Gravitational-wave selection effects using neural-network classifiers

We present a novel machine-learning approach to estimate selection effects in gravitational-wave observations. Using techniques similar to those commonly employed in image classification and pattern recognition, we train a series of neural-network classifiers to predict the LIGO/Virgo detectability of gravitational-wave signals from compact-binary mergers. We include the effect of spin precession, higher-order modes, and multiple detectors and show that their omission, as it is common in large population studies, tends to overestimate the inferred merger rate in selected regions of the parameter space. Although here we train our classifiers using a simple signal-to-noise ratio threshold, our approach is ready to be used in conjunction with full pipeline injections, thus paving the way toward including actual distributions of astrophysical and noise triggers into gravitational-wave population analyses.

Figure 1

Comparison between the semianalytic and neural-network detectability. The black dashed line shows the semianalytic estimate of $p_{\det }$ obtained from the cumulative distribution function of the projection parameter $\omega$, evaluated at $\omega ={\rho }_{\mathrm{thr}}/{\rho }_{S\mathrm{opt}}$ with ${\rho }_{\mathrm{thr}}=8$. Trivially, one has $p_{\det }=0$ for $\omega >1$. The blue circles mark the corresponding estimate obtained by numerically marginalizing over the prediction of our nonspinning neural network evaluated on the validation set. We use $N_{\mathrm{MC}}={10}^4$ Monte Carlo samples of $p(\lambda )$. A long tail at large values of $\omega$ is omitted for clarity.

Figure 2

Neural-network performances during the training process. Solid (dashed) lines indicate the accuracy evaluated over the validation (training) data set. Colors indicate three different neural networks trained considering nonspinning binaries, dominant emission mode, and single detector (blue); precessing binaries, higher-order modes, and single detector (orange); precessing binaries, higher-order modes, and three detectors (green).

Figure 3

Probability distribution of the difference between the detectability $p_{\det }$ estimated using our neural networks and the semianalytic calculation. The latter is only justified in the case of nonprecessing sources and a single detector (blue histogram). In the other three cases, the assumptions underlying the analytic calculation are not valid. Here “validation” (blue and orange histograms) refers to the parameter distributions used in the nominal validation process (cf. Fig. 2), while powerlaw and log-uniform (green and red histograms) refer to two test populations similar to those used in Refs. [1, 2] to estimate BH merger rates.

Figure 4

Deviation between the numerical and semianalytical estimates of $p_{\det }$ as a function of redshifted mass $M_z$ and mass ratio $q$. This figure is produced using the validation sample of our model trained on precessing systems, higher harmonics, and the full LIGO/Virgo network. For large values of $M_z$ and low values of $q$, we predict, on average, larger value of $p_{\det }$ compared to the semianalytic estimate.