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The chaos and the doubt

Probability is said to be born of the correspondence between Pierre de Fermat and Blaise Pascal, some time in the middle of the 17th century. Somewhat surprisingly, many texts retrace the history of the concept up until the 20th century; yet it has gone through major transformations since then. Probability always describes what we don't know about the world, but the focus has shifted from the world to what we don't know.

Henri Poincaré investigates in Science et Méthode (1908) why chance would ever happen in a deterministic world. Like most of his contemporaries, Poincaré believed in absolute determinism, there is no phenomenon without a cause, even though our limited minds may fail to understand or see it. He distinguishes two flavors of randomness, of which he gives examples.

If a cone stands on its point we know that it will fall but we do not know which way (...) A very small cause, which escapes us, determines a considerable effect that we can not but see, and then we say that this effect is due to chance.

And a little bit later he continues.

How do we represent a container filled with gas? Countless molecules...

The reverend’s gambit

Two years after the death of Reverend Thomas Bayes in 1761, the famous theorem that bears his name was published. The legend has it he felt the devilish nature of his result and was too afraid of the reaction of the Church to publish it during his lifetime. Two hundred and fifty years later, the theorem still sparkles debate, but among statisticians.

Bayes theorem is the object of the academic fight between the so-called frequentist and Bayesian schools. Actually, more shocking than this profound disagreement is the overall tolerance for both points of view. After all, Bayes theorem is a theorem. Mathematicians do not argue over the Pythagorean Theorem: either there is a proof or there isn't. There is no arguing about that.

So what's wrong with Bayes theorem? Well, it's the hypotheses. According to the frequentist, the theorem is right, it is just not applicable in the conditions used by the Bayesian. In short, the theorem says that if $(A)$ and $(B)$ are events, the probability of $(A)$ given that $(B)$ occurred is $(P(A|B) = P(B|A) P(A)/P(B))$. The focus of the fight is the term $(P(B...

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