= Inferring a binary from a detector network
For detector $k$, model the calibrated data as $d_k(t)=h_k(t;\theta)+n_k(t)$, with a shared source parameter vector $\theta$, separate detector responses, propagation delays and <noise> $n_k$. Under an independent stationary <Gaussian process> approximation with one-sided <power spectral density> $S_k(f)$, define the noise-weighted <inner product> and <likelihood function>
$$
(a|b)_k=4\operatorname{Re}\int_{f_{\min}}^{f_{\max}}\frac{\widetilde a(f)^*\widetilde b(f)}{S_k(f)}\,df,\qquad
\log\mathcal L(\theta)=-\frac12\sum_k(d_k-h_k|d_k-h_k)_k+\mathrm{constant}.
$$
Strain <power spectral density> has units $\mathrm{Hz}^{-1}$; the displayed <inner product> is dimensionless when the <Fourier transform> integrates over seconds. A <matched filter> rewards agreement over the full evolving <wave phase>, weighting frequency bands by their noise. One conspicuous peak is insufficient: a candidate should also have the right distribution of signal across frequency bands and compatible projections in the network.
The parameter paper used coherent <Bayesian inference>, <numerical relativity>-informed inspiral–merger–ringdown models and explicit calibration uncertainties. It found a strong distance–inclination correlation; source-frame <masses> additionally require converting the detector-frame masses by a cosmological <redshift>.<Properties of the Binary Black Hole Merger GW150914 by B. P. Abbott and others>[II Parameter estimation, equations (1)–(5), II.A waveform models; III Results, p. 7, Fig. 2]{ref} Conceptually, multiply the <likelihood function> by a <prior distribution> to obtain a <posterior distribution>. Changes in waveform families, residual structure after subtraction, and checks with a generic transient reconstruction test parts of the modeling assumptions. Detection significance and parameter uncertainty answer different questions.
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