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Reasoning Under Uncertainty
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 d is the
diagnosis (hypothesis) under consideration, then the
probability
P(d|e)
is interpreted as the probabilistic measure of belief or
strength that the hypothesis d holds 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
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
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
Model of Evidential Strength
• Quantification scheme for modeling inexact reasoning
• The concepts of belief and disbelief as 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 expert’s
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 expert’s belief in h at any given time. Thus 1 - P(h)
reflects expert’s disbelief regarding the truth of h
• If P(h|e) > P(h), then it means that the observation of e
increases the expert’s belief in h, while decreasing his/her
disbelief regarding the truth of h
• In fact, the proportionate decrease in disbelief is given by
the following ratio
P(h|e) - P(h)
1 - P(h)
• The ratio is called the measure of increased belief in h
resulting from the observation of e (i.e. MB[h,e])
Probability and Evidential Model
• On the other hand, if P(h|e) < P(h), then the observation of
e would decrease the expert’s belief in h, while increasing
his/her disbelief regarding the truth of h. The proportionate
decrease in this case is
P(h) - P(h|e)
P(h)
• Note that since one piece of evidence can not both favor
and disfavor a single hypothesis, that is when MB[h,e] >0
MD[h,e] = 0, and when MD[h,e] >0 then MB[h,e] = 0.
Furthermore, when P(h|e) = P(h), the evidence is
independent of the hypothesis and MB[h,e] = MD[h,e] = 0
Definitions of Evidential Model
MB[h,e] =
{
1
if P(h) = 1
max[P(h|e), P(h)] - P(h)
otherwise
1 - P(h)
MD[h,e] =
{
1
if P(h) = 0
min[P(h|e), P(h)] - P(h)
otherwise
- P(h)
CF[h,e] = MB[h,e] - MD[h,e]
Characteristics of Belief Measures
• Range of degrees:
0 <= MB[h,e] <= 1
0 <= MD[h,e] <= 1
-1 <= CF[h,e] <= +1
• Evidential strength of mutually exclusive hypotheses
– If h is shown to be certain P(h|e) = 1
MB[[h,e] = 1
MD[h,e] = 0
CF[h,e] = 1
– If the negation of h is shown to be certain P(~h|e) = 1
MB[h,e] = 0
MD[h,e] = 1
CF[h,e] = -1
Characteristics of Belief Measures
• Suggested limits and ranges:
-1 = CF[h,~h] <= CF[h,e] <= C[h,h] = +1
• Note: MB[~h,e] = 1 if and only if MD[h,e] = 1
• For mutually exclusive hypotheses h1 and h2,
if MB[h1,e] = 1, then MD[h2,e] = 1
• Lack of evidence:
– MB[h,e] = 0 if h is not confirmed by e (i.e. e and h are independent
or e disconfirms h)
– MD[h,e] = 0 if h is not disconfirmed by e (i.e. e and h are
independent or e confirms h)
– CF[h,e] = 0 if e neither confirms nor disconfirms h (i.e. e and h are
independent)
More Characteristics of Belief Measures
• CF[h,e] + CF[~h,e] =/= 1
• MB[h,e] = MD[~h,e]
The Belief Measure Model as an
Approximation
• 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 measurements
and moreover, we need to keep MBs and MDs
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-] <= CF[h, e- & e+] <= CF[h, e+]
• MB[h,e+] = 1 then MD[h, e-] = 0 and CF[h, e+] = 1
• MD[h, e-] = 1 then MB[h, e+] = 0 and CF[h, e-] = -1
• The case where MB[h, e+] = MD[h, e-] = 1 is contradictory
and hence CF is undefined
Defining Criteria for Approximation
• If s1 & s2 indicates an ordered observation of evidence,
first s1 then s2 then:
– MB[h, s1&s2] = MB[h, s2&s1]
– MD[h, s1&s2] = MD[h, s2&s1]
– CF[h, s1&s2] = CF[h, s2&s1]
• If s2 denotes a piece of potential evidence, the truth or
falsity of which is unknown:
– MB[h, s1&s2] = MB[h, s1]
– MD[h, s1&s2] = MD[h, s1]
– CF[h, s1&s2] = CF[h, s1]
Combining Functions
MB[h, s1&s2]
MD[h, s1&s2]
=
=
{
{
0
If MD[h, s1&s2] = 1
MB[h, s1] + MB[h, s2](1 - MB[h, s1])
0
otherwise
If MB[h, s1&s2] = 1
MD[h, s1] + MD[h, s2](1 - MD[h, s1])
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])
otherwise
Probabilistic Reasoning and Certainty Factors
(Revisited)
• Of methods for utilizing evidence to select diagnoses or
decisions, probability theory has the firmest appeal
• The usefulness of Bayes’ theorem is limited by practical
difficulties, related to the volume of data required to
compute 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 under
uncertainty.
• The MYCIN model tries to combine these two areas
(probabilistic, CFs) by providing a semi-formal bridge
(theory) between the two areas
A Simple Probability Model
(The MYCIN Model Prelude)
• Consider a finite population of n 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
and
n(e1 & e2 & h) = n(e1 & h) * n(e2 & h)
n(h)
n(h)
n(h)
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)
P(ei)
* [ 1 - P(h | e’)]
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]