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B6/B7: longer-chain identified covariance geometry — research pilot

No Bayesian inferential qualification

This is an unpublished draft-branch scientific diagnostic experiment, not a validation of nominal population-covariance intervals. Stable eyetrajectoriespy 1.1.0 and protected main remain unchanged. Scientific blocker #255 remains open.

Why this experiment is needed

The earlier 36-fit joint-loading score-marginal elliptical-slice study was a negative result: B6 covariance median R-hat/ESS was 1.084/22.5 for the original collapsed-mean Gibbs and 1.141/15.4 for joint-loading ESS; B7 paired 1.407/4.3 vs 1.378/4.9, asynchronous 1.149/11.4 vs 1.418/4.4. None met the prespecified covariance R-hat ≤1.01 and bulk ESS ≥400 screen. A scientifically sound loading update can nevertheless mix poorly; green CI is not posterior coverage evidence.

Design frozen before the run

One new independently prior-generated rank-one dataset for each of B6 sparse univariate, B7 paired planar and B7 asynchronous planar, retaining original timestamps, common Gaussian priors and known per-channel noise.

On each dataset, fit:

  1. Native original partially collapsed mean Gibbs: four chains, 1,000 warmup + 800 retained draws each.
  2. Native score-marginal loading elliptical-slice comparator under exactly the same priors/noise: four chains, 1,000 warmup + 800 retained draws each.
  3. Independently constructed analytic-score-marginal PyMC NUTS: two chains, 1,200 tuning + 800 retained draws each, using the independent likelihood derivation, rather than native conditionals.

Total: nine actual posterior fits (three datasets × three backends) with zero tolerable silent failures. Fit errors and PyMC divergences are retained.

Identifiable outputs and diagnostics

Compute at the common-grid midpoint: population x mean, B6 x covariance, and B7 joint x/y cross-covariance, all invariant under loading sign changes. Do not screen using raw loading R-hat alone.

  • Rank-normalized R-hat, bulk/tail and mean ESS, and genuine ArviZ spectral mean MCSE rather than marginal SD divided by bulk ESS.
  • Per-chain posterior means and chain-mean range relative to posterior SD; first/second-half mean shifts are descriptive only, not formal convergence tests.
  • Posterior mean, SD and marginal 90% interval, one known generating truth per dataset, zero-divergence diagnostics and MCSE magnitude. Compare source/posterior differences cautiously when either chain is mixing poorly.
  • Report the prespecified rank R-hat ≤1.01 and bulk ESS ≥400 screen as diagnostic, never as a coverage or release gate.

This pilot is intentionally limited to rank one. Rank-two independent posterior references, simulation-based calibration of posterior ranks across many fresh prior-generated datasets and nominal interval coverage require separate predeclared larger experiments. They are not inferred from three NUTS fits or nine total posterior attempts.

Completed original-source results — 10 October 2026

The original scientific fitting workflow #38039822342 completed 7/7 jobs, 9/9 actual posterior fits, zero fit exceptions and a successful cross-design SHA256 evidence audit. Retained original study artifacts:

Source provenance: original fitting was executed at PR #269 parent-study head b0e64de579d72a40ace71c1ef3fe3dfe84db6a55; the subsequent tests/doc-only corrections at 024779aaaa516681f34855577d65c54e7280b561 did not change the posterior algorithm. The corrected-head general CI finished 24/24 green. Results below are the originally emitted JSON scientific rows, not refits on the doc-update commit.

Nine source-verified fitted B6/B7 covariance R-hat and bulk ESS results

Original evidence figure: recreated directly from the source-run's nine original JSON result rows, with the specified source workflow and artifact provenance. The graphic is a visualization of the original evidence, not a fourth estimator or a refit.

Identifiable covariance diagnostics

Study Sampling architecture Rank R-hat Bulk ESS Tail ESS Spectral mean MCSE Exploratory R-hat ≤1.01 and ESS ≥400?
B6 rank 1 Original partially collapsed Gibbs 1.0901 45.6 32.6 0.001677 No
B6 rank 1 Score-marginal elliptical slice 1.2049 13.4 41.2 0.003107 No
B6 rank 1 Independent PyMC NUTS 1.0060 399.6 340.6 0.000702 Borderline threshold, formally no
B7 rank-1 paired Original partially collapsed Gibbs 1.2928 11.7 74.6 0.002090 No
B7 rank-1 paired Score-marginal elliptical slice 1.4250 8.5 24.3 0.001355 No
B7 rank-1 paired Independent PyMC NUTS 1.0005 705.7 838.3 0.000224 Yes, screen only
B7 rank-1 asynchronous Original partially collapsed Gibbs 1.1594 19.9 108.4 0.001514 No
B7 rank-1 asynchronous Score-marginal elliptical slice 1.6015 6.8 32.7 0.001909 No
B7 rank-1 asynchronous Independent PyMC NUTS 0.9997 699.2 959.7 0.000205 Yes, screen only

Original source diagnostic functionals are the B6 midpoint x variance and the B7 midpoint x/y cross-covariance. The independent NUTS runs reported zero divergences in each study. All three samplers' single-dataset marginal 90% intervals contained their respective generating covariance truth. This one-of-one observation per design does not estimate nominal repeated-sampling posterior coverage.

Posterior agreement and its limits

Study Original Gibbs posterior mean Elliptical slice posterior mean Independent NUTS posterior mean
B6 variance 0.03154 0.03337 0.03514
B7 paired x/y covariance −0.01757 −0.01500 −0.01762
B7 asynchronous x/y covariance 0.02066 0.01848 0.01897

Native posterior means lie in the vicinity of the independent reference, but apparent numerical agreement is not reliable agreement when native bulk ESS is only 6.8–45.6 and between-chain differences are substantial. On the covariance functional, chain-mean ranges as fractions of pooled posterior SD reached 0.97/1.48/0.018 for B6 original/ESS/NUTS, 0.93/1.27/0.038 for B7 paired, and 0.76/1.54/0.041 for B7 asynchronous. NUTS population mean itself is generally better mixed than native covariance, but reference quality must be assessed across more functionals/datasets.

Computational efficiency not established: native Gibbs retained 4 × 800 draws versus NUTS 2 × 800. The source study did not measure per-method wall time; therefore absolute ESS ratios and mean MCSE differences are not speed or ESS-per-second comparisons.

Scientific decision: evidence across three newly prior-generated rank-one datasets disfavors further unmotivated native Gibbs transition variants as the first priority. A separate validated four-chain independent NUTS reference, rank-2 diagnostic study, many-dataset rank-based SBC, known-truth nominal coverage and compute-budget-matched timing remain required before selecting a production posterior backend or promoting any uncertainty interface. Blocker #255 and all release/inferential flags remain unchanged.

Reproducibility

Source script: scripts/run_b6_b7_longchain_identified_geometry.py; dedicated CI .github/workflows/science-b6-b7-longchain-covariance-geometry.yml. The original source study creates per-design cases.json, evidence.json, SHA256SUMS, independently verifies the complete experiment in a combined job, and preserves failures. The preceding original workflow and retained artifacts document completion; do not attribute the fitting run to a later docs-only head. Do not change the underlying sampler default or promote posterior inferential validity based on these results.