all the intersections X, Y are not empty then m3 is computed
by
using only the upper part of the fraction above (I.e. normalize
by
dividing by 1)
If there are intersections of X, Y that are empty the upper part
of the
fraction is normalized by 1-k (where k is the sum of the m1*m2
on the
X,Y elements that give empty intersection
