Conclusions

Reading Gauss' work, there are no doubts that the Prince Mathematicorum had clear ideas on how to tackle inverse probability problems, i.e. what goes presently under the name Bayesian inference. In particular, he presented as original what is now called Bayes factor, i.e the factor to update the odds in favor of an hypothesis with respect to the alternative one, in the light of a new observation. However, it is curious that, as far as I could find, this result is not acknowledged in the current literature. For example, his name appears only once in the Sharon Mcgrayne rather comprehensive book on the history of Bayesian reasoning[13], as being cited by Enrico Fermi, who was teaching his students data analysis methods derived from his Bayes' theorem.¹⁵

At this point a long discussion could follow on the question if Gauss could be classified as a Bayesian and why, later on in his book, he did not proceed applying consistently the probabilistic reasoning he had setup, getting the joint probability distribution of the values of the orbital elements given the observed geocentric measurements, but he derived, instead, the least square method to get (relatively) simple formulae for the most probable values (this aim was clearly stated). And all this in the same text, just a few pages after, and not in a later stage of his life.

Well, I am not an historian, and therefore I can only state my impressions based on a limited amount of reading. Gauss appears in the section of the book upon which this modest note is based not only as the genius he is famous to be, but also a very practical scientist going straight to his goals. Trying to set a multi-dimensional inference to write down the joint pdf of parameters of a non-linear problem and exploiting it at best, something that we can do nowadays, thanks to unprecedented computing power and novel mathematical methods, would have just been a waste of time two centuries ago. We have also seen that he didn't even care to state the general rule to update probability ratios, which would have required just a couple of lines of text, because he had in mind a problem for which the priors were reasonable `flat'. Moreover, he was also well aware of the practical meaning and limits of the mathematical functions, as when, later in the same section, he commented in `article' 177 on the “defect” of his error function, because “the function just found cannot, it is true, express rigorously the probabilities of the errors”. Indeed, the `error function' $\varphi()$ was not specified up to the end of 'article' 176. Only in the following article he showed that a good candidate for it was, under well stated conditions, ... the Gaussian, a function having the “defect” of contemplating values ranging from minus infinity to plus infinity. Then other interesting articles follow,¹⁶but I don't want to spoil you the pleasure of the reading.¹⁷

Finally, someone might be intrigued about what Gauss meant by probability. “Probabilitas”. What else?¹⁸