Role of the priors and sequential update of $f(r)$ as new observations are considered

The very essence of the so called probabilistic inference (`Bayesian inference') is given by Eq. (). The rest is just a question of normalization and of extending it to the continuum, that in our case of interest is

$$f(r\,\vert\,x,T,I) \propto f(x\,\vert\,r,T,I)\cdot f(r\,\vert\,I)\,.$$ (33)

It is evident the symmetric role of $f(x\,\vert\,r,T,I)$ and $f(r\,\vert\,I)$, if the former is seen as a mathematical function of $r$ for a given (`observed') $x$, that is $x$ playing the role of a parameter. This function is known as likelihood and commonly indicated by ${\cal L}(r\,;\, x,T)$.<sup>19</sup>Indicating the second factor of Eq. (), that is the `infamous' prior that causes so much anxiety in practitioners [8], by $f_0(r)$, we get (assuming $I$ implicit, as it is usually the case)

$$f(r\,\vert\,x,T,I) \propto {\cal L}(r;x,T)\cdot f_0(r)\,,$$ (34)

which makes it clear that we have two mathematical functions of $r$ playing symmetric and peer roles. Stated in different words, each of the two has the role of `reshaping' the other [1]. In usual `routine' measurements (as watching your weight on a balance) the information provided by ${\cal L}(\ldots)$ is so much narrower, with respect to $f_0(\ldots)$,²⁰that we can neglect the latter and absorb it in the proportionality factor, as we have done above in Sec. . Employing a uniform `prior' is then usually a good idea to start with, unless $f_0(\ldots)$ arises from previous measurements or from strong theoretical prejudice on the quantity of interest. It is also very important to understand that `the reshaping' due to the priors can be done in a second step, as it has been pointed out, with practical examples, in Ref. [1].

Let us now see what happens when, in our case, the Bayes rule is applied in sequence in order to account for several results on the same rate $r$, that is assumed to be stable. Imagine we start from rather vague ideas about the value of $r$, such that $f_0(r)=k$ is, in practice, the best practical choice we can do. After the observation of $x_1$ counts during $T_1$ we get, as we have learned above,

$$f(r\,\vert\,x_1,T_1) = \frac{T_1^{\,x_1+1}\cdot r^{\,x_1}\cdot e^{-T_1\,r}}{x_1!}.$$ (35)

Then we perform a new campaign of observations and record $x_2$ counts in $T_2$. It is clear now that in the second inference we have to use as `prior' the piece of knowledge derived from the first inference. So, we have, all together, besides irrelevant factors,

$$f(r\,\vert\,x_1,T_1,x_2,T_2) \propto f(x_2\,\vert\,r,T_2)\cdot f(r\,\vert\,x_1,T_1)$$ (36)

$$\propto r^{\,x_2}\cdot e^{-T_2\,r}\cdot r^{\,x_1}\cdot e^{-T_1\,r}$$ (37)

$$\propto r^{\,x_1+x_2}\, e^{-(T_1+T_2)\,r}\,,$$ (38)

that is exactly as we had done a single experiment, observing $x_{tot} = x_1+x_2$ counts in $T_{tot}=T_1+T_2$. The only real physical strong assumption is that the intensity of Poisson process was the same during the two measurements, i.e. we have being measuring the same thing.

This teaches us immediately how to `combine the results', an often debated subject within experimental teams, if we have sets of counts $x_i$ during times $T_i$ (indicated all together by ' $\underline x$' and ` $\underline T$'):

$$f(r\,\vert\,\underline{x},\underline{T}) = \frac{(\sum_i T_i)^{\,\sum_ix_i+1}\cdot r^{\,\sum_ix_i}\cdot e^{-(\sum_iT_i)\,r}}{(\sum_ix_i)!}\,,$$ (39)

without imaginative averages or fits. But this does not mean that we can blindly sum up counts and measuring times. Needless to say, it is important, whenever it is possible, to make a detailed study of the behavior of $f(r\,\vert\,x_i\,T_i)$ in order to be sure that the intensity $r$ is compatible with being constant during the measurements. But, once we are confident about its constancy (or that there is no strong evidence against that hypothesis), the result is provided by Eq. (), from which all summaries of interest can be derived.²¹