Inferring $p_s$

If priors are uniform then, Eq. (41) becomes

$$\hspace{-6.0mm}f(p_s\,\vert\,x,\,n,\,\lambda_,\,p_b) \propto \sum_{n_s,\,n_b} f_{2{\cal B}}(x\,\vert\,n_s,\,p_s\,n_b,\,p_b) \, f(n_b\,\vert\,{\cal P}_{\lambda_b})\,\delta_{n,\,n_s+n_b}\,.$$ (44)

Figure 10 gives the result for $x=9$, $n=12$, and assuming several hypothesis for $\lambda_b$ and $p_b$.

Figure: Inference about $p_s$ for $n=12$ and $x=9$, depending on the expected background [$\lambda _b=0$, 1, 2, 4, 6, 8, 10, 14, as (possibly) indicated by the number above the lines]. The three plots are obtained by three different hypotheses of $p_b$.

• The upper plot is for $p_b=0.75$, equal to $x/n$. The curves are for $\lambda_b=0,$ 1, 2, 4, 6, 8, 10, 12 and 14, with the order indicated (whenever possible) in the figure. If the expected background is null, we recover the simple result we already know. As the expected background increases, $f(p_s)$ gets broader, because the inference is based on a smaller number of objects attributed to the signals and because we are uncertain on the number of events actually due the background. In a very noisy environments ( $\lambda_b \approx n$, or even larger), the data provide very little information about $p_s$ and, essentially, the prior pdf (dashed curve) is recovered. Note also that for all values of $\lambda_b$ the posterior $f(p_s)$ is peaked at $x/n=0.75$. This is due to the fact that $p_b$ was equal to the observed ratio $x/n$, therefore, for any hypothesis of $n_b$ attributed to the background, $x_b=p_b \,n_b$ counts are in average `subtracted' from $x$ (this is properly done in an automatic way in the Bayes formula, followed by marginalization).
• The situation gets more interesting when $p_b$ differs from $x/n$.

  The middle plot in the figure is for $p_b=0.25$. Again, the case $\lambda _b=0$ gives the the pdf we already know. But as soon as some background is hypothesized, the curves start to drift to the right side. That is because high background with low $p_b$ favors large values of $p_s$.

  The opposite happens if we think that background is characterized by large $p_b$, as shown in the bottom plot of the figure.