Conclusions

The issue of fits has been approached from probability first principles, i.e. using throughout the rules of probability theory, without external ad hoc ingredients. It has been that the main task consists in building up the inferential model, that means in fact to properly factorize the joint probability density function of all variables of the problem. We have seen that this factorization, based on the so called chain rule of probability theory, has a very convenient graphical representation, that takes the name of Bayesian (or belief/causal/influence) network. Modeling the problem in terms of such networks not only helps to understand the problem better, but, thanks the huge amount of mathematical developments relates to them, it becomes the only way to get a (numerical) solution when problems get complicated.

We have also seen how to recover well known formulas, obtained starting from other approaches, under well defined conditions, thus indicating that other methods can be seen as approximations of the most general one, and that are therefore applicable if the conditions of validity hold.

The linear case with errors on both axis and extra variance of the data has been shown with quite some detail, giving un-normalized formulas for the pdf. In particular, going to the pretext to write this paper, we can see that Eq. (43) of Ref. [17] is not reproduced. In fact, if I understand it correctly, that equation should have the same meaning of Eq. (53) of this paper. However, Eq. (43) of Ref. [17] contains an extra factor $\sqrt{1+m^2}$ (using the notation of this paper), that it is a bit odd, for several reasons (besides the fact that I do not get it - but this could be judged a technical argument by the hurry reader). The first reason is just dimensionality: $m\,x$ is homogeneous with $y$ and for this reason $m\,\sigma_x$ can be combined (quadratically) to $\sigma_y$, but $m^2$ cannot be added tout court to 1. The second is that if there was such a factor in Eq. (53), then one cannot reproduce Eqs. (58), (60) and (61), that one can be obtained in simpler ways (and that give rise to the likelihoods shown in Section 6, some of them rather well known). Note that the addition of a term $\sqrt{1+m^2}$ in Eq. (53) has the net effect of overestimating $m$, an effect that is consistent with the claim by [1] of a slope larger than that obtained by [14].⁸