Appendix E. Selection Effects

In this section, we discuss how to carry out inference while taking into account selection effects, which arise from the fact that some events are easier to detect than others. We loosely follow the arguments from Abbott et al. (Reference Abbott2016); however, see also Mandel et al. (Reference Mandel, Farr and Gair2018); Fishbach et al. (Reference Fishbach, Holz and Farr2018).

Some gravitational-wave events are easier to detect than others. All else equal, it is easier to detect binaries if they are closer, higher mass (at least, up until the point that they start to go out of the observing band), and with face-on/off inclination angles. More subtle selection effects arise due to black hole spin (see, e.g., Ng et al., Reference Ng, Vitale, Zimmerman, Chatziioannou, Gerosa and Haster2018). Typically, a gravitational-wave event is said to have been detected if it is observed with a matched-filter signal-to-noise ratio—maximized over extrinsic parameters $\theta_{\text{extrinsic}}$ —above some threshold $\rho_{\text{th}}$

(1) \begin{align} \rho^{\prime}_{\text{mf}} \equiv \max_{\theta_{\text{extrinsic}}} \left(\rho_{\text{mf}}\right) > \rho_{\text{th}} .\end{align}

Usually, $\rho_{\text{th}}=8$ for a single detector or $\rho_{\text{th}}=12$ for a $\geq2$ detector network.

Selection effects are characterised by $p_{\text{det}}$ , the probability that a signal exceeds the detection threshold. There are different ways to calculate $p_{\text{det}}$ in practice. The probability density function for $\rho_{\text{mf}}$ given $\theta$ —the distribution of $\rho_{\text{mf}}$ arising from random noise fluctuations—is a normal distribution with mean $\rho_{\text{opt}}$ and unit variance

(2) \begin{align} p(\rho^{\prime}_{\text{mf}}|\theta) = \frac{1}{2\pi} \exp\left(-\frac{1}{2}\Big(\rho^{\prime}_{\text{mf}}-\rho_{\text{opt}}(\theta)\Big)^2\right) ,\end{align}

see Fig. 1. Thus,

(3) \begin{align} p_{\text{det}}(\theta) = & \int_{\rho_{\text{th}}}^\infty dx \frac{1}{\sqrt{2\pi}} \exp\left({-\frac{1}{2}\Big(x-\rho_{\text{opt}}(\theta)\Big)^2}\right) \end{align}

(4) \begin{align} = & \frac{1}{2}\,\text{erfc}\left(\frac{\rho_{\text{th}}-\rho_{\text{opt}}(\theta)}{\sqrt{2}}\right) .\\[10pt]\nonumber\end{align}

Alternatively, one may express $p_{\text{det}}$ as the ratio of the “visible volume” $\mathcal{V}(\theta)$ to the total spacetime volume $\mathcal{V}_{\text{tot}}$

(5) \begin{align} p_{\text{det}}(\theta) = \frac{\mathcal{V}(\theta)}{\mathcal{V}_{\text{tot}}} .\end{align}

The visible volume is typically calculated numerically with injected signals.

Figure 1.

The distribution of matched filter signal-to-noise ratio maximized over phase for the same template in many noise realisations (blue). The distribution peaks at $\rho_{\text{opt}}=7.6$ (dashed black). The theoretical distribution (Eq. 2) is shown in orange.

Given a population of N events,

(6) \begin{align}\mathcal{L}(d,N|\Lambda,\text{det}) = \frac{1}{p_{\text{det}}(\Lambda|N)} \mathcal{L}(d,N|\Lambda,R) .\end{align}

In analogy to Eq. 5, the $p_{\text{det}}$ normalization factor can be calculated using the visible volume as a function of the hyper-parameters $\Lambda$

(7) \begin{align} \mathcal{V}(\Lambda) \equiv \int d\theta \mathcal{V}(\Lambda) \pi(\theta|\Lambda) .\end{align}

Naively, one might expect that

(8) \begin{align} p_{\text{det}}(\Lambda|N) = \left(\frac{\mathcal{V}(\Lambda)}{\mathcal{V}_{\text{tot}}}\right)^N ,\end{align}

but this expression is incorrect because it does not marginalize over the Poisson-distributed rate, which ends up changing the answer. Marginalizing over the rate, we obtain

(9) \begin{align} p_{\text{det}}(\Lambda|N) = & \int dR \left(\frac{\mathcal{V}(\Lambda)}{\mathcal{V}_{\text{tot}}}\right)^N \pi(N|R) \pi(R) \nonumber\\ = & \int dR \left(\frac{\mathcal{V}(\Lambda)}{\mathcal{V}_{\text{tot}}}\right)^N \left[e^{-R \mathcal{V}(\Lambda)} \frac{\mathcal{V}(\Lambda)^N R^N}{N!} \right] \pi(R) \nonumber\\ = & \left(\frac{\mathcal{V}(\Lambda)}{\mathcal{V}_{\text{tot}}}\right)^N \left[ \int dR \, e^{-R \mathcal{V}(\Lambda)} \frac{\mathcal{V}(\Lambda)^N R^N}{N!}\right] \pi(R) .\end{align}

Note that $p_{\text{det}}$ depends on our prior for the rate R. If we choose a uniform-in-log prior $\pi(R)\propto1/R$ , we obtain

(10) \begin{align} p_{\text{det}}(\Lambda|N) \propto & \left(\frac{\mathcal{V}(\Lambda)}{\mathcal{V}_{\text{tot}}}\right)^N ,\end{align}

which reproduces the results from Abbott et al. (Reference Abbott2018). Note that

(11) \begin{align} \mathcal{L}(d|\Lambda,\text{det}) \neq \int d\theta \mathcal{L}(d|\theta,\text{det}) \, \pi(\theta | \Lambda) .\end{align}