Image Processing Reference

In-Depth Information

Appendix A

Probabilities: A Historical Perspective

This appendix is largely inspired by [BLO 96].

Among the different methods for representing knowledge, numerical methods,

which attempt to model the imprecision and uncertainty of data and knowledge, are

widely used for problems as diverse as multi-criteria aggregation, combining testi-

monies, or fusing heterogenous images. Probabilistic methods certainly are the most

popular, but still give rise to a number of controversies, particularly between frequen-

tist or objectivist methods and subjectivist methods. Although subjectivists seem to

be taking over in many fields, frequentist concepts are still of great practical use, par-

ticularly when it comes to learning a law based on large samples, for example, to

recognize cultivations in an aerial image.

A historical overview of the different meanings of probability can help explain the

causes of these controversies and show that the choice of a method can be thought

through and justified by the problem at hand and by our interpretation of probability.

section A.1 will focus on this historical perspective and section A.2 on the charac-

terization of different classes of probabilities. This is largely based on review articles

cited as references.

It seems remarkable that the hypothesis of the additivity of probabilities
1
, which is

widely recognized today, only appeared so late. This hypothesis is stated as an axiom

1. The additivity relation expresses the fact that for two exclusive events
A
and
B
, the probabil-

ity of the union denoted by

A
+
B

p
(
A
+
B
)=
p
(
A
)+
p
(
B
)

is equal to

. Particularly, we infer

p
(
A
)+
p
(
A
)=1

A

A

, where

denotes the opposite of

(or its complement in set theory terms).

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