Resolution power

Having to turn the qualitative judgment regarding the `separation' of the distributions of $f_P$ for different $p$, as it results from Figs. -, into a resolution power, one needs some convention. First, we remind that, unless $p$ is very small, we have good theoretical reasons, confirmed by Monte Carlo simulations, that $f_p$ is about Gaussian, at least in the range of a few standard deviations around its mean value. But Gaussian curves are, strictly speaking, never separated from each other, because they have as domain the entire real axis for all $\mu$'s and $\sigma$'s. In fact this is a “defect” of such distribution, as Gauss himself called it [9] and some grain of salt is required using it. In order to form an idea of how one could define conventionally `resolution', the upper plot of Fig.  shows

Figure: Upper plot: Examples of Gaussians whose $\mu$ parameters are separated by 1 $\sigma$. Bottom plot: resolution power in $p$, defined by Eq. (), for $\kappa=3$ (lines between points just to guide the eye). Filled (blue) circles for $p=0.1$ and open (cyan) circles for $p=0.5$. Solid lines for $\pi_2=0.115\pm 0.022$, dashed lines for $\pi_2=0.115\pm 0.007$ and dotted lines for $\pi_2=0.022\pm 0.007$ ( $\pi_1=0.978\pm 0.007$ in all cases).

some Gaussians having unitary $\sigma$, with $\mu$'s differing by one $1\sigma$.

We see that a `reasonable separation' is achieved when they differ by a few $\sigma$'s - let us say, generally speaking, $\kappa\, \sigma$, although absolute separation can never occur, for the already quoted intrinsic “defect” of the distribution. Having to choose a value, we just opt arbitrarily for $\kappa=3$, corresponding to the two solid lines of the figure, although the conclusions that follow from this choice can be easily rescaled at wish. Moreover, as we can see from Figs. - (and as it results from the approximated formulae)

• the standard deviation of the distributions varies smoothly with $p$;
• the mean value depends linearly on $p$ for obvious reasons.⁴²

Therefore the resolution power in the interval $[p,p+\Delta p]$ can be evaluated by a simple proportion

$${\cal R}(p,p+\Delta p) \approx \frac{\Delta p} {\left.\mbox{E}(f_P)\right\vert _{p+\Delta p}-\le... ...left.\sigma(f_P)\right\vert _{p+\Delta p/2}\,. % \label{eq:resolution_power_0}$$ (77)

For example, using the numbers of the Monte Carlo evaluations shown in Fig. , for $p=0.1$ and $n_s=300$ we get $0.1 / (0.288-0.201) \times 3 \times 0.0305 = 0.105$, reaching at best $\approx 0.022$ in the case of $n_s=10000$ shown in Fig. . The resolution power at a given value of $p$, is obtained in the limit ` $\Delta p\rightarrow 0$':

$${\cal R}(p) \approx \frac{\Delta p} {\left.\mbox{E}(f_P)\right\vert _{p+\Delta p}-\le... ...\cdot \left.\sigma(f_P)\right\vert _p\, \hspace{0.8cm}(\Delta p \rightarrow 0).$$ (78)

The bottom plot of Fig.  shows the variation of the resolution power in $p$ for the same values of $n_s$ of Figs. - and for the usual cases of $\pi_{1}$ and $\pi_{2}$ of those figures (in the order: solid, dashed and dotted line - the lines are drawn just to guide the eye and to easily identify the conditions). The resolution power has been evaluated using the approximated formulae, for $\kappa=3$, around $p=0.1$ (blue filled circles) and around $p=0.5$ (cyan open circles), using for the gradient $\Delta p=0.01$ (the exact value is irrelevant for the numerical evaluation, provided it is small enough). $[$Obviously, if one prefers a different value of $\kappa$ (in particular one might like $\kappa=1$), then one just needs to rescale the results.$]$