Combining ratios of rates

Let us end the work with the related topic of `combining several values' of $\rho$, a problem we have already slightly touched above. Let us start phrasing it in the terms we usually hear about it. Imagine we have in hand $N$ instances of $(X_1,T_1,X_2,T_2)$. From each of them we can get a value of $\rho$ with `its uncertainty'. Then we might be interested in getting a single value, combining the individual ones.

The first idea that might come to the mind is to apply the well known weighted average of the individual values, using as weights the inverses of the variances. But, before doing it, it is important to understand the assumptions behind it, that is something that goes back to none other than Gauss, and for which we refer to Refs. [29,44]. The basic idea of Gauss was to get two numbers (let us say `central value' and standard deviation - indeed Gauss used, instead of the standard deviation, what he called `degree of precision' and `degree of accuracy' [44], but this is an irrelevant detail) such that they contain the same information of the individual values. In practice the rule of combination had to satisfy what is currently known as statistical sufficiency. Now it is not obvious at all that the weighted average using E$(\rho)$ and $\sigma(\rho)$ satisfies sufficiency (see e.g. the puzzle proposed in the Appendix of Ref. [44]).

Therefore, instead of trying to apply the weighted average as a `prescription', let us see what comes out applying consistently the rules of probability on a suitable model, restarting from that of Fig. . It is clear that if we consider meaningful a combined value of $\rho$ for all instances of $(X_1,T_1,X_2,T_2)$ it means we assume $\rho$ not depending on a quantity $v$. However, $r_2$ could. This implies that the values of $r_2$ are strongly correlated to each other.³⁵Therefore the graphical model of interest would be that at the top of Fig. .

Figure: Possible reductions of the model of Fig.  for the `combination' of $\rho$ (see text).

Again, at this point there is little more to add, because what would follow depends on the specific physical model.

A trivial case is when both rates, and therefore their ratio, are assumed to be constant, although unknown, yielding then the graphical model shown in the bottom diagram of Fig. , whose related joint pdf, evaluated by the best suited chain rule, is an extension of Eqs. ()-()

$$f(\ldots) \!\! = \left[\,\prod_{j=1}^N f(x_{2_j}\,\vert\,r_2,T_{2_j})\right]\!\cdo... ...})\right] \!\cdot\! f(r_1\,\vert\,r_2,\rho)\!\cdot\! f_0(\rho)\,,\ \ \ \ \ \ \$$ (142)

from which the unnormalized joint pdf follows:

$$\tilde f(\ldots) \!\! \propto \left[\,\prod_{j=1}^N r_2^{x_{2_j}}\, \cdot e^{-T_{2_j}\,r_2}\rig... ...}\,r_1}\right]\!\cdot \delta(r_1-\rho\cdot r_2)\cdot f_0(\rho) \ \ \ \ \ \ \ \$$ (143)

$$ \propto \left[ r_2^{x_{2_{tot}}}\, \cdot e^{-T_{2_{tot}}\,r_2}\right]\!\c... ...r_1}\right]\!\cdot \delta(r_1-\rho\cdot r_2)\cdot f_0(\rho)\,. \ \ \ \ \ \ \ \$$ (144)

We recognize the same structure of Eq. (), with $x_1$ replaced by $x_{1_{tot}}=\sum_j x_{1_j}$ $T_1$ by $T_{1_{tot}}=\sum_j T_{1_j}$, $x_2$ by $x_{2_{tot}}=\sum_j x_{2_j}$ and $T_2$ by $T_{2_{tot}}=\sum_j T_{2_j}$. We get then the same result obtained in Sec.  if we use the total numbers of counts in the total times of measurements. This is a simple and nice result, close to the intuition, but we have to be aware of the model on which it is based.