Methods and limitations

Coefficient-draw selection

Use a finite matrix with draws in rows and coefficients in columns. Remove the intercept. Keep coefficient units unchanged when comparing thresholds.

S2M follows section 2.2.1 of Li and Pati (2017). It recursively splits absolute coefficients and retains the saved lower-mean noise candidate at stopping. Recursion continues while the center gap is greater than b; it stops when the gap is at most b, or a small/constant candidate boundary applies. The signal count is the mode across draws; tied modes favor the smaller count.

Basic 2-M uses the smaller cluster size, which need not be the lower-mean cluster. Both rules rank variables by median absolute coefficient, with larger column indices winning equal-median ties. Sparse spike draws can give zero medians for all variables; a ranking tie can then select a never-included variable. Use inclusion summaries when inclusion probability is the target.

small_set="legacy" retains recursive candidate pairs. "refine" splits a distinct pair. Initial constant magnitudes have no defined split. The explicit all-zero convention assigns zero signals; S2M requires zero_draws="empty". These are boundary choices, not a new selection guarantee. Continuous-prior S2M draws retain at least one signal. No automatic threshold is provided.

Gaussian fitting

The model is

$$ Y = X\beta + \epsilon, \qquad \epsilon \sim N(0,\sigma^2 I). $$

Here Y is the response vector for n observations, X is the model matrix, $\beta$ denotes the coefficient vector and sigma² is the residual variance. The mathematical model includes an intercept; the fitting functions add it internally.

Predictors and response are centered to integrate a flat intercept prior. They are not standardized. The coefficient priors below exclude the intercept, whose prior is flat. The residual variance prior is inverse-gamma with shape 3/2 and scale 3/2. This intercept-prior convention is explicit; the final paper does not supply every fitting detail.

Prior	Coefficient distribution given scales and sigma²
Ridge	$N(0,\sigma^2\mathtt{prior_scale}^2)$
Horseshoe	$N(0,\sigma^2\tau^2\lambda_j^2)$; $\tau,\lambda_j$ independent half-Cauchy(0,1)
Spike-and-slab	Point mass at zero or $N(0,\sigma^2\omega_j^2)$; $\omega_j^2\sim\mathrm{IG}(3/2,3/2)$; inclusion probability $\sim\mathrm{Beta}(1,15)$

The fit stores separate chain and retained-draw axes. draws means retained draws per chain; burnin is additional and is discarded. These differ from the paper's convention of reporting total iterations. Seeds define local NumPy generators and do not modify NumPy's global random state.

A coefficient interval describes the fitted posterior under these assumptions. It is not a repeated-dataset coverage guarantee. These Gaussian fitters do not handle right-censored outcomes as ordinary complete outcomes. Binary, count and survival likelihoods are not implemented.

SAVS and DSS

The signal adaptive variable selector (SAVS) retains coefficient j only when $\lVert X_j\rVert^2|\hat\beta_j|^3>1$, where $X_j$ is predictor column j and $\hat\beta_j$ is its supplied estimate. Equality sets the coefficient to zero. Its threshold depends on predictor units. See Ray and Bhattacharya (2018).

For decoupling shrinkage and selection (DSS), let $\bar\beta$ be the supplied estimate and $\lambda$ the penalty. The sparse coefficient vector minimizes

$$ n^{-1}\lVert X\bar\beta-X\gamma\rVert_2^2 +\lambda\sum_j\frac{|\gamma_j|}{|\bar\beta_j|}. $$

For positive $\lambda$, $\bar\beta_j=0$ constrains $\gamma_j=0$. At $\lambda=0$, the supplied estimate is returned. This is the Hahn–Carvalho adaptive-L1 approximation, not exact minimization of their variable-count penalty. Cyclic coordinate descent reports a relative convex optimality error and warns when the tolerance is unmet. With dependent predictors, sparse coefficients and selected sets may not be unique.

Evidence boundaries

The Python package uses the repaired local R implementation as a reference for fixed draw summaries. This is distinct from comparison with the released CRAN version. General selection accuracy, high-dimensional sampler mixing and full paper reproduction require separate evidence. These checks do not establish false-discovery control.

Numerical limits

Fitting stops if a covariance factorization fails or its estimated reciprocal condition number is at most 1e-12. It does not clip prior scales or silently retry a failed chain. Extreme predictor units and prior scales can trigger this limit. Summary and prediction methods reject nonfinite arithmetic; very wide ranges may be unrepresentable in float64. Integer arrays are accepted only when conversion preserves their values.