Image Processing Reference
Fuzzy sets, for example, can perfectly account for the partial belonging phenomenon
that is in fact observed. Relaxing the probability additivity constraint is not enough
to solve the problem: the solution does in fact involve the modeling of a completely
different phenomenon. We will discuss this further in the next two chapters.
Another limitation stems from the difficulty of adding, to our reasoning system,
knowledge that cannot be simply expressed with probabilities.
Along the same idea, it is difficult to model the absence of knowledge, imprecise
knowledge (unlike uncertain knowledge, which is naturally represented by probabil-
ities), or also what we do not know about a phenomenon. The insufficient reason
principle is not enough for taking into account what is not known and can lead to
contradictions depending on how it is expressed.
The same type of problem arises with the maximum entropy principle. Shafer's
well-known example about the probability for the existence of life on planet Sirius is
a good illustration [SHA 76] 7 . These drawbacks, which are not better solved using a
subjective version of probabilities, emerge whenever the objective is to model human
reasoning that involves decisions based on data that is at the same time imprecise and
uncertain, partial, not completely reliable, conflicting, and constraints and objectives
that are not always very precise.
[BAR 85] B ARNETT S., C AMERON R., Introduction to Mathematical Control Theory , Oxford
University Press (Applied Mathematics Series), 1985.
[BAZ 93] B AZARAA M., S HERALI H., S HETTY C., Non Linear Programming Theory and
Algorithms , Wiley, New York, 1993.
[CHA 92] C HAN Y. , R UDNICKI S., “Bearings-Only and Doppler-Bearing Tracking Using
Instrumental Variables”, IEEE Transactions on AES , vol. 28, no. 4, p. 1076-1083, 1992.
[CHA 95] C HAUVIN S., Evaluation des théories de la décision appliquées à la fusion de cap-
teurs en imagerie satellitaire, PhD Thesis, Ecole Nationale Supérieure des Télécommuni-
cations and Nantes University, 1995.
7. We will not present here Shafer's original example, which is debatable, but an example given
by Dubois that is close. Not knowing whether life exists on Sirius is typically expressed in
terms of probability by p(life) = p(no life) = 0.5. If we express the problem in another way, by
assuming that there are three possibilities, plant life, animal life or no life, the expression of not
knowing leads us to assign a probability of 1/3 to each of the three hypotheses. We then get
p(life) = 2/3. Likewise, we can obtain as many different values as there are ways of expressing
the problem. Similar examples can be found in image processing, particularly in classification