Numerical validation

Fixed calculations

Analytic checks cover exact two-means partitions, S2M stopping, SAVS thresholding, DSS convex optimality, Gaussian conditional draws and the conjugate ridge posterior. Fixed selection and sparse-coefficient results also agree with the repaired installed R VsusP 1.0.0.9000 on R 4.5.0. This reference is not the released CRAN version. Python's default clustering is deterministic; R's is not.

Native posterior checks

Four fixed seven-observation inputs compare Gibbs draws with numerical integration. Horseshoe checks use one predictor with weak or strong fitted signal. Spike-and-slab checks use two orthogonal or correlated predictors. Each uses four chains with 2000 burn-in and 4000 retained draws per chain.

Check	Maximum rank R-hat	Minimum bulk ESS	Minimum tail ESS
Horseshoe, weak signal	1.0017	2415	2560
Horseshoe, strong signal	1.0000	12322	13246
Spike-and-slab, orthogonal	1.0011	2504	2504
Spike-and-slab, correlated	1.0074	413	400

ESS means effective sample size. Compared quantities include coefficient first and second moments, mean residual variance and, for spike-and-slab, inclusion and complete-model probabilities. Quadrature refinements changed the targets by less than 5e-11. Every discrepancy was within the prespecified audit tolerance: the larger of 0.01 and five estimated Monte Carlo standard errors. These broad audit thresholds diagnose large errors; they do not prove exact sampling.

Posterior moment differences divided by the audit tolerance, with the acceptable band from minus one to one.

The plotted band is a numerical acceptance rule, not a confidence interval.

Quadrature and sampled complete-model probabilities for the correlated input, next to four coefficient traces.

Error bars show ±1.96 estimated Monte Carlo standard errors. Every twentieth retained draw is shown in the traces; all draws enter summaries. The model labels give the two inclusion states. See the numeric results and plot values.

Interpretation

These are low-dimensional posterior checks. They do not establish general selection accuracy, repeated-dataset interval coverage or high-dimensional mixing. The full Li–Pati simulations and lymphoma analysis have not been reproduced. Platform checks are recorded separately from statistical checks.