Introduction to Evidential Reasoning

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Introduction to Evidential Reasoning

• Belief Functions

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Ignorance Types
Incompleteness Combinatory John is
married, but his wife's name is not given
Combinatory All computer scientists like
pizza, but their names are not available.
Imprecision Combinatory John's wife is
Jill or Joan. Combinatory Jill is not John's
wife. Interval theory Paul's height is between
170 and 180. Fuzzy sets Paul is tall.
Possibility Theory the possibility for Paul's
height to be about 175 cm. (physical
form) Uncertainty Probability Theory
Upper-Lower Probabilities Possibility
Theory the possibility that Paul's height is
about 175 cm. (epistemic form) Subjective
Probabilities Belief functions (Credibility)
the chance of it being "heads" when tossing a
coin.
my degree of belief that cancer X is due to a
virus.
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A murder case
A somehow reliable witness testifies that the
killer is a male. -Testimony reliability a
0.7 -A priori equal belief that the killer is a
male or a female. Is the killer Male? M the
killer is a male
Classical probability analysis P(M)
P(M/reliable)P(reliable) P(M/reliable)P(reliab
le) P(M)1x0.7 0.5x0.3 0.85
Evidence Theory- DS Theory bel(M) 0.7.
0.3
0.7
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Bayesian approach Answering what is the
belief in A? as expressed by the unconditional
probability that A is true given evidence, e ?
Assumption precise probabilities can be
assessed for all events. Too rare Why believe
one hypothesis other than that provided by the
evidence?? The evidence have to be re-organised
so that probabilities sum to unity. Pros -rules
of probability calculus uncontroversial,
constant conclusions with the probability
assessments. -Bayesian theory is easy to
understand. Cons It is least suited to problems
where there is -partial or complete
ignorance -limited or conflicting information
due to assumptions made(e.g. equi-probability) C
annot deal with imprecise, qualitative or natural
language judgements such as if A then
probably B.
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DempsterShafer approach Answering the question
what is the belief in A, as expressed by
the probability that the proposition A is
provable given the evidence? An alternative to
traditional probabilistic theory for the
mathematical representation of uncertainty. Where
as a Bayesian approach assesses probabilities
directly for the answer, the DempsterShafer
approach assesses evidence for related questions.
-combination of evidence obtained from multiple
sources and the modeling of conflict between
them. -Allocation of a probability mass to sets
or intervals Pros -Ability to model various
types of partial ignorance, limited or
conflicting evidence -more flexible model than
Bayes theorem. -computationally simpler than
Bayes theorem. -No assumption regarding the
probability of the individual constituents of the
set or interval. -evaluation of risk and
reliability in engineering applications when it
is not possible to obtain a precise measurement
from experiments, or when knowledge is obtained
from expert elicitation. Cons -Can produce
conclusions that are counter-intuitive.
DempsterShafer is most suited to situations
where beliefs are numerically expressed and where
there is some degree of ignorance, i.e. there is
an incomplete model.
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Belief functions ? the frame of
discernment(elements of the set ? are called
worlds) One actual world ?0. But which? An
agent can only express the strength of his/her
opinion (called degree of belief) that the actual
world belongs to this or that subset of
?. Shafer belief function bel 2? ? 0, 1
bel(A) denotes the strength of Agents belief
that ?0?A. bel satisfies the following
inequalities Other useful functions
(1-1 with bel) 1.basic belief assignment (bba)
m 2? ? 0, 1 defined as m(A) for A ? ?
is called the basic belief mass (bbm) given to A.
It may happen that m(?) gt 0. The relation from m
to bel is given by 2. plausibility function
pl 2? ? 0, 1 is defined as Shafer bel is
normalized gt closed world assumptiongt
bel(?)1, pl(?)1,m(?) 0.
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Observations in belief functions
Not supporting any strictly more specific
propositions A basic belief mass given to a set A
supports also that the actual world is in every
subsets that contains A. The degree of belief
bel(A) for A quantifies the total amount of
justified specific support given to A. We say
justified because we include in bel(A) only the
basic belief masses given to subsets of A.
m(x,y) given to x,y could support x if
further information indicates this.However given
the available information the basic belief mass
can only be given to x,y. We say specific
because the basic belief mass m(Ø) is not
included in bel(A) as it is given to the subset Ø.
Entertained beliefs and beliefs in a decision
context Uncertainty induces beliefsgraded
dispositions that guide our behavior rationala
gent behavior described within decision
contexts It has been argued that decisions are
rational only if we use a probability measure
over the various possible states of the nature
and compute with it the expected utility of each
possible act, the optimal act being the one that
maximizes these expected utilities (DeGroot,
1970 Savage, 1954). beliefs can only be
observed through our decisionsgtuse of
probability functions to represent quantified
beliefs 2 categories of beliefsEntertained
beliefs and beliefs in a decision
context Entertained beliefsgtprovide the
quantified belief of the Agent (use of Justified
Evidences) Beliefs in a decision contextgtprovide
a method for rational decision making(probability
function).
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Dempster Rule of Combination
Zadeh provides a compelling example of erroneous
results. 1 patient with neurological symptoms, 2
physicians Doctor1 meningitis 0.99 brain tumor
0.01 Doctor2 concussion 0.99 brain tumor 0.01.
Using Dempster m (brain tumor) Bel (brain
tumor) 1 !!! Complete support for a very
unlikely diagnosis problem when strongly
conflicting evidence
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Problem of Dempster when highly conflicting
evidence
System failure, 2 experts. Failure caused by
Component A, B or C. Expert 1 m1(A) 0.99
(failure due to Component A) m1(B) 0.01
(failure due to Component B) Expert 2 m2(B)
0.01 (failure due to Component B) m2(C) 0.99
(failure due to Component C) Dempsters
Rule combination of the masses 1. To
calculate the combined basic probability
assignment for a particular cell, simply multiply
the masses from the associated column and row. 2.
Where the intersection is nonempty, the masses
for a particular set from each source are
multiplied, e.g., m12(B) (0.01)(0.01)
0.0001. 3. Where the intersection is empty, this
represents conflicting evidence and should be
calculated as well. For the empty intersection of
the two sets A and C associate with Expert 1 and
2, respectively, there is a mass associated with
it. m1(A) m2(C)(0.99)(0.99) (0.9801). 4. Then
sum the masses for all sets and the conflict. 5.
The only nonzero value is for the combination of
B, m12(B) 0.0001. In this example there is only
one intersection that yields B, but in a more
complicated example it is possible to find more
intersections to yield B. 6. For K, there are
three cells that contribute to conflict
represented by empty intersections. K
(0.99)(0.01) (0.99)(0.01) (0.99)(0.99)
0.9999 7. Calculate the joint, m1(B) m2(B)
(.01)(.01) / 1-0.9999 1 Bel (B) 1!!!
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Other combination Rules
Yagers rule almost same matrix as Dempsters
rule. Exceptions in the nomenclature and
allocation of conflict 1. Ground probability
assignments (q) instead of basic probability
assignments (m) 2. q(Ø) instead of using K (but
q(Ø)K) Not normalization by factor
(1-K). significant reduction of the value for
Belief -gt counterintuitive results
sometimes large expansion of Plausibility.
Inagakis Rule The matrix same as Dempsters.
-ground probability functions like Yager.
m12(B) depends on the value of k which is now a
parameter. k experimentally or by expert
expectation When k 0 gt Yagers Rule. When k
1/(1- q(Æ)) gt Dempsters rule m12(B) ? 1,
because sums of all masses must be equal to 1. k?
gt ? filtering of the evidence.
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Other combination Rules
Zhangs Rule measure of intersection based on the
cardinality of the sets. Problems with Zhangs
measure of intersection 1. The equivalence with
Dempsters rule when the cardinality is 1 for all
relevant sets or when the CAB in the
circumstance of conflicting evidence.
(This should not pose a problem if there is no
significant conflict.) 2. If the cardinality of B
was greater than 1, even completely overlapping
sets will be scaled. Mixing The formulation for
mixing in this case corresponds to the sum of
m1(B)(1/2) and m2(B)(1/2). m12(A) (1/2)(0.99)
0.445 m12(B) (1/2)(0.01) (1/2) (0.01)
0.01 m12(C) (1/2)(0.99) 0.445 Dubois and
Prades Disjunctive Consensus Pooling Unions of
multiple sets
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Which model to use depends on the specific
application