Information Fusion in Signal and Image Processing: Major Probabilistic and Non-Probabilistic Numerical Approaches

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 testimonies, or fusing heterogenous images. Probabilistic methods certainly are the most popular, but still give rise to a number of controversies, particularly between frequentist or objectivist methods and subjectivist methods. Although subjectivists seem to be taking over in many fields, frequentist concepts are still of great practical use, particularly 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 characterization 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 probabilities¹, which is widely recognized today, only appeared so late. This hypothesis is stated as an axiom in Kolmogorov theory. However, the works of Cox show that these axioms can ...