Application to the inference of unknown values of physical quantities

In fact, immediately after the proof of his theorem, Gauss continues:

“Now, so far as we suppose that no other data exist for the determination of the unknown quantities besides the observations $V=M$, $V'=M'$, $V''=M''$ etc., and, therefore, that all systems of values of these unknown quantities were equally probable previous to the observations, the probabilities, evidently, of any determinate system subsequent to the observations will be proportional to $\Omega$. This is to be understood to mean that the probability that the values of the unknown quantities lie between the infinitely near limits $p$ and $p+$d$p$, $q$ and $q+$d$q$, $r$ and $r+$d$r$, $s$ and $s+$d$s$,, etc. respectively, is expressed by

$$\ \ \ \ \ \lambda\,\Omega\,$$d$$p\,$$d$$q\,$$d$$r\,$$d$$s\,\cdots,\$$   etc.$$,$$   (G3)$$$$

where the quantity $\lambda$ will be a constant quantity independent of $p$, $q$, $r$, $s$, etc.: and, indeed,, $1/\lambda$ will, evidently, be the value of the integral of order $\nu$ ,

$$\int^\nu\!\Omega\,$$d$$p\,$$d$$q\,$$d$$r\,$$d$$s\,\cdots,\$$   etc.$$,$$   (G4)$$$$

for each of the variables $p$, $q$, $r$, $s$, etc, extended from the value $-\infty$ to the value $+\infty$.”

As we can see, it is well stated the assumption of `flat priors', as we use to say nowadays (with the original words of Gauss, in Latin: “valorum harum incognitarum ante illa observationes aeque probabilia fuisse”).¹³

It is, instead, less clear how he uses the result of his theorem (the quote at the beginning of this section follows immediately the end of the proof of the theorem, with no single word in between). The implicit intermediate step is

$$P(H\,\vert\,E) \propto P(E\,\vert\,H)\,,$$ (15)

extended to set of continuous uncertain values (`uncertain vector') $\theta$ as

$$P( \mbox{\boldmath$\theta$} \,\vert\, ) \propto P( \,\vert\, \mbox{\boldmath$\theta$} )\,.$$ (16)

Then, remembering that $\Omega$ was the joint pdf of the observations $[$see Eq. (G1)$]$, which we have rewritten in more compact notation as Eq. (6), we have

$$f( \mbox{\boldmath$\theta$} \,\vert\, \mbox{\boldmath$V_m$} - \mbox{\boldmath$V$} ) \propto f( \mbox{\boldmath$V_m$} - \mbox{\boldmath$V$} \,\vert\, \mbox{\boldmath$\theta$} )$$

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$$f( \mbox{\boldmath$\theta$} \,\vert\, \mbox{\boldmath$V_m$} - \mbox{\boldmath$V$} ) = \lambda\cdot f( \mbox{\boldmath$V_m$} - \mbox{\boldmath$V$} \,\vert\, \mbox{\boldmath$\theta$} )\,,$$

being $\lambda$ just the normalization constant, i.e.¹⁴

$$\frac{1}{\lambda} = \int_{\mathbb{R}^\nu}f( \mbox{\boldmath$V_m$} - \mbox{\boldmath$V$} \,\vert\, \mbox{\boldmath$\theta$} )\, \mbox{\boldmath$\theta$} \,.$$ (17)