Relative belief updating ratio

Let us consider again Eq. () and focus on the role of the likelihood to reshape $f_0(r)$. Being multiplicative factor irrelevant, it can be useful to rewrite that equation as

$$f(r\,\vert\,x,T,I) \propto \frac{{\cal L}(r\,;\, x,T)}{{\cal L}(r_R\,;\, x,T)}\cdot f_0(r)$$ (40)

with $r_R$ a reference value, in principle arbitrary, but conceptually very interesting if properly chosen. In fact, the ratio in the above formula acquires the meaning of relative belief update factor [13,17,23],²²and the updating Bayes' rule can be rewritten as

$$f(r\,\vert\,x,T,I) \propto {\cal R}(r\,;\,x,T,r_R)\cdot f_0(r) \,.$$ (41)

This way of rewriting the Bayes' rule is particularly convenient when the likelihood is not 'closed', that is it does not go to zero when the quantity of interest is `very large' or `very small'.

To be clear, let us make the example of having observed zero counts, that is the experiment was indeed performed, but no event of interest was found during the measurement time $T$. If we use a flat prior and only stick to the summaries, we have that the most probable value is zero, with E$(r)=\sigma(r)=1/T$: the larger is the measuring time, the more the distribution of $r$ is squeezed towards zero. But this does not give a complete picture of what is going on. Since ${\cal L}(r; x=0,T)$ goes to 1 for $r\rightarrow 0$, the likelihood is opened in the left side. Figure shows ${\cal R}$

Figure: Relative belief updating factor ${\cal R}(r;x=0,T,r_0=0)$ for different observation times.

functions for this case, for different $T$, although in this very simple case ${\cal R}$ is mathematically equivalent to the likelihood.²³If our beliefs about $r$ were above O( $100\,$s$^{-1}$), the observation of zero events practically rule them out, even with $T=1\,$s (`1 s' is arbitrarily chosen in this hypothetical example, just to remind that both $T$ and $r$ have physical dimensions).

If we run the experiment longer and longer, keeping observing zero events, the possible values of $r$ gets smaller and smaller. What is mostly interesting, in this plot, is the region in which ${\cal R}$ is flat: it means that if our beliefs are concentrate there, then the experiment does not teach us more than what we already believed: the experiment looses sensitivity in that region and then reporting `probabilistic' upper limits makes no sense and it can be highly misleading (even more reporting `C.L. upper limits`) [13,27].