Relative Binning — Implementation Status

Companion to the design note. This page tracks what has been implemented, tested, committed, and pushed for the Fourier- and time-domain relative-binning (heterodyned) likelihoods, and what remains.

All code lives in jaxpe/gw/likelihood/; every stage was committed only after its unit tests and its performance-comparison test passed, and pushed to origin/master.

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Stage ledger

Phase	Commit	Content	Validation	Speedup (compile excluded)
RB-1 FD dominant	4f4cb40	RelativeBinningFDLikelihood (detector network)	exact-at-fiducial; parity vs FDNetworkLikelihood	~12× vs full FD
RB-2 FD higher modes	96b26fe	RelativeBinningFDLikelihoodHM (per-mode/pair, diagonal covariance)	parity vs dense multi-mode FD (fd_dense_loglikelihood_modes)	—
RB-3 Toeplitz infra	2f034bc	toeplitz.py: ACF, matvec, Gohberg–Semencul C⁻¹v (JAX)	vs dense to 1e‑9…1e‑11; round-trip; jit+vmap	~3300× vs dense solve
RB-4 TD dominant	15269dd	RelativeBinningTDLikelihood + dense reference	exact-at-fiducial; parity (intrinsic/extrinsic/noisy)	~370× vs dense
RB-4 TD higher modes	3406365	RelativeBinningTDLikelihoodHM (Appendix-A cross-mode tensors)	parity; single-mode HM == dominant (<1e-9)	~74× (2 modes)
RB-4 TD network	007e37b	RelativeBinningTDNetwork (block-diagonal sum)	vs summed dense	—
RB-4 t_c	28e1b79	edge_times + shifted-sampling support	parity vs dense at shifted t_c	—
RB-5 posterior-level	8705cab	grid posterior-recovery validation	JS divergence < 1e-3, matching mean/width, peaks at truth	—
design note	398c770, abc11cc, …	full FD+TD design, verified vs both source papers	—	—

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Validation method

Every accuracy test compares against the exact reference on the same data:

• exact-at-fiducial (<1e-6) — the tight anchor: with the trial equal to the fiducial, the ratio is identically 1 and the heterodyned likelihood must equal the full/dense likelihood to machine precision;
• parity to the Zackay error model β·(1+|lnL|) — the discrepancy grows with distance from the fiducial in likelihood units, so the principled tolerance is proportional to (1+|lnL|), not a fixed absolute; checked across intrinsic, extrinsic, noisy, higher-mode, and network cases;
• convergence — finer bins reduce the error;
• performance-comparison tests — identical likelihood computed with and without relative binning; the two match numerically and the binned version is much faster, with JAX compile time excluded (warm-up before timing);
• posterior-level (RB-5) — the heterodyned and dense posteriors over an intrinsic parameter agree in mean/width and have Jensen–Shannon divergence < 1e-3 (the paper’s acceptance criterion is agreement at the posterior level, not just pointwise).

References: FD — Zackay, Dai & Venumadhav 2018 (arXiv:1806.08792), bilby RelativeBinningGravitationalWaveTransient; TD — Sharma, Vijaykumar & Kumar 2026 (arXiv:2601.11239), incl. Appendices A/B/C.

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Public API (jaxpe.gw.likelihood)

• Fourier domain: RelativeBinningFDLikelihood, RelativeBinningFDLikelihoodHM, frequency_bin_edges, fd_dense_loglikelihood_modes.
• Time domain: RelativeBinningTDLikelihood, RelativeBinningTDLikelihoodHM, RelativeBinningTDNetwork, time_bin_edges, td_dense_loglikelihood, td_dense_loglikelihood_hm, extrinsic_coefficient.
• Covariance: toeplitz.autocorrelation_from_psd, toeplitz.toeplitz_matvec, toeplitz.inverse_generator, toeplitz.inverse_matvec.

Tests: test_relative_binning_fd.py, test_toeplitz.py, test_relative_binning_td.py — 44 tests.

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What remains

• Full production end-to-end PE. The remaining work is not a unit-testable commit-after-tests increment: it requires wiring a production higher-mode waveform (e.g. NRSur7dq4/JaxNRSur) to generate modes at the (non-uniform) bin edges, integrating a sampler / GPry surrogate, and running a percentile–percentile injection campaign (hours–days of compute). The method itself is already validated at the posterior level on a synthetic problem (JS < 1e-3), the strongest proxy short of a full run. This belongs to a dedicated effort with its own design choices (which waveform model, which sampler).

Also noted in the design: the summary-data setup is the real cost for long TD signals (the N_bins × N_modes Gohberg–Semencul solves — amortised over ~10⁷ evaluations), and the speedup assumes the waveform can be evaluated at arbitrary (bin-edge) times; EOB-type models that cannot must generate on a dense grid and downsample, keeping the covariance speedup but not the waveform one.