Neural posterior estimation with guaranteed exact coverage: The ringdown of GW150914

We analyze the ringdown phase of the first detected black-hole merger, GW150914, using a simulation-based inference pipeline based on masked autoregressive flows. We obtain approximate marginal posterior distributions for the ringdown parameters, namely the mass, spin, and the amplitude and phases of the dominant mode and its first overtone. Thanks to the locally amortized nature of our method, we are able to calibrate our posteriors with injected simulations, producing posterior regions with guaranteed (i.e., exact) frequentist coverage of the true values. For GW150914, our calibrated posteriors provide only mild evidence ($\sim 2\sigma$) for the presence of an overtone, even if the ringdown is assumed to start at the peak of the amplitude.

Figure 1

Results for the GW150914 ringdown analysis. Left panels: marginal posterior 68%, 95% regions from the uncalibrated simulation-based analysis (solid/black) and calibrated regions with guaranteed exact coverage (68% orange, 95% yellow) for mass and spin (top panel) and QNM amplitudes (bottom panel). The red/dot-dashed contours are from a likelihood-based analysis. Right panels: maps of required credibility (for 68% guaranteed coverage) used to obtain the calibrated regions on the left. The black contours are as in the left panel, for reference.

Figure 2

68% and 95% marginal posteriors for the GW150914 ringdown analysis, from the uncalibrated simulation-based analysis (solid/black) and the likelihood-based analysis (red/dashed).