Numerical reference checks

The test fixtures contain six scalar cases from installed R ashr 2.2-63 and five Gaussian regression cases from installed mr.ash.alpha 0.1-43 (commit 8e257fdb73589f280192b1c99a176d82196be298). Three multivariate cases also use installed mashr 0.2.79. All exports used R 4.5.1. Archive and exporter hashes are stored with each fixture.

Comparison	Quantities	Absolute and relative tolerance
Gaussian scalar, fixed prior	Mean, SD, null/sign probabilities, weights	2e-11
Student-t/uniform scalar, fixed prior	Same quantities	2e-7
Gaussian scalar, estimated weights	Same quantities	2e-6
Gaussian regression, converged fit	Coefficients, intercept, residual variance, weights	1e-5
Gaussian regression, one sweep	Same quantities	2e-10
Multivariate, fixed prior or one component	Means, SDs, covariance, null/sign probabilities, weights	2e-10
Multivariate, estimated mixture	Same quantities	5e-6

Regression comparisons use explicit grids, zero initial coefficients, common initial weights/variance, an intercept and no standardization. R uses method_q="sigma_indep_q", matching the fixed-factor variance update. Its sigma_dep_q option uses a different update and need not agree after one sweep. Stopping rules also differ: AShrink checks changes in the evidence lower bound (ELBO), whereas R checks coefficient or mixture-weight changes. Matching results in these cases does not establish global optimization, interval coverage or prediction performance.

Recreate the fixtures

R is needed only to regenerate fixtures. Install the pinned reference packages and jsonlite in an isolated R library, selected through R_LIBS_USER. Use the ashr 2.2-63 source archive and the pinned mr.ash.alpha Git commit; install mashr 0.2.79 for its exporter. Installing unpinned current versions does not reproduce these references.

Rscript tools/export_ashr_reference.R ashr-reference.json
Rscript tools/export_regression_reference.R regression-reference.json
Rscript tools/export_mashr_reference.R multivariate-reference.json
python -m pytest tests/test_r_reference.py

The exporters write runtime/library provenance along with the fixed cases. Repository fixtures omit machine-specific library paths. The tests read the committed fixtures and do not invoke R. To check a regenerated export, compare its case values with the matching fixture; local namespace paths may differ.

The pinned regression package requests C++11. With current RcppArmadillo headers, the reference build required a C++17 compiler setting and paths to installed Fortran libraries. The reference source was unchanged. Compiler and BLAS differences can affect final rounding.

R warns that half-uniform priors without a null-biased penalty can give misleading sign probabilities. Those fixed-prior cases check arithmetic only. The regression reference also warns when substantial weight reaches the largest prior component. The comparison keeps the explicit grid unchanged; an applied analysis should inspect its adequacy.

Conjugate count models and fixed-pattern multivariate models have independent analytic tests. Their inclusion does not imply full ashr or mashr coverage. Logistic and Poisson-regression tests check null-only likelihoods and the stationarity equations for a single normal factor. Random-intercept tests compare fixed-variance posterior means with a joint Gaussian block solution. These checks address the implemented updates; they do not establish interval coverage or prediction performance across repeated datasets.

Multivariate comparisons use explicit covariance patterns, correlation matrices, unit scale multipliers and no automatic covariance normalization. One case checks a single full-rank component, one estimates mixture weights, and one reconstructs fixed-weight summaries with singular patterns and a zero atom. The fixed-weight case uses mash(..., fixg=TRUE) in R and public result reconstruction in Python. It checks posterior calculations rather than fitting new mixture weights. R's one-component optimizer reports that no optimization is needed. Learning covariance patterns is not supported.