Relative Binning in Gravitational-Wave Inference

• Relative binning is a method that approximates gravitational-wave likelihoods by compressing full-resolution frequency data into coarse frequency bins using a nearby fiducial waveform.
• The technique leverages the smooth variation of the waveform ratio to reduce expensive inner-product computations, achieving speedups up to 10^4 compared to full FFT grid evaluations.
• Its effectiveness is demonstrated in GW170817 analyses and extended to handle precession, higher harmonics, and lensed signals, ensuring reliable and efficient posterior inference.

Relative binning is a fast likelihood-evaluation method for gravitational-wave parameter estimation that replaces repeated full-resolution frequency-domain inner products by a compressed calculation around a fiducial waveform. Its central premise is local: during Bayesian inference, waveforms with appreciable posterior weight are already close to the best fit, so the ratio of a candidate waveform to a suitably chosen reference waveform varies smoothly with frequency even when the waveform itself is rapidly oscillatory. By exploiting that smoothness in coarse frequency bins, relative binning makes reliable posterior inference and evidence calculation practical for long compact-binary signals, including binary neutron-star inspirals such as GW170817, using very moderate computational resources (Zackay et al., 2018, Dai et al., 2018).

1. Concept and computational setting

In standard gravitational-wave parameter estimation, detector strain data are modeled in the frequency domain, and Bayesian inference requires repeated evaluation of the Gaussian-noise likelihood for many trial waveforms. The dominant cost is the repeated computation of noise-weighted inner products over the full Fourier grid. This is especially severe for long signals such as binary neutron-star inspirals, for which long data stretches, high sampling rates, and many thousands of in-band cycles produce very large frequency arrays. For a GW170817-like analysis with $T=2048$ s and a sampling rate of $4096$ Hz, the full grid contains of order $\sim 10^7$ frequency-domain points; for GW170817 specifically, the signal spends roughly $\sim 4000$ cycles in band from $23$ Hz upward (Zackay et al., 2018, Dai et al., 2018).

Relative binning addresses that bottleneck by comparing each candidate waveform $h(f)$ to a fiducial waveform $h_0(f)$ chosen near the high-likelihood region. The rapidly varying phase of the waveform itself is not interpolated directly. Instead, the method works with the relative waveform

$r(f)\equiv \frac{h(f)}{h_0(f)},$

whose amplitude and phase corrections vary much more slowly with frequency when $h_0$ is close to the best fit. Frequency space is then partitioned into coarse bins, and $r(f)$ is approximated inside each bin by a low-order function, typically linear in $4096$0. The dense full-grid calculation is thereby replaced by a sum over a much smaller number of bins.

The method is inherently event-specific rather than global. It does not attempt to represent an entire waveform family over a broad parameter domain. Instead, it compresses the likelihood locally around a fiducial waveform for a particular signal. This locality is the reason it can be much more compact than methods that seek a broad reduced representation.

2. Likelihood formalism and compressed overlaps

The accelerated likelihood is the standard stationary Gaussian-noise frequency-domain likelihood. Using the overlap notation of the original method,

$4096$1

with $4096$2 the detector data, $4096$3 the model waveform, $4096$4 the one-sided noise power spectral density, and $4096$5 the analyzed duration, the log likelihood can be written as

$4096$6

This is equivalent to the standard noise-weighted inner product formulation

$4096$7

or, in continuum form,

$4096$8

The parameter-dependent cost is therefore concentrated in $4096$9 and $\sim 10^7$0 (Zackay et al., 2018).

Relative binning rewrites the waveform as

$\sim 10^7$1

and approximates the ratio within each bin $\sim 10^7$2 by

$\sim 10^7$3

where $\sim 10^7$4 is a representative bin frequency. The full-resolution data and fiducial waveform are then compressed once into per-bin summary data,

$\sim 10^7$5

$\sim 10^7$6

With these summary data, the overlaps are approximated by short bin sums: $\sim 10^7$7

$\sim 10^7$8

Operationally, the expensive part is moved into preprocessing. For each trial waveform, one evaluates the waveform ratio only at sparse frequencies, reconstructs $\sim 10^7$9 and $\sim 4000$0, and combines them with the stored summary data. The approximation therefore changes the cost structure from full-grid matched-filter-like sums to a small number of per-bin contractions. The same overlap structure also supports analytic maximization or marginalization over constant phase and overall amplitude normalization, because both real and imaginary parts of $\sim 4000$1 remain available.

3. Bin construction, accuracy control, and computational scaling

The accuracy of relative binning depends on the smoothness of the relative waveform inside each bin. The original binning rule is phase-based: the waveform phase is written as a sum of power laws,

$\sim 4000$2

with examples including chirp mass ($\sim 4000$3), symmetric mass ratio ($\sim 4000$4), leading spin effects ($\sim 4000$5), tidal effects ($\sim 4000$6), and coalescence time shift ($\sim 4000$7). A parameterized bound on coefficient variation then defines a worst-case cumulative phase perturbation, and bins are chosen so that the maximum differential phase change across each bin remains below a tolerance $\sim 4000$8. In this construction, $\sim 4000$9 controls how broad a region of parameter space should be covered around the fiducial waveform, while $23$0 controls the interpolation accuracy inside each bin (Zackay et al., 2018).

The original paper also states that likelihood error behaves approximately as

$23$1

with target $23$2. For a GW170817-like binary neutron star signal, using $23$3 and $23$4 rad, about $23$5 to $23$6 bins suffice to achieve $23$7. The application paper for GW170817 uses an explicit empirical target

$23$8

and reports that about $23$9 frequency bins satisfy that bound; increasing the number of bins from about $h(f)$0 to about $h(f)$1 leaves the posteriors unchanged (Dai et al., 2018).

The resulting dimensionality reduction is large. For long inspirals, the effective number of frequency-domain evaluations is reduced from the full grid to a number comparable to the number of bins. The original comparison is:

Method	Required waveform evaluations
Relative binning	$h(f)$2
Reduced order quadrature	$h(f)$3
Multi-band interpolation	$h(f)$4
Full FFT grid	$h(f)$5

For frequency-domain waveform models, the reported speedups are $h(f)$6 relative to naive matched filtering on the full FFT grid and $h(f)$7 relative to reduced order quadrature (Zackay et al., 2018). The same work notes that if more accuracy is needed, one may use more bins or increase the interpolation order beyond linear. Empirically, if the phase error within each bin is below about one radian, increasing polynomial order is preferable; if it is larger, one should use more bins.

4. GW170817 as the first full application

The first major application of relative binning was parameter estimation for GW170817 using publicly available LIGO data. The analysis used $h(f)$8 s of data from each LIGO detector, sampled at $h(f)$9 Hz, with the likelihood restricted to $h_0(f)$0. The waveform model was IMRPhenomD_NRTidal from LALSuite; intrinsic parameters included detector-frame chirp mass $h_0(f)$1, symmetric mass ratio $h_0(f)$2, aligned spins $h_0(f)$3, and tidal deformabilities $h_0(f)$4, while in-plane spin effects were neglected. The implementation analytically maximized over detector amplitudes and phases and treated detector extrinsic parameters independently to reduce effective dimensionality (Dai et al., 2018).

The workflow was iterative but simple. A fiducial waveform was first chosen from crude estimates, summary data were computed, the relative-binning likelihood was maximized, the fiducial waveform was updated to improved best-fit parameters, and the summary data were recomputed. In practice, one update was sufficient. Using about $h_0(f)$5 bins, the method supported both emcee Markov-chain Monte Carlo and pyMultiNest nested sampling, with statistically similar posteriors. The emcee run collected about $h_0(f)$6 samples, of which $h_0(f)$7 were independent, within about $h_0(f)$8; pyMultiNest produced about $h_0(f)$9 samples using even fewer computational resources than the MCMC run. Posterior samples and a tutorial Python implementation were released publicly at https://bitbucket.org/dailiang8/gwbinning/ (Dai et al., 2018).

Methodological validation and astrophysical conclusions were separated clearly in that analysis. The posteriors were reported to be in good agreement with Abbott et al. for GW170817 and the emcee and MultiNest posteriors were in excellent agreement, supporting the robustness of the compressed likelihood. Scientifically, the results did not favor non-zero aligned spins at a statistically significant level, with

$r(f)\equiv \frac{h(f)}{h_0(f)},$0

and showed no significant evidence for non-zero tidal deformability as quantified by the Bayesian evidence, whether or not high-spin or low-spin priors were adopted. For $r(f)\equiv \frac{h(f)}{h_0(f)},$1 Hz the posterior in $r(f)\equiv \frac{h(f)}{h_0(f)},$2 was bimodal, with a major peak near $r(f)\equiv \frac{h(f)}{h_0(f)},$3 and a minor peak near $r(f)\equiv \frac{h(f)}{h_0(f)},$4; for $r(f)\equiv \frac{h(f)}{h_0(f)},$5 Hz it became singly peaked near $r(f)\equiv \frac{h(f)}{h_0(f)},$6. The paper emphasized that structure above $r(f)\equiv \frac{h(f)}{h_0(f)},$7 Hz should not be overinterpreted because the contribution to the log likelihood from that range was only about $r(f)\equiv \frac{h(f)}{h_0(f)},$8, too small to cleanly distinguish real signal information from random noise fluctuation (Dai et al., 2018).

5. Extensions to precession, higher harmonics, software frameworks, lensing, and 3G detectors

Subsequent work generalized relative binning beyond the dominant-mode, non-precessing setting. The central difficulty is that for waveforms with spin-orbit precession and multiple radiation harmonics, the ratio of the full detector strain to a fiducial strain becomes jagged because it is the ratio of a sum of interfering terms. "Mode-by-mode Relative Binning" (MRB) addresses this by decomposing the strain into mode contributions and interpolating smoother mode-level quantities instead of the rough total-waveform ratio. In benchmarks on IMRPhenomXPHM, the original relative binning setup in those comparisons used $r(f)\equiv \frac{h(f)}{h_0(f)},$9 bins, while MRB reached comparable or better likelihood accuracy with $h_0$0–$h_0$1 bins depending on the case; the headline claim is up to an order-of-magnitude reduction in the number of waveform model calls per frequency compared to the previously used relative binning scheme, for the same tolerable absolute log-likelihood accuracy of order $h_0$2 (Leslie et al., 2021).

A separate development integrated relative binning into Bilby through the class RelativeBinningGravitationalWaveTransient, together with source models lal_binary_neutron_star_relative_binning and lal_binary_black_hole_relative_binning. That implementation includes a fully coherent network statistic, extrinsic parameters such as distance, sky location, inclination, and polarization, and phase and luminosity-distance marginalization. Its bin construction uses five power-law indices $h_0$3, and analyses in that paper commonly used $h_0$4 rad and $h_0$5, producing about $h_0$6 bins. A flagship demonstration reproduced GW170817 in about $h_0$7 hours on a single-core CPU, using 256 s of strain data from Livingston, Hanford, and Virgo, IMRPhenomPv2_NRTidal, Dynesty sampling, and a refined fiducial waveform; the maximum Jensen–Shannon divergence against GWTC-1/LALInference samples was $h_0$8, and a p-p test on $h_0$9 binary-neutron-star injections in Gaussian noise showed no significant inference bias and good posterior calibration (Krishna et al., 2023).

Relative binning has also been extended to complete binary-black-hole inference with precession and higher-order modes, and then to joint inference for strongly lensed gravitational-wave signals. In that framework, the detector-frame waveform is decomposed mode-by-mode, summary data are generalized to mode and cross-mode quantities $r(f)$0, $r(f)$1, $r(f)$2, and $r(f)$3, and the same event-specific compression is used for single-event and lensed analyses. For two lensed images, the parameter space increases to $r(f)$4 parameters, with relative lensing parameters $r(f)$5, $r(f)$6, and $r(f)$7. The implementation is optimized with Numba, integrated with Bilby, uses Dynesty for relative-binning and regular single-event analyses and Golum for regular lensed joint analyses, and employs IMRPhenomXPHM with modes $r(f)$8. Reported speed-ups are about $r(f)$9 times faster than current techniques for binary black holes with total mass larger than $4096$00, about a factor of $4096$01 on average for lower masses, up to $4096$02 for LIGO-Virgo analyses, and up to $4096$03 for third-generation analyses, while recovered posteriors match those found with traditional techniques (Narola et al., 2023).

These extensions preserve the original conceptual distinction of relative binning: they remain local compressions around a fiducial waveform rather than universal reduced models over the full parameter space. The extensions mainly change what is interpolated. Standard relative binning interpolates the ratio of total strain waveforms; MRB and higher-mode/precession formulations interpolate smoother mode-resolved structures before recombining them in the likelihood.

6. Scope, assumptions, limitations, and common misunderstandings

Relative binning works best when there exists a fiducial waveform close to the region of high posterior probability and when the trial waveforms of interest are sufficiently probable, meaning sufficiently close to the fiducial waveform that the ratio $4096$04 remains smooth and accurately representable in the chosen bins. This is why the original application updated the fiducial waveform after an initial maximization step, and why the Bilby implementation supports fiducial refinement via optimization. If the fiducial waveform is poor, if the sampler explores parameter regions far from the reference point, or if the waveform develops sharp frequency-domain features unresolved by the bins, the approximation can degrade (Zackay et al., 2018, Krishna et al., 2023).

A common misunderstanding is to treat relative binning as a universally accurate reduced representation of an entire waveform family. The literature does not support that interpretation. The method is a local approximation around a fiducial waveform, and its efficiency depends on that localization. This suggests that very broad exploratory searches, multimodal posteriors, or unusual waveform morphologies may require more conservative binning, multiple fiducials, or specialized variants such as mode-by-mode decompositions. The original papers also note that higher modes require separate binning for each mode because different modes oscillate strongly relative to one another, and that extensions to time-domain models are possible only with complications (Zackay et al., 2018, Leslie et al., 2021).

Another misunderstanding is to equate accurate likelihood compression with the creation of new physical information. The GW170817 application explicitly cautioned against overinterpreting posterior structure in $4096$05 above $4096$06 Hz because the signal-to-noise there is negligible owing to rapidly rising detector noise. Relative binning can evaluate the likelihood accurately, but it cannot create information absent from the data (Dai et al., 2018).

Practical limitations also remain in software realizations. The Bilby implementation concludes that time marginalization is not practically useful in this framework because the available routes either lose the $4096$07 savings or incur large $4096$08 precomputation overhead. In higher-mode and precessing analyses, implementation complexity increases because mode-resolved waveform data and twisting ingredients are required. In third-generation scenarios, longer signal duration generally increases the benefit of relative binning, but very high SNR can demand denser binnings and partially reduce the gain; one reported $4096$09 injection with network SNR $4096$10 yielded a speed-up below 1 (Krishna et al., 2023, Narola et al., 2023).

Within those limits, relative binning has become a family of likelihood-compression methods for gravitational-wave inference. Its defining structure remains unchanged across variants: choose a fiducial waveform near the likelihood peak, precompute summary data on the full frequency grid, approximate candidate-to-fiducial ratios on coarse bins, and evaluate the likelihood through compressed overlaps. The method’s significance lies in turning event-specific local smoothness into large computational savings without changing the statistical target of the analysis.