Image Processing Reference
In-Depth Information
each problem to tackle
6
. For example, the “Bayesian” diagnosis can be formalized as
follows:
p
A
i
|
p
O
A
i
p
A
i
p
O
A
i
p
A
i
O
=
|
|
j
p
O
A
j
p
A
j
p
O
A
i
p
A
i
+
p
O
A
i
p
A
i
=
,
[6.56]
|
|
|
where
p
(
A
i
|
O
) refers to the probability for a patient to have the pathology
A
i
,given
a set of observations
O
(clinical examinations, images, etc.),
p
(
O
A
i
) refers to the
conditional probability of the observations given the pathology and
p
(
A
i
) is the
a pri-
ori
|
O
). The use of this
formula requires either that all of the pathologies are known, or to have statistics con-
necting the observations to
A
i
(“non-pathology
A
i
”). Both solutions seem unrealistic.
Furthermore, it has to be possible for all of the probability distributions involved in the
formula to be estimated. The problem then is the limit to the statistical tests connecting
symptoms or observations with pathologies and the difficulty of having an estimate of
the
a priori
probabilities. These limits are of course general and are not specific to this
particular example.
probability of
A
i
. The decision is made based on the
p
(
A
i
|
Probabilistic modeling can only deal with singletons that represent the different
hypotheses, under the closed world's constraint. We saw in the previous example of
medical diagnosis how this hypothesis does not fit reality. Furthermore, singletons
cannot be used to represent complex situations. Let us take the case of images affected
by the partial volume effect (a common situation in medical imaging). The usual mod-
els in other works for representing this phenomenon consist of assigning to a point
probabilities of belonging to the types of tissue it is comprised of, which are propor-
tional to the amount of each type of tissue in the volume represented by this point.
However, this does not correspond to anything real. This type of probabilistic model
implies that we are faced with an uncertainty regarding the class to which the point
belongs (we know that the point can belong to several classes but we do not know
which one), whereas in fact it belongs to several different classes simultaneously. To
us, this seems to be the typical example of probabilistic models that are used but do
not properly model the observed phenomenon.
Fuzzy sets or belief function theory (or Dempster-Shafer theory) allow us to better
describe the reality of certain problems and to find less disputable interpretations.
6. Frequentist methods lead to the opposite situations, particularly Fisher's theory, which is
essentially automatic: “faced with a new situation, the statistician can apply maximum like-
lihood in an automatic fashion, with little chance (in experienced hands) of going far wrong
and considerable chance of providing a nearly optimal inference. In short, he does not have to
think a lot about the specific situation in order to solve his problem” [EFR 86]. It is a theory of
archetypes, that allows us to obtain reasonable solutions by separating the different problems,
in cases where the Bayesian approach, which deals with everything “all at once”, would be too
complex. This automatic nature is certainly one of the reasons behind the popularity of Fisher's
theory, despite the disadvantages of the frequentist methods.
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