Signal discovery in sparse spectra: A Bayesian analysis

A Bayesian analysis of the probability of a signal in the presence of background is developed, and criteria are proposed for claiming evidence for, or the discovery of a signal. The method is general and, in particular, applicable to sparsely populated spectra. Monte Carlo techniques to evaluate the sensitivity of an experiment are described. As an example, the method is used to calculate the sensitivity of the GERDA experiment to neutrinoless double beta decay.

Figure 1

Analysis chain. The Monte Carlo generator (MC) generates a pool which consists of signal and background events. An ensemble is defined as a set of events representing a possible outcome of an experiment. The numbers of events are randomly chosen according to the parameters $S_0$ and $B_0$. From each ensemble a spectrum is extracted and subsequently analyzed. The probability $p(H_1|\mathrm{\text{data}})$ for each spectrum is depicted here as the outcome of the analysis.

Figure 2

The spectrum (top, left) was randomly generated as described in the text. The signal events are indicated by the open histogram, the background events by the shaded histogram. The probability density for $S$ (top, right) from the same spectrum is shown. The distribution of $S^{*}$ (bottom, left) as well as the distribution of the $\log p(H_1|\mathrm{\text{data}})$ (bottom, right) are calculated from ensembles generated under the same assumptions.

Figure 3

The spectrum (top, left) was randomly generated as described in the text. No signal events are present in the spectrum. The probability density for $S$ for the same spectrum (top, right) is shown. The distribution of the limit (90% probability) of the signal contribution (bottom, left) as well as the distribution of the $p(H_1|\mathrm{\text{data}})$ (bottom, right) are calculated from ensembles generated under the same assumptions.

Figure 4

The upper plot shows the expected 90% probability lower limit on the half-life for neutrinoless double beta decay versus the exposure under different background conditions. Also shown is the half-life for the claimed observation [8]. The lower plot shows the expected 90% probability upper limit on the effective Majorana neutrino mass versus the exposure. The effective Majorana neutrino mass for the claimed observation is also shown. The mass values were determined from the half-life using the matrix element reported in 11.

Figure 5

Top: the half-life for which 50% of the experiments would report a discovery is plotted versus the exposure under different background conditions. Bottom: the effective Majorana neutrino mass for which 50% of the experiments would report a discovery versus the exposure. The mass was determined from the half-life using the matrix element reported in 11.