Counting and confusion: Bayesian rate estimation with multiple populations

We show how to obtain a Bayesian estimate of the rates or numbers of signal and background events from a set of events when the shapes of the signal and background distributions are known, can be estimated, or approximated; our method works well even if the foreground and background event distributions overlap significantly and the nature of any individual event cannot be determined with any certainty. We give examples of determining the rates of gravitational-wave events in the presence of background triggers from a template bank when noise parameters are known and/or can be fit from the trigger data. We also give an example of determining globular-cluster shape, location, and density from an observation of a stellar field that contains a nonuniform background density of stars superimposed on the cluster stars.

Figure 1

The mean foreground rate uncertainty, Eq. (45), as a function of threshold for data sets with ${\mathrm{\Gamma }}_f=100$ (solid line), ${\mathrm{\Gamma }}_f=10$ (dashed line), and ${\mathrm{\Gamma }}_f=1$ (dash-dotted line). Recall that ${\mathrm{\Gamma }}_f$ is the mean number of foreground events above $x_{\mathrm{th}}=x_1=8$. The total background rate, $R_b(x_{\mathrm{th}})$, is shown by the dotted line; we fix $R_b(x_{\mathrm{th}}=x_1=8)=1$, so on average there is one background event above $x=8$. For $x_{\mathrm{th}}\gtrsim x_1$, increasing the threshold tends to increase the foreground rate uncertainty because the rate is foreground dominated and fewer events are included in the data set. For $x_{\mathrm{th}}\lesssim x_1$, the background rate dominates at small loudness, and the foreground rate uncertainty asymptotes to the counting error on the events that stand out from the background, $\mathrm{\Delta }{R}_f/R_f\simeq 1/\sqrt{{\mathrm{\Gamma }}_f}$.

Figure 2

The cumulative distribution of the ranking statistics for the synthetic data used to test the formalism on the model from Sec. 5a. The solid line gives the cumulative distribution of the synthetic data; the dashed line gives the theoretical cumulative distribution for the models in Eqs. (56) and (57) combined with $R_f=10.4$ and $R_b=95.1$.

Figure 3

The marginalized posterior densities for $R_f$ (solid line) and $R_b$ (dashed line) for the analytic model discussed in Sec. 5a. The vertical lines indicate the “true” values used to generate the synthetic data set. Both the true foreground and background rates lie well within the probability envelope for $R_f$ and $R_b$.

Figure 4

Foreground probability for each event in the synthetic data set of Sec. 5a marginalized over all other parameters. True foreground events are in dark grey, background events in light grey. Even though our method cannot identify the status of most events with confidence, it can still correctly estimate the rates (Fig. 3).

Figure 5

Posteriors on foreground rate obtained using the method described in this paper, the loudest-event statistic and the foreground-dominated analysis for the data set from Sec. 5a. For the loudest-event statistic, we present the posterior with and without an upper limit on the background rate, $R_b$; in both cases the rate posterior is significantly wider than the one obtained with the method described in this paper. For the foreground-dominated statistic, the limits $x_{\min }=3.82$ and $x_{\min }=4.07$ give likelihood ratios of $\widehat{f}/\widehat{b}=0.5$ and 0.99. For this data set, the thresholds in fact include 19 and 7 background events, respectively, so the corresponding rate estimates are significantly biased. An omniscient threshold of $x_{\min }=4.38$ would produce exactly 7 foreground and zero background events, and the resulting posterior is essentially indistinguishable from the curve for the full analysis.

Figure 6

The foreground (solid lines) and background (dashed lines) rate posterior, marginalized over all flags and the $N_{\mathrm{eff}}$ parameter, for the gravitational-wave template detection scenario with overlapping templates discussed in Sec. 5b. The true values of the rates, $R_f=10.4$ and $R_b=95.1$, are indicated with vertical lines. The distributions are not significantly wider than those of Fig. 3, in spite of the extra parameter.

Figure 7

The posterior on the number of effective templates, $N_{\mathrm{eff}}$, for the model and data discussed in Sec. 5b, marginalized over all state flags and rates. The true value, $N_{\mathrm{eff}}=1000$, is indicated by the vertical line.

Figure 8

Density contours and synthetic data for the example in Sec. 5d. The contours describe the true density profile with the parameters in Eq. (62). The points are the realization of this density profile used as synthetic data in Sec. 5d; the dashed line encloses one Plummer scale radius about the true cluster center. Because the peak cluster density is equal to the background density at the cluster center, the cluster is barely apparent to the eye.

Figure 9

Contours of the posterior probability distribution for the center of the cluster, ${\overrightarrow{x}}_0$, for the example from Sec. 5d. The center $(x,y)=(x_0,y_0)$ is determined to within about 5% of the structural radius of the cluster, $r_0$ [see Eq. (62)].

Figure 10

Posterior density for the scale parameter for the cluster, $r_0$, for the example from Sec. 5d. The true value is indicated by the vertical line [see Eq. (62)].

Figure 11

Posterior densities for the number of stars in the cluster ($R_f$) and in the field ($R_b$) in the example from Sec. 5d. Vertical lines indicate the true values [see Eq. (62)].