References

Published

2024-09-28

Believe nothing, O monks, merely because you have been told it, or because it is traditional, or because you yourselves have imagined it. Do not believe what your teacher tells you merely out of respect for the teacher.     (Attributed to Gautama Buddha)


But in the natural sciences, whose conclusions are true and necessary and have nothing to do with human will, one must take care not to place oneself in the defense of error; for here a thousand Demostheneses and a thousand Aristotles would be left in the lurch by every mediocre wit who happened to hit upon the truth for himself.     (Galileo Galilei)


The notions, ideas, and rules that you have learned in this course have been presented in such a way as to appear plausible and intuitively understandable. In some places we gave sketch of proofs.

But that is not enough.

A “data mechanic” (see preface) might be excused for using incorrect formulae, and might simply say “this is the procedure is was taught”. You instead, as a data engineer and data scientist, have the duty to check the validity of the theory and principles that you use in developing new algorithms, code, solutions.

In particular, you cannot accept theories or methods simply because

  •   they are commonly used, or used by the majority of some community

  •   some “authority” or known scientist says they are correct

In fact, Science began when the two criteria above were discarded as not valid. This is said very explicitly in Galileo’s quote above, for example. Imagine if Einstein had said “all scientists see no problem with the notion of simultaneity, so it must be correct”, or “great scientists like Maxwell or Poincaré did not see any problem with the notion of simultaneity, so it must be correct”. There is no scientific progress with this kind of wrong reasoning.

Instead, the only two scientific criteria you have to decide on the validity of a method or theory are

  •   experimental corroboration

  •   logical proof

which you must do as much as possible by yourself. The more verification you delegate to others, to majority or “authority”, the less you are doing science.


For this reason you have, at some point, go and check for yourself the validity of what you’ve learned in this course. You might in fact find out that something was not correct! Then you’ll correct it and make science advance. Throughout the course We have given references where many proofs can be found. Here are some final references containing the main proofs of what you have learned here; you should check and validate them at some point.

Foundations of the probability calculus

The four fundamental rules of the Probability Calculus have been at least since Laplace in the 1700s, essentially in their present form. Laplace used them to infer properties of planets and their orbit (with results still valid today). Proof of their logical foundation and necessity started to appear in the 1940s, a formal milestone being the proof by R. T. Cox in 1946. They have been tightened and reformulated in different ways since. Here are some old and recent works on the foundations (as opposed to works that simply mention the rules and apply them). Cox’s and Jaynes’s are probably the first ones to be checked:


Foundations of Decision Theory

Decision Theory is much younger than the Probability Calculus, and its foundations probably still needs to be tightened here and there. Here are old and recent works on its foundations: