Numerical methods and limits

The scalar implementation estimates finite-mixture weights for the adaptive-shrinkage models described by Stephens (2017). Gaussian component posteriors are analytic. Student-t component posteriors use adaptive one-dimensional quadrature and record integration diagnostics. Quadrature can be substantially slower than the analytic Gaussian calculation.

Mixture fitting uses expectation–maximization (EM) with fixed component grids, followed by constrained optimization when the optimality residual remains large. Diagnostics retain the objective history, iteration count, stopping rule and a Karush–Kuhn–Tucker (KKT) optimality residual. A small residual is a numerical stopping criterion; it does not establish statistical calibration or uniqueness of the fitted prior. Results retain nonconvergence information rather than treating every returned fit as reliable.

The R reference version is ashr 2.2-63. Uniform widths and normal standard deviations have different meanings. Student-t inputs are scale parameters. The normal-prior/Student-t combination uses an independent integration method; the inspected R implementation rejects that combination. Ranked summaries include all tied local probabilities at a cutoff, which can differ from R's order-dependent cumulative averages within ties.

Count models use conjugate Gamma–Poisson and Beta–binomial mixtures sharing a specified mode, with an optional point mass at that mode. Multivariate models use finite mixtures of zero-mean normals with supplied covariance patterns, following the fixed-pattern part of Urbut et al. (2019). AShrink does not implement all methods in ashr or mashr.

Posterior probabilities and credible intervals condition on estimated mixture weights. They omit uncertainty from fitting those weights. Dependence between scalar observations, misspecified likelihoods and unsuitable prior components can affect inference. The reference checks describe the numerical comparisons and their limits.

Gaussian regression uses cyclic variational coordinate ascent with residual-variance-scaled normal-mixture priors, following Kim et al. (2024). The objective is an evidence lower bound, rather than the exact marginal likelihood. Coefficient marginals factorize; their variances need not equal those from the exact joint posterior. Mean predictions reuse fitted preprocessing. See the regression guide for inputs and limits.

Binary/binomial regression uses a quadratic logistic lower bound. Poisson regression uses analytic exponential moments of independent mixture-normal factors, with guarded inner Newton optimization. Gaussian random-intercept regression uses independent coefficient and group factors. These extensions have focused analytic and stationarity checks, not repeated-dataset calibration studies. Prediction applies the fitted mean coefficients; nonlinear predictions do not integrate their uncertainty. See the regression guide.