Indirect measurements

The case of quantities measured indirectly is conceptually very easy, as there is nothing to `think'. Since all values of the quantities are associated with random numbers, the uncertainty on the input quantities is propagated to that of output quantities, making use of the rules of probability. Calling $\mu_1$, $\mu_2$ and $\mu_3$ the generic quantities, the inferential scheme is:

$$\begin{array}{l} f(\mu_1\,\vert\,{data}_1) \\ f(\mu_2\,\vert\... ...[\mu_3=g(\mu_1,\mu_2)] {} f(\mu_3\,\vert\,{data}_1,{data}_2)\,.$$ (2.4)

The problem of going from the probability density functions (p.d.f.'s) of $\mu_1$ and $\mu_2$ to that of $\mu_3$ makes use of probability calculus, which can become difficult, or impossible to do analytically, if p.d.f.'s or $g(\mu_1,\mu_2)$ are complicated mathematical functions. Anyhow, it is interesting to note that the solution to the problem is, indeed, simple, at least in principle. In fact, $f(\mu_3)$ is given, in the most general case, by

$$f(\mu_3) = \int\,f(\mu_1)\cdot f(\mu_2)\cdot \delta(y_3-g(\mu_1,\mu_2)) \, \mu_1 \mu_2\,,$$ (2.5)

where $\delta()$ is the Dirac delta function. The formula can be easily extended to many variables, or even correlations can be taken into account (one needs only to replace the product of individual p.d.f.'s by a joint p.d.f.). Equation () has a simple intuitive interpretation: the infinitesimal probability element $f(\mu_3)\,d\mu_3$ depends on `how many' (we are dealing with infinities!) elements $d\mu_1 d\mu_2$ contribute to it, each weighed with the p.d.f. calculated in the point $\{\mu_1, \mu_2\}$. An alternative interpretation of Eq. (), very useful in applications, is to think of a Monte Carlo simulation, where all possible values of $\mu_1$ and $\mu_2$ enter with their distributions, and correlations are properly taken into account. The histogram of $\mu_3$ calculated from $\mu_3= g(\mu_1,\mu_2)$ will `tend' to $f(\mu_3)$ for a large number of generated events.^2.16

In routine cases the propagation is done in an approximate way, assuming linearization of $g(\mu_1,\mu_2)$ and normal distribution of $\mu_3$. Therefore only variances and covariances need to be calculated. The well-known error propagation formulae are recovered (Section ), but now with a well-defined probabilistic meaning.