B6/B7 independent posterior: population-functional convergence and MCSE¶

The first successful independent reference posterior comparison in original workflow #37988738287 used three exact-prior matched B6/B7 rank-one datasets and produced 3/3 actual PyMC NUTS fits (source artifact 11645572128, zero divergences) after replacing the overlarge PyTensor expression with a participant-vectorized rank-one marginal likelihood.

That first comparison is not a successful convergence qualification. PyMC worst raw-parameter Rhat was 1.836 (B6 rank one), 1.020 (B7 rank-one paired), and 1.033 (B7 rank-one asynchronous); native Gibbs worst monitored Rhat was respectively 1.381, 1.109 and 1.691. Some native bulk ESS minima were 3.2–14.3. The reference B6 loading may exhibit an unresolved \(\lambda\mapsto-\lambda,\ z_i\mapsto-z_i\) symmetry: raw loading Rhat is not the right scientific convergence criterion for the identified population covariance \(L L^\top\). This hypothesis must be tested rather than assumed to explain bad Rhat.

Source-verified original posterior comparisons¶

All equal-tailed 90% posterior intervals overlap. Nevertheless, both implementations' cross-covariance intervals missed the known-generating cross-covariance in the one paired B7 dataset. The asynchronous B7 cross-covariance discrepancy is nontrivial relative to its independent-reference posterior SD, but the native MCMC has exceptionally poor ESS. These facts are not evidence of adequate calibration or same-posterior equality.

Additional independent convergence diagnostics¶

This focused research follow-up calculates rank-normalized split Rhat, bulk ESS, tail ESS and approximate Monte Carlo standard error of the posterior mean, calculated as marginal posterior SD / sqrt(bulk ESS) for portability across ArviZ versions (not ArviZ spectral MCSE) on each identified midpoint functional in each actual sampler: B6 population mean and variance; B7 x population mean and x/y cross-covariance. It also reports the difference between native and independent posterior means in units of their combined MCSE, with a conspicuous warning that MCSE and marginal overlap cannot qualify a nonconverged chain.

For transparency, raw-parameter worst NUTS Rhat is still retained even though it can reflect factor-sign nonidentifiability rather than marginal covariance mixing. The test suite deliberately creates two chains with opposite rank-one loading signs but matching loading-squared distributions, asserting that the scientific Rhat is computed on the identified covariance, not raw factors.

A prespecified diagnostic flag requires both population-functional Rhat \(\leq 1.01\) and bulk ESS \(\geq 400\) per pair before even considering the summaries numerically trustworthy; this is a diagnostic screen, not a scientific inference or posterior coverage qualification. No failed fit is excluded from the attempt ledger.

Future statistical qualification still required¶

Before opening the native B6/B7 gate: distinguish nonidentifiability from nonconvergence; assess MCSE of posterior interval endpoints, not only means; run independent multi-dataset prior predictive simulation-based calibration for identified functionals; evaluate rank-two loading geometry, heterogeneous irregular clocks and measurement noise; and demonstrate held-out participant predictive coverage. Any appropriate computational threshold must be specified before looking at the numerical results, never retrospectively adjusted to obtain green CI.

All work remains in stacked draft PRs, with production main, stable 1.1.0, public root exports and release flags unchanged.