Changes in version 0.9.6 Highlights - Multi-response lmb(): lmb() now handles both univariate and multivariate responses with a single unified interface, mirroring the behaviour of R's lm(). When the response has a single column the result is an lmb object (unchanged from prior releases). When the formula specifies multiple response columns (e.g. cbind(y1, y2) ~ x), lmb() fits a separate Bayesian linear model per response column and returns a named list with class mlmb. For the multi-response case, pfamily must be a list of pfamily objects with exactly one entry per response column; passing a single pfamily object is an error. Summary, print, and coefficient methods for mlmb objects are included. - Conjugate GLM priors (Poisson, binomial, Gamma): New closed-form IID sampling paths for intercept-only models with identity links. dBeta() with rBeta_reg() supports Beta–Binomial(identity) conjugate updates; dGamma(Inv_Dispersion = FALSE) with rGamma_Conjugate_reg() supports Gamma–Poisson(identity) and Gamma–Gamma(identity) rate priors. Prior_Setup() can calibrate conjugate hyperparameters for these families (weighted Poisson rate and binomial probability defaults). See ?dBeta, ?dGamma, and the Chapter 02 / Chapter 07–11 vignettes. - Vignette structure: Reworked Chapter 00 as a roadmap across five main parts plus technical appendices. Chapter 02 is now a conceptual introduction to single-parameter conjugacy; worked examples move to Chapter 02-S01 through Chapter 02-S05 (Beta–Binomial, Normal–Normal, Gamma–Poisson, exposure-weighted Poisson, and related topics). A Companion textbooks section in Chapter 00 indexes optional Bayes Rules! and LearnBayes appendices tied to the main GLM chapters. - opencltools import: Core host/runtime OpenCL discovery and diagnostics (detect_*, PATH helpers, environment checks) now live in the opencltools package (Imports, >= 0.8.0). glmbayes keeps package-specific entry points (glmbayesCore_has_opencl(), diagnose_glmbayes()) that report compile-time OpenCL status for this build while delegating shared GPU/runtime checks—reducing duplicated maintenance in glmbayes. - Bayes Rules! companion examples: Optional vignette appendices reproduce book datasets and published posterior summaries using lmb(), glmb(), Prior_Setup(), and dNormal() (suggested package bayesrules for data only). Coverage includes bikes (Ch. 03), weather_perth (Ch. 08–09), equality_index (Ch. 10), Gamma–Poisson conjugacy (Ch. 02-S04), and a scope note for Gamma regression (Ch. 11). Comparison tables use printed book values, not live rstanarm fits. See Chapter 00 § Companion textbooks. - LearnBayes examples: Chapter 02-S04, Appendix A, maps the hearttransplants example from Albert (2009) / LearnBayes (exposure-weighted Gamma–Poisson conjugacy) to glmb() with analytic Albert posteriors for verification (suggested package LearnBayes). Other changes - Expanded testthat coverage for dBeta() / binomial(identity) conjugate paths and related glmb() integration. Changes in version 0.9.5 - Tests / CRAN: All OpenCL-specific testthat blocks now call skip_on_cran() (in addition to skip_if_no_opencl()), consistent with existing Boston/Cleveland OpenCL tests. OpenCL coverage remains for local checks and source builds with OpenCL; CRAN checks avoid parallel/GPU-heavy tests that could trigger CPU time vs elapsed time NOTES. Changes in version 0.9.4 - Vignettes: A vignette that previously used the notangle engine now uses the standard R Markdown vignette machinery (knitr / rmarkdown::html_vignette), so builds align with CRAN expectations and vignette index ordering should be consistent with the rest of the package. - OpenCL sources (inst/cl): Removed unused or superseded material, consolidated kernels and library fragments, and aligned .cl layout and dependency tagging with the conventions used in 'openclport' and 'nmathopencl' (prelude, shims, nmath/ stems, family kernels under src/). See inst/cl/README.md for how the assembled program is stitched. - OpenCL program assembly: Reworked loading so the full OpenCL program is built from explicit fragments (global header, nmath closure, family/link kernels) rather than ad hoc concatenation—clearer ownership of what enters GPU compilation and easier parity with CPU paths. - Tests: Added and expanded testthat coverage aimed at OpenCL code paths (including binomial examples that exercise GPU envelope evaluation), complementing existing Cleveland-style checks. - Bug fix — binomial OpenCL: Binomial f2_f3 OpenCL kernels now evaluate the data log-likelihood with the same proportion × trial-count semantics as dbinom_glmb on the CPU (round successes and trials, clamped probability). This fixes envelope / PLSD failures for aggregated binomial data (e.g. cbind(successes, failures) / MASS::menarche) where the previous kernels treated y like a raw success count. Changes in version 0.9.3 - Published on CRAN. - Version bump in response to CRAN resubmission feedback. Changes in version 0.9.2 - Version bump in preparation for resubmission incorporating CRAN review feedback. Changes in version 0.9.1 - Wrapped OpenCL-dependent examples in \donttest{} for CRAN compliance. - Reduced iteration counts in rlmb Gibbs sampler example to stay within CRAN example time limits on slower check machines. Changes in version 0.9.0 First CRAN submission. This release is a stable pre-release with a near-complete feature set relative to earlier development builds. Highlights Bayesian Generalized Linear (glmb) and Linear (lmb) modeling functions: glmb() is a Bayesian analog for the classical glm() function while lmb() covers Gaussian models. Calls largely mirror those for the classical functions but leverage pfamilies for prior specifications. Method functions largely mirror those for the classical functions. Samples generated by the functions are largely iid samples (no MCMC convergence dignostics are needed). Implemented Likelihood families/ link functions: Most of the families implemented in the glm() function are also implemented in the glmb() function (the lmb() function covers only gaussian() families). Link functions that lead to log-concave likelihood functions are generally implemented. Specifically, we have the following: Supported likelihoods: gaussian (identity), Poisson / quasi-Poisson (log), binomial / quasi-binomial (logit, probit, cloglog), Gamma (log). Prior Family functions: pfamily constructors are used to specify priors and play the same kind of role for the prior specifications as family constructors and link functions play for the likelihoods. Specifically, we have the following: Supported Priors: Normal (all families/links), Normal–Gamma and independent Normal–Gamma (gaussian families), and Gamma-on-precision (gaussian and Gamma families). Prior_Setup function: The package comes with a convenient Prior_Setup() function that provides default prior input parameters for each of the implemented models. Basic calls (without tailoring) mirror traditional calls to the glmb() and lmb() functions respectively and only require the user to provide the model formula and (if not the gaussian family) the family/link function. The function can also be used to easily adjust prior specifications (see documentation for details). Extensive Method functions: The package comes with extensive method functions that mirror those for the classical functions. These include dedicated print(), summary(), predict() and simulate() functions. Lower Level Modeling functions: The package comes with lower level modeling/simulation functions that advanced users can use to implement block Gibbs samplers. These generally come with less overhead than the glmb() and lmb() functions and are called internally by the the higher level modeling functions. RcppParallel and OpenCL GPU Acceleration Implementations Some of the simulation functions comes with use_parallel and use_opencl options that speed up simulation for higher dimensional models. Extensive help files, vignettes, examples and demos The package also comes with extensive help files for the varios functions that are complemented with a rich set of vignettes. A large number of examples and demos are also availabel (see the READM.md file for a sample). Earlier development history (0.1.x series) The notes below summarize major work during the initial development series before the 0.9.0 pre-release. OpenCL and GPU acceleration - Completed the OpenCL-based grid construction framework for large models. - Added GPU-aware envelope sizing and improved OpenCL failure handling. - Introduced diagnostic utilities to assess OpenCL availability and performance. - Improved configure scripts to detect OpenCL and provide informative messages. - Expanded OpenCL documentation and added a dedicated vignette chapter. Parallel CPU sampling (RcppParallel) - Enabled parallel envelope construction and parallel iid sampling. - Added pilot functions for large-dimension grid estimation. - Implemented thread-safe parallel sampling for independent normal-gamma models. Core statistical improvements - Migrated to an improved independent normal-gamma simulation algorithm. - Added theoretical derivations for independent normal-gamma regression. - Improved UB2 and RSS minimization routines, including scaling corrections. - Enhanced Prior_Setup() to support family-specific prior construction. - Added dedicated envelope evaluation and sizing functions. Package infrastructure - Significant cleanup to remove NOTES and improve CRAN readiness. - Improved configure and Makevars files for portability. - Added testthat tests, including OpenCL-specific tests. - Consolidated envelope-building functions into a cleaner structure. Documentation - Major updates to README and package-level documentation. - Added multiple new vignettes and expanded existing ones. - Improved examples for lmb(), rlmb(), and OpenCL models. Bug fixes (0.1.x era) - Corrected scaling in UB2 minimization. - Improved error handling for missing OpenCL functionality. - Fixed various small issues uncovered during parallelization work. Changes in version 0.1.0 - Forked from glmbayesCore as the full-featured backend for lmebayes: lmebayesCore is a history-preserving fork of glmbayesCore (created 2026-07-15) that keeps the complete glm/envelope engine and the two-block Gibbs mixed-model stack (model_setup(), Prior_Setup_lmebayes(), rlmerb()/rglmerb(), rLMM_reg/rGLMM_reg routes, etc.). glmbayesCore itself is being stripped down to only the glm/envelope engine that glmbayes needs; lmebayes now depends on lmebayesCore instead. Package identity (DESCRIPTION, NAMESPACE, GLMBAYES_R_NS, .Call()/DLL registration) was renamed accordingly; the internal C++ glmbayes:: namespace was left unchanged as an implementation detail. First time builds - Per-group dGamma_list() prior now uses the §3.3.4 marginal rate: dGamma_list.lmebayes_prior_setup() feeds each group's dGamma() the Chapter A12 §3.3.4 marginal ING rate (beta integrated out) instead of the §3.3.5 fixed-beta rate_gamma. This is the theoretically correct choice for the Block~1 ING sampler, which draws sigma2_j from the marginal law and then b_j | sigma2_j (beta is never held fixed at a point estimate during that draw). rate_gamma remains on ing_prior_measurement_group for diagnostic comparison only (printed by a dev-only table in Prior_Setup_lmebayes() whenever dispformula requests per-group dispersion); nothing downstream consumes it. Truncation bounds (disp_lower/disp_upper, blup_infl, R_lo/R_hi) are unaffected -- they were already mean-matched at sigma2_hat, which is now also the new rate's exact prior mean (previously it was not, for the most BLUP-inflated groups). See inst/DGAMMA_LIST_MARGINAL_AND_BOUNDS.md Parts I-II. - GLMM router (.lmebayes_run_glmm_engine()): non-Gaussian rglmerb() dispatches through REG_ROUTE_TABLE to rGLMM_reg_known_vcov() or rGLMM_reg_estimated_vcov() (replacing a direct rGLMM_reg() call). LMM routing (.lmebayes_run_lmm_engine()) uses the same table for four Gaussian routes. - GLMM engines split (rGLMM_reg.R): monolithic rGLMM() replaced by rGLMM_reg_known_vcov(), rGLMM_reg_estimated_vcov(), and dispatcher rGLMM_reg() (shared help ?rGLMM_reg). Non-Gaussian models always run a pilot stage (unless n_pilot = 0L); the two routes differ in eigenvalue-bound complexity (fixed dNormal τ² vs ING disp_lower conservatism), not in whether a pilot runs. - LMM engines merged (rLMM_reg.R): rLMMNormal_reg.R and rLMMIngNormal_reg.R are one module with shared help ?rLMM_reg (aliases for six exports). There is no standalone rLMM() export — matrix Gaussian LMMs use rLMMNormal_reg_* / rLMMindepNormalGamma_reg_* routes; formula GLMMs use rGLMM_reg via rglmerb(). - rlmerb() four-route routing: .lmebayes_run_lmm_engine() dispatches to rLMMNormal_reg_known_vcov(), rLMMNormal_reg_estimated_vcov(), rLMMindepNormalGamma_reg_known_vcov(), or rLMMindepNormalGamma_reg_estimated_vcov() from fixed vs ING Block~2 and fixed vs dGamma σ². Legacy rLMMindepNormalGamma_reg() (outer σ² loop) remains exported but is not the default rlmerb() path. - ICM at fixed variance components: lmerb_posterior_mean() / glmerb_posterior_mode() iterate Block~1 / Block~2 hyperparameters at fixed (\tau^2_k) and (\sigma^2) plug-ins (.two_block_tau2_plug_in_from_pfamily(), rate/(shape−1)). Removed joint posterior-mode τ² iteration (two_block_joint_posterior_mode() stack). - Scale-invariant ICM stopping rule: glmerb_posterior_mode() now measures the Block~2 delta used for the tol/converged check as a per-component Mahalanobis distance (\sqrt{(\gamma_k^{new}-\gamma_k)^\top P_{\gamma_k}^{\mathrm{post}} (\gamma_k^{new}-\gamma_k)}) in each RE component's own posterior-precision metric, maximized over components (k) — not the previous raw (\ell_\infty) change in fixef. The old criterion depended on the arbitrary units of each hyper-covariate, so rescaling or whitening an X_hyper[[k]] column changed convergence behavior even though the fitted model and posterior were unchanged; the new one does not. The per-block posterior precision (post_P_list) is now also hoisted out of the ICM loop since it does not depend on the iteration state. (Superseded for lmerb_posterior_mean() by the exact closed-form solve below, which has no stopping rule at all.) - lmerb_posterior_mean() is now an exact closed-form solve, not ICM: the Gaussian Block~1/Block~2 target is exactly jointly Gaussian, and Block~1's conditional mean per group is affine in the shared hyperparameter vector (\gamma) with no direct coupling between groups. Substituting that affine relationship into the Block~2 update eliminates every group's random effect algebraically (a Schur-complement/Henderson-mixed-model-equations elimination), leaving one small linear system in (\gamma) alone (dimension = total hyperparameter count, independent of the number of groups (J)). Solving it once gives the exact joint mean — no alternating iteration, tol, maxit, or non-convergence warning is possible for this model, and it costs (O(J)) (never a (J \times J) or (J p_{re})-dimensional matrix), so it scales to large numbers of groups. tol/maxit remain accepted (for interface parity with glmerb_posterior_mode(), which is unchanged and still iterates for non-Gaussian families) but are unused; the return always has converged = TRUE, iterations = 1L, delta = 0. This also restores the exact D0 = 0 (start at the true posterior mean) assumption that two_block_tv_bound()/two_block_l_for_tv()'s sweep-count guarantee relies on, which a non-converged ICM start could silently violate. - two_block_l_for_tv() no longer errors when the search exceeds l_max: it now issues a single warning() (when warn = TRUE) and returns a practical uncertified fallback capped at 200 inner sweeps (l <= 199), not l_max = 1e6. The old l_max return caused integer overflow in pilot cost optimization (n_pilot * m_convergence_pilot) and invalid n_pilot values. Internal repeated calls (pilot cost search) pass warn = FALSE to avoid warning spam. Mode-gap pilot sweep calibration and inner-sweep counts are likewise capped at 200 via .two_block_cap_inner_sweeps() / .two_block_m_pilot_from_gap(). l_max/m_min calibration is inherently a best-effort setup step (choosing a burn-in sweep count), not part of the returned draws, so a near-degenerate rate$lambda_star (close to 1) should not abort the whole lmerb()/rlmerb()/glmerb()/rglmerb() call; all internal calibration call sites inherit this automatically. - dGamma_list() gains disp_center = c("sigma2_hat", "dispersion2"): the default ("sigma2_hat") reproduces the existing per-group truncation window unchanged (mean-matched at sigma2_hat_j, upper tail widened by disp_upper_anchor). The new opt-in "dispersion2" mean-matches both bounds symmetrically at an EnvelopeCentering()-style dispersion estimate that integrates over the random effect's own posterior uncertainty (new n_rss_iter argument, default 10L, controls its fixed-point iteration count) instead of BLUP-inflating sigma2_hat_j; disp_upper_anchor is ignored in this mode. Tends to produce narrower, better-centered upper tails for groups with large BLUP/OLS RSS inflation. New helper .lmebayes_group_dispersion2_envelope_centering() (R/mixed_rmerb_helpers.R); window_diagnostics gains a dispersion2 column. See inst/DGAMMA_LIST_MARGINAL_AND_BOUNDS.md Part III. - Rate helper rename: two_block_rate_v2() removed; use two_block_rate_from_pfamily_list() (R/two_block_ergodicity.R) for the pfamily_list adapter around two_block_rate(). - Two-block ergodicity consolidation: rate, TV-bound, and mode-weight helpers merged into R/two_block_ergodicity.R. two_block_mode_weights() is no longer exported (still used internally by rGLMM() for non-Gaussian rate calibration). - C++ R callbacks via registered namespace: envelope and simulation C++ now resolve EnvelopeOpt(), EnvelopeSort(), glmbfamfunc(), rNormal_reg.wfit(), and rgamma_ct() from the glmbayesCore namespace (R_interface.h / GLMBAYES_R_NS), so downstream packages (e.g. lmebayes) no longer need to re-export them for search-path lookup. - LMM engine split: rLMMNormal_reg() samples with fixed observation dispersion (\code{prior_list = list(dispersion = sigma2)}); P is a separate argument. rLMMindepNormalGamma_reg() implements an outer two-block Gibbs sampler (dispersion via rGamma_reg(), fixed effects and random effects via full rLMMNormal_reg() runs). - rGLMM() pilot defaults: non-Gaussian models now run pilot + main by default. New argument gap_tol (default 0.0196) derives n_pilot when n_pilot = NULL; n_pilot = 0L or gap_tol = NULL skips the pilot. Gaussian models never run a pilot. tv_tol now defaults to 0.01. Helper .two_block_resolve_n_pilot() centralises the policy. - Matrix LMM / GLMM ICM: when start = NULL (default), rLMM_reg routes and rGLMM() compute Block~2 starts via lmerb_posterior_mean() / glmerb_posterior_mode() at fixed variance-component plug-ins, using .two_block_measurement_prior_list() (and .two_block_tau2_plug_in_from_pfamily() for τ²). Outputs include ranef.mode and icm_info. Non-Gaussian rGLMM() still requires b_start when start is user-supplied. - Matrix LMM replicate chains: Gaussian LMM sampling is exported as six rLMM_reg engines (four direct rlmerb() routes plus two dispatchers). Formula-level fitting remains in lmebayes (rlmerb() / lmerb()). - Restored rGLMM(): matrix-level GLMM replicate-chain orchestration (TV calibration, pilot chi-squared, post-pilot eigenvalue upper bound, main-stage sampling via rGLMM_sweep) is exported again as rGLMM(). Replaces the earlier C++-staged implementation; returns the fixef.* namespace. Formula-level fitting remains in lmebayes (rglmerb() / glmerb()). - Candidate counts surfaced by the two-block v2 sampler: two_block_rNormal_reg_v2() now returns iters_fixef_draws, an n x p_re matrix of the total number of Block 2 candidates generated per stored draw, summed over the m_convergence inner sweeps. dIndependent_Normal_Gamma components count the envelope accept-reject candidates until acceptance (the iters_out already produced by rIndepNormalGammaReg, previously discarded by the Gibbs loop); dNormal components count exactly one conjugate draw per sweep, so their columns equal m_convergence. Dividing by m_convergence gives the average number of candidates per accepted draw (roughly the reciprocal envelope acceptance rate), matching the iters semantics of rglmb-style samplers. Reading the counts consumes no RNG, so draws are bitwise-identical to the previous version under the same seed. - Prior-vs-data guard for dIndependent_Normal_Gamma sampling: rindepNormalGamma_reg() now rejects calls where the Gamma (precision) part of the prior carries more effective prior observations than the data supply: inverting the Prior_Setup() calibration shape = (n_prior + 1 + p)/2, sampling requires n_prior <= n_w = sum(weights) (equivalently a prior weight pwt <= 0.5). Rationale: the dispersion envelope caps its log-tilt at n_w/2 - the data contribution to the posterior Gamma shape (Remark 4.1.3 of the ING vignette) - a strengthening of the validity condition lm_log2 < shape2 that presumes a likelihood-dominated regime. Prior-dominated calls could previously bind that cap on every envelope build (console UB3A mean slope warnings) and silently degrade the envelope. Note that n_prior here is the effective sample size of the Gamma component specifically; under the Prior_Setup() calibration the Gamma and coefficient parts share a common n_prior, so the two are not fully independent. - Same guard in the two-block v2 sampler: two_block_rNormal_reg_v2() enforces n_prior <= J per dIndependent_Normal_Gamma component (with J = length(group_levels), the Block 2 hyper-regression observation count and q_k = length(mu): 2*shape - 1 - q_k <= J, i.e. pwt_disp <= 0.5). Calibration-only paths (two_block_rate_v2()) are exempt since they use the disp_lower plug-in without sampling. - pfamily_list() generic: New S3 generic for building a named list of pfamily objects from a prior-specification container. Downstream packages provide methods (e.g. lmebayes for Prior_Setup_lmebayes() objects, mapping each random-effect component to dNormal() or dIndependent_Normal_Gamma()). - Convergence rate for the two-block sampler: New two_block_rate() computes the eigenvalues of A = P11^{-1/2} P12 P22^{-1} P21 P11^{-1/2} (Nygren 2020, Remark 8) for the joint Gaussian posterior targeted by two_block_rNormal_reg(), without ever forming the J*p_re x J*p_re Block 1 precision: the cross moment is accumulated per group with p_re x p_re solves followed by a single q x q symmetric eigendecomposition. The maximal eigenvalue lambda* is the geometric TV contraction rate of the sampler; m_for_tol(tol) returns the implied number of inner Gibbs sweeps. For non-Gaussian families explicit IRLS-style weights give a local-Gaussian heuristic. Validated against a dense brute-force construction of the joint precision and against the observed contraction of the ICM mean recursion (lmerb_posterior_mean()), which contracts at exactly lambda*. - Likelihood precision at the posterior mode: New two_block_mode_weights() evaluates per-observation likelihood precisions (IRLS/Fisher weights) at a supplied random-effects value - typically the joint posterior mode from glmerb_posterior_mode() - and assembles the per-group likelihood precision blocks Z_j' W_j Z_j. Weights are computed generically from the family object (w_i = wt_i mu'(eta_i)^2 / (V(mu_i) phi)): exact observed Hessian for canonical links (gaussian, poisson-log, binomial-logit), expected (Fisher) information otherwise - including correct probit/cloglog/Gamma-log weights where glmbfamfunc()$f7 carries copy-pasted logistic weights. The weights component feeds two_block_rate(weights = ) directly, providing the local-Gaussian heuristic input for extending the TV-rate analysis to non-Gaussian glmerb models. Validated against f7 on its correct branches and against the exact Gaussian rate path. - Explicit TV convergence bounds: New two_block_tv_bound() evaluates the total-variation bound between the l-step kernel and the target (Nygren 2020) from the two_block_rate() spectrum, two ways: method = "theorem3" computes the exact per-eigendirection terms d_i^(l) using the closed form erf_n(x) = pchisq(2 x^2, n) with r_i^(l) = (1 - a_{i-1}^{2l})/(1 - a_i^{2l}); method = "corollary1" evaluates the looser geometric envelope with explicit constants. With the chain started at the exact posterior mean (as lmerb does), the mean term vanishes identically (D0 = 0 default) and only the variance-convergence sum remains, which decays like lambda*^{2l} - twice the exponent of the crude (lambda*)^m proxy. two_block_l_for_tv() inverts the bound to give the number of inner Gibbs sweeps required for a target tolerance, and print.two_block_rate() now tabulates proxy vs Theorem 3 vs Corollary 1 sweeps. On the lmerb big_word_club example (lambda* = 0.839): TV <= 1e-3 needs 16 sweeps (Theorem 3) / 23 (Corollary 1) vs 40 for the proxy. - Two-block Gibbs loop in C++: The main loop of two_block_rNormal_reg() (Block 1 random-effects update, Block 2 hyperparameter update, m_convergence inner steps, replicate sampling) now runs entirely in C++ (two_block_rNormal_reg_cpp_export in src/twoBlockGibbs.cpp), eliminating per-iteration R/C++ round trips. This is a port-only change: the R wrapper still performs input validation, glmbfamfunc() resolution, and output assembly, and the C++ driver calls the same per-block samplers (rNormalGLM envelope sampler, rNormalReg) in the same order as the previous R loop. Draws are statistically equivalent but not bit-reproducible against the old R loop because the C++ rejection sampler uses its own RNG stream (compare averages over many draws, not individual draws). - Faster GLM block sampling: block_rNormalGLM() now performs block partitioning and prior payload assembly in C++ (block_rNormalGLM_cpp_export), removing per-call R overhead in block Gibbs loops (e.g. Block 1 of the lmebayes two-block sampler). The sampling algorithm itself is unchanged: each block still calls the existing rNormalGLM() envelope sampler serially. Posterior modes are numerically identical to the previous R-prep path; individual draws follow the same distribution but are not bit-reproducible against the old path (compare means over longer runs). Present-but-NULL prior elements (e.g. dispersion = NULL) are treated as absent, matching R is.null() semantics.