Joint multi-likelihood nested Laplace approximation

Outer-grid nested Laplace driver for joint models – multiple response arms sharing one latent prior block, parameterized as a per-arm field amplitude (sigma) on a unit-precision latent.

Supported priors:

"bym2" – outer grid over (sigma, rho [, alpha]). Latent: phi (n_s) | theta (n_s) with unit-precision ICAR + iid components.
"icar" – outer grid over (sigma [, alpha]). Latent: phi (n_s) with unit-precision ICAR.
"car_proper" – outer grid over (sigma, rho_car [, alpha]). Latent: phi (n_s) with Q = D - rho_car * W.

Other backends (NNGP, HSGP, RW1/2, AR1) follow the same interface and land under Phase 3.

Usage

tulpa_nested_laplace_joint(
  responses,
  prior,
  copy = NULL,
  phi_grid = NULL,
  prior_sigma = NULL,
  prior_alpha = NULL,
  prior_phi = NULL,
  cell_coupling = "separable",
  control = list()
)

Arguments

responses

A named list of arm specs (length >= 1). Each arm:

y – numeric [N_arm] response.
n_trials – integer [N_arm] (use rep(1L, N_arm) for non-binomial).
X – numeric matrix [N_arm x p_arm] fixed-effects design.
spatial_idx – integer [N_arm], 1-based map obs -> spatial unit.
re_idx – optional numeric [N_arm] 1-based RE group index; defaults to rep(0, N_arm) (no RE).
n_re_groups – optional integer (default 0L).
sigma_re – optional numeric (default 1); ignored when n_re_groups == 0.
family – one of "binomial", "gaussian", "poisson", "neg_binomial_2", "beta", "lognormal", "gamma", "inverse_gaussian". For "lognormal", y is on the natural scale and the linear predictor eta = E[log y] (identity link on the log scale); the -log(y) Jacobian is included in the kernel's log_lik.
phi – numeric dispersion (gaussian/lognormal residual SD, negbin size, beta precision); default 1.
field_coef – optional per-arm field coefficient controlling this arm's multiplier on the shared latent field's amplitude. One of: * numeric scalar (default 1) – constant multiplier. 0 means the arm carries NO field at all (the per-row scatter eta += field_coef * sigma * z is skipped for that arm). * character of length 1 – names an outer-grid hyperparam axis (currently "alpha"); the coefficient varies across the grid. * list(name = , grid = ) – embedded axis declaration, equivalent to declaring the axis and naming it on this arm. At most one arm may declare a hyperparam-driven axis (the cover hurdle and occu_cover both need only one). Shared axes across multiple arms are deferred. A single-block copy coefficient is declared here, on the arm – not through a separate copy argument.

prior

A list describing the shared latent prior block. Required field type. Backend-specific fields:

bym2: n_spatial_units, adj_row_ptr, adj_col_idx, n_neighbors, scale_factor (default 1); optional sigma_grid (donor-arm field amplitude, default 5 log-spaced values in [0.1, 3]), rho_grid (default c(0.2, 0.5, 0.8, 0.95, 0.99, 0.999)).
icar: n_spatial_units, adj_row_ptr, adj_col_idx, n_neighbors; optional sigma_grid (default 5 log-spaced values in [0.1, 3]).
car_proper: same as icar plus rho_car_grid (default c(0.5, 0.8, 0.95, 0.99)).

sigma_grid's default is a starting axis, not a hard ceiling: when the fitted field-SD posterior mode rails the top node (pareto_k_regime = "collapsed_edge", see below), the driver re-centres the axis on a mode-Hessian and refits (up to two attempts, the second adding a light default PC(U=3, alpha=0.01) prior on sigma unless prior_sigma was pinned – see there), so a sparse or strongly-identified species is not silently truncated at 3.0. This engages whether or not control$diagnose_k computed the full outer Pareto-k diagnostic: the mode-Hessian is reused from the diagnostic when it ran, or computed on its own (one extra batched finite-difference solve, only when the grid actually collapsed) when it did not – so diagnose_k = FALSE, the default, does not leave a railed axis stuck. A sigma_grid the caller PINNED always wins: auto-recenter engages when the field is left NULL, when it is marked with auto_grid() (how a wrapper package declares an axis it defaulted rather than one the user chose), or when its nodes are exactly the engine's own default axis. Declines gracefully (keeps the fixed-grid fit) when another axis in the same grid has unguessable support (car_proper's rho_car); whichever way it declines, outer_grid_recenter_declined says which (see below).

copy

Multi-block copy specification (multi-block prior only). For a single-block fit there is no copy argument: declare the copy coefficient on the arm via responses[[X]]$field_coef = list(name = "alpha", grid = G). On the multi-block path copy is an unnamed list of specs – list(list(arm, block, alpha_grid),...) – coupling N distinct shared latent fields, each onto its own arm with its own $\alpha$ axis, integrated over the product outer grid. Each spec must name a distinct block.

The $\alpha$ axis takes EITHER alpha_grid, which states its nodes, OR alpha_n (field_coef$n on the single-block path), which re-reads the engine's own axis at a higher RESOLUTION – more nodes between the same bounds, keeping the atom at 0 that gives the "no copy" base model posterior mass. They are different requests and supplying both is an error. Use alpha_n to integrate the same axis more accurately: it is the only outer axis a copy fit could not otherwise raise, since stating nodes for it also restates the prior structure it carries (gcol33/tulpa#633). The copy block may be any of icar / bym2 / car_proper / rw1 / rw2 / ar1 / iid; blocks with their own per-arm scaling (lf, hsgp_mo) or a precomputed precision (tgmrf) do not take a copy. A copy block's own sigma_grid (the donor field amplitude, same default ceiling as the single-block prior$sigma_grid above) auto-recenters on collapsed_edge the same way, one block per attempt.

phi_grid

Optional list specifying per-arm dispersion axes on the outer grid. Accepts either a named list (keys = arm names) or a positional list of length n_arms. Each entry is one of:

NULL or scalar – no axis for that arm; the kernel uses the parse-time scalar responses[[k]]$phi.
numeric vector of length > 1 – adds a new outer-grid axis phi_<arm> taking those values; the kernel rewrites arms[k].phi at each grid point before the inner Newton solve.

Family-specific interpretation of arm$phi (the parse-time scalar and the grid values):

gaussian – residual VARIANCE, the one engine convention (the SD is sqrt(phi)). Use phi_grid to estimate it as a hyperparameter instead of pinning it pre-fit.
lognormal – residual variance on the log scale; identical kernel parameterization as gaussian plus the -log(y) Jacobian.
neg_binomial_2 – dispersion (variance is mu + mu^2/phi).
beta – precision (variance is mu(1-mu)/(1+phi)).
gamma, inverse_gaussian – shape / dispersion.
binomial, poisson – ignored.

Each phi_<arm> axis is appended to the Cartesian product and varies slowest (within-spatial warm starts hold). The axis appears as a regular hyperparameter in theta_grid, theta_mean, and theta_sd, and participates in adaptive-grid refinement.

prior_sigma, prior_alpha

Optional regularizing hyperpriors on the donor field amplitude $\sigma$ and on the copy coefficient $\alpha$. Each is NULL (flat, default) or a list of the form list(family, params):

list("pc.prec", c(U, alpha)) – Penalized Complexity prior, calibrated by P(theta > U) = alpha. Closed-form density lambda * exp(-lambda * theta) with lambda = -log(alpha)/U. Drop-in for the weakly-identified small-n_pos regime . Pick U at the upper end of plausible values so the prior shrinks the tail without biasing the modal cell when the data identifies it: default-friendly choice on $\sigma$ is c(U = 1.0, alpha = 0.01) (donor amplitude); on the dimensionless copy coefficient $\alpha$ the recommended choice is c(U = 4.0, alpha = 0.01). What counts as strong is set by the measure the axis already carries, and on the copy scale that is not flat: with no prior_alpha it takes the engine's own exponential slab, whose rate puts 5 % of the prior above the largest declared node (control$copy_slab). A prior_alpha REPLACES that slab, so it shrinks harder than the default only once its own lambda = -log(alpha)/U beats the slab's – on a copy grid reaching alpha = 2, lambda = 1.50, the strength of U = 3.07. Too small a U over-shrinks the copy coefficient past the modal cell and, through the alpha * sigma copy axis, inflates the coupled donor amplitude sigma: on a fixture with truth $\alpha = 1$ and sigma = 0.6, c(U = 2.0, alpha = 0.01) pulls the $\alpha$ geometric mean to 0.93 and lifts sigma to 0.68, and c(U = 0.25, alpha = 0.01) to 0.51 and 0.94.
list("half_normal", scale) – half-normal with scale scale > 0. Sharper tail decay than PC; use when stronger regularization is desired and the truth is well inside the prior. The contribution is added to log_marginal cell-by-cell at the kernel-call boundary, so refinement passes (adaptive grid, var-of-means consistency) see the regularized posterior. When the data identifies the parameter (e.g. n_pos >= ~200) the prior is essentially harmless – the lever is tail-shrinkage at small n_pos. prior_alpha only applies when copy is active; prior_sigma applies on any sigma-named axis.

WHICH AXES ONE SPEC REACHES. A list-of-blocks prior can carry the same axis on several blocks at once – two copied fields carry b1.alpha and b2.alpha, and each copy block carries its own donor b<k>.sigma. One spec applies to EVERY block carrying the axis it names, each block taking its own independent copy of the density (its atom / continuum split is normalised against its own axis, so nothing is shared between blocks). To regularize the blocks differently, or only some of them, key the prior by block the way a multi-block copy spec does:

prior_alpha = list(
  list(block = 1, prior = list("pc.prec", c(4.0, 0.01))),
  list(block = 2, prior = list("half_normal", 2.0))
)

A block the list does not name carries no prior on that axis. An entry naming a block that carries no such axis is an error, and the message names the blocks that do. The per-block shape is a multi-block one: on a single-block prior, where the axis is unique, pass the spec itself. A fit reports what it applied in hyperprior_axes: one record per role with the scope the caller stated it at ("every_block" / "per_block"), the block indices reached, and the grid columns those are.

prior_sigma also interacts with the auto-recenter above: its second attempt engages the engine's own weakly-informative PC(U = 3, alpha = 0.01) prior, and a prior_sigma the caller PINNED suppresses that (the caller's prior stands). Pinning is decided by provenance, not presence: a spec marked with auto_grid(), or one equal by value to the engine's own default, is a default and does not suppress the escalation. When it does, the fit carries outer_grid_prior_declined = "prior_pinned", so a second attempt that changed only the grid geometry is legible rather than looking like the full escalation.

prior_phi

Optional regularizing hyperprior on the per-arm dispersion axes declared through phi_grid (e.g. a Beta precision on a cover arm, a negbin dispersion, a Gaussian residual SD). Same families as prior_sigma – NULL (flat over the phi grid, default), list("pc.prec", c(U, alpha)), or list("half_normal", scale). A single spec re-weights every phi_<arm> axis on the grid, the way prior_sigma re-weights any sigma-named axis; with no phi_grid it is a no-op. Without it the phi grid carries an implicit flat prior over its bounds. The PC scale is the dispersion's own units (a precision for beta, a size for neg_binomial_2), so pick U at the upper end of plausible values.

cell_coupling

Character scalar naming a per-cell coupled likelihood registered against tulpa's process-global registry (see src/cell_coupling_registry.h). Defaults to "separable", the arm- separable per-obs sum every existing joint fit uses. Consumer packages (e.g. tulpaObs) compile a tulpa::CellCouplingSpec subclass in their own src/ and register it from R_init_<pkg> via the tulpa_register_cell_coupling C callable; the R driver validates the name against the registry and the inner Newton routes the per-cell contribution through evaluate_cell() when the spec couples at least one arm.

control

Optional list of perf/numerical tuning knobs (statistical arguments stay top-level), following the control convention of tulpa(). Recognised elements (defaults in parentheses):

max_iter (50L), tol (1e-6) – inner Newton iteration budget and tolerance.
n_threads (1L) – inner-loop OpenMP threads (per-observation scatter, compute_eta, log-likelihood reduction). For typical joint workloads (N in the hundreds to a few thousand) inner parallelism is overhead-dominated; prefer n_threads_outer, which stacks better on many-core hardware. Capped at the physical performance-core count by default (see n_threads_scatter), as the inner per-observation loops oversubscribe a hybrid CPU's efficiency cores past that point. A fit is reproducible bit for bit at a GIVEN n_threads: the per-observation sum cuts its range into that many contiguous chunks and adds the chunk sums in chunk order, so nothing about the answer is left to the OpenMP runtime. Chunking imposes its own association, so two different n_threads agree only to floating-point tolerance (measured at 6e-14 on log_marginal over four families).
n_threads_scatter (performance-core count) – cap on the inner per-observation threads. Overrides the default performance-core cap on n_threads; raise it to use all logical cores or lower it to leave headroom. No effect where the core topology cannot be resolved (off Windows), where n_threads is used as requested.
n_threads_outer (1L) – outer-grid OpenMP threads. When > 1, a pilot Laplace at the centre cell warm-starts the remaining cells, each dispatched across n_threads_outer threads with its own CHOLMOD solver and NewtonScratch (inner OpenMP auto-disabled). 1L is serial, chained warm-starts – bitwise identical to the pre-speedup driver. Recommended on multi-core workstations: parallel::detectCores() - 1L. It is a REQUEST: the driver clamps it to the team the OpenMP environment hands out (omp_get_max_threads(), OMP_THREAD_LIMIT, and the check farm's core cap), because the per-cell block cache sizes its slot array from that same number. A job script exporting OMP_NUM_THREADS=1 to pin the INNER threads therefore runs the outer grid serially whatever is asked for here; the fit says so once, in a message, and records the pair in n_threads_outer_requested / n_threads_outer_realised.
tile_warm (TRUE) – when n_threads_outer > 1 and a copy block is present, group outer cells into tiles sharing every axis except the copy coefficient alpha, solve one warm Tier-2 per tile from the centre pilot, and warm-start the rest from their tile pilot. Falls back to the single-tier path when no copy block / single tile / n_threads_outer <= 1L. FALSE recovers the pre-tiling behaviour (e.g. regression testing).
prune (FALSE), prune_tol (1e-3) – opt-in cheap-pass screening. When prune = TRUE, the driver sweeps the outer lattice running a short inner Newton per cell, each warm-started from the previous screened cell's quasi-mode (lattice-adjacent), computes a screening Laplace log-marginal, softmax-normalises, and skips the full inner Newton on cells whose normalised weight is < prune_tol. The neighbour-warm-start sweep keeps every cheap mode near its cell's true mode, so the cheap ranking is faithful to the full-solve ranking even when the inner latent mode moves substantially across the grid. Pruned cells get log_marginal = -Inf, n_iter = 0, and inherit the pilot mode; the pilot cell is never pruned, and at least prune_min_keep cells (5) are solved in full whatever the tolerance says – the highest-ranked dropped cells are restored up to that floor, because the outer grid placement pass reads a finite-difference curvature off the cells that were SOLVED and a kept set of one carries none. Two fallbacks to the full grid, each with a warning: the cheap-pass safety gate (the cheap-screen argmax disagrees with the full-solve argmax, or the kept posterior collapses onto a cell the screen mis-estimated by more than the margin it discarded cells by), and a screened fit whose grid placement declined for want of curvature. Neither bounds the error from a cell that was discarded and never solved: the gate compares the screen against the cells it kept. Stacks with n_threads_outer. prune_tol must be in [0, 1). Default FALSE (the full grid is correct).
prune_log_gap (unset) – the screening cut in nats: keep every cell within this many nats of the best screened cell. prune_tol is a normalised weight, so what it cuts at is the gap -(log(prune_tol) + log(Z)), and the whole range a caller would type is a few tens of nats – on an outer surface spanning thousands every setting returns the same kept set. It reaches the kernel as prune_tol = exp(-gap), so the realised cut is the requested gap less log(Z); the fit reports the realised value in prune_log_gap_cut. Setting both this and prune_tol is an error.
screen_iters (5L) – inner Newton steps the cheap screen takes per cell, read only when prune = TRUE. The screen only has to RANK the cells, and each one is warm-started from its already-screened lattice neighbour, so its quasi-mode is near the cell's own mode after very few steps. Every step is paid on every cell of the grid, including the cells the screen keeps and then solves in full, so a depth above what the ranking needs makes screening cost more than the solves it avoids. Must be a single integer >= 1.
x_init (NULL) – warm-start for the first grid point's inner solve.
verbose (FALSE) – when TRUE, announce the engaged outer integrator for a multi-block prior in one line at selection time (see integration), and report each CCD decline reason.
store_Q (FALSE) – also return the per-grid joint precision Q (lower triangle, CSC) as Q_csc_p_per_grid, Q_csc_i_per_grid, Q_csc_x_per_grid, Q_csc_n, letting callers compute INLA-style total-variance posterior moments (Var-of-means + Mean-of-Var) on inner latent coordinates such as fixed-effect betas.
keep_grid_hessians (TRUE) – retain the per-grid-point fixed-effect mode and marginal precision as $grid_modes / $grid_hessians, which is what lets summary.tulpa_fit(), confint.tulpa_fit() and vcov.tulpa_fit() report the grid-marginalized fixed-effect covariance instead of NA. Memory is O(n_fixed^2) per cell. When the retention is not available the reason is recorded on $grid_fixed_declined.
adaptive_grid (FALSE), adaptive_grid_edge_thresh (0.02), adaptive_grid_max_passes (1L) – when adaptive_grid = TRUE, a mode-tracked 1D refinement pass triggers on any axis whose marginal boundary weight exceeds adaptive_grid_edge_thresh. New points are appended on that axis (interior densification + outward log-spaced extension) paired with the boundary cell's modal other-axis values, each carrying a calibration term so it contributes on the marginal scale – O(n_new_points) kernel solves, not the full cartesian product. The edge score is max(marginal_weight_at_boundary, exp(max_log_marginal_at _boundary - max_log_marginal_overall)), catching both boundary pile-up and integrand truncation; 0.02 is ~4 log units of decay. adaptive_grid_max_passes caps the passes (one usually suffices). Fixes posterior CI under-coverage when truth sits near a grid edge.

How far a given axis may be moved depends on where its nodes came from. An axis whose nodes the CALLER wrote down – copy$alpha_grid / field_coef$grid, an entry of phi_grid – states where the fit integrates, so refinement densifies it and never places a node past either end node; a mode outside that range shows up as mass at the edge. An axis the ENGINE placed (alpha_n / field_coef$n, or no nodes given) carries no such statement, so refinement may follow the posterior out past its ends. Nodes a wrapper package computed as a default of its own are declared with auto_grid() and count as engine-placed.
axis_refine (NULL) – per-axis override of the rule above, named by outer-grid axis ("alpha", "phi_<arm>"): "none" (the axis takes no refinement nodes), "densify" (nodes inside the declared span only) or "extend" (nodes past the end nodes too). A single unnamed value applies to every refinable axis. An unknown axis name, or a request for nodes on an axis this driver does not refine, is an error rather than a silent no-op. Read only when adaptive_grid = TRUE or var_of_means_consistency = TRUE.
var_of_means_consistency (TRUE) – run a post-integration consistency pass on the variance of the per-arm posterior means and attach var_of_means_consistency_info.
inner_factorization ("auto") – which factorization the dense inner Newton applies to the Hessian it assembled: "auto" by the latent dimension, "sparse" for CHOLMOD, "dense" for the dense Cholesky. Independent of force_sparse, which selects the assembly path.
force_sparse (FALSE) – linear-algebra backend for the inner joint solve. TRUE / FALSE select the sparse or dense path outright, regardless of the dense/sparse heuristic. "auto" selects by the latent dimension the fit will actually build, taking the sparse path above n_x > 1000 and the dense one at or below it, where the sparse symbolic-analysis and indirection overhead outweighs the fill-in saving. A model whose latent dimension cannot be determined resolves to dense.
integration ("auto") – outer-grid node layout for a multi-block prior. A central composite design (CCD) integrates the hyperparameter posterior on 1 + 2d + 2^d nodes oriented by the Cholesky of the posterior covariance at the joint hyperparameter mode – far fewer inner solves than the d-dimensional tensor product (25 vs 81 at d = 4). "auto" (the default) uses the CCD only at >= 4 transformable axes, where the tensor product's k^d blow-up bites hardest, and keeps the cheaper, more ridge-robust tensor grid at <= 3 axes; "ccd" lowers the CCD threshold to >= 3 axes; "grid" always forces the full tensor product. "grid_adaptive" is the low-dimensional companion to the CCD: it seeds a coarse subsample of the SAME tensor lattice (latent block axes and phi axes together), floods outward from the posterior mode on the fine lattice, and evaluates only the cells within a log-density cutoff of the peak – a strict, uniform-weight subset of the dense tensor, so its posterior matches the dense grid to that cutoff at fewer inner solves when the hyperparameter posterior concentrates (a sharply-identified field SD / precision). It declines back to the dense tensor on a diffuse posterior (kept region would rival the tensor) or a degenerate lattice, so it never costs accuracy; tune it with adaptive_grid_cutoff / adaptive_grid_stride / adaptive_grid_max_frac. The CCD auto-falls back to the tensor grid for an axis whose support is not safely transformable (a CAR_proper rho_car or a non-BYM2 rho), or a flat / ridged / degenerate outer mode-find or Hessian; an active phi_grid rides as a tensor axis crossed on top of the CCD. Single-block joint priors always use the tensor grid. The CCD mode-find runs cheap warm-started inner solves and does not write to the checkpoint file. Under verbose = TRUE the engaged integrator is announced in one line at selection time (e.g. outer integration: CCD (4 latent axes, 25 nodes) or tensor grid (72 cells), or CCD declined -> tensor grid), so the auto switch to the CCD at >= 4 axes is never silent. The resolved integrator is also returned on the joint result as $integration, alongside $integration_requested and $integration_declined (see Value), so a fallback is on the fit itself and not only in a verbose message.
ccd_budget (1) – ceiling on what PLACING a CCD may spend, as a multiple of the tensor grid the design replaces. Every mode-find probe is one full inner Newton solve, the same solve an outer grid cell costs, and a placement round is 1 + 2d + 4 C(d, 2) of them (33 at d = 4), so at the round caps alone a placement can spend far more than the integration it avoids. At 1 a placement plus its design costs no more than that tensor grid; the fit declines to the tensor grid with integration_declined = "placement_budget" when the ceiling is reached, and reports what it spent as ccd_modefind_evals (see Value). Inf places without a ceiling. ccd_budget_floor (TRUE) exempts the designed first attempt – the eight rounds from the grid-median seed, whose cost is set by the axis count and not by the grid – so the ceiling bounds the escalation (grid seed, step calibration, the 30-round rescue mode-find) rather than the path the CCD was built for; FALSE makes ccd_budget a hard cap on the whole placement.
local_ccd (NULL) – local CCD refinement of a multi-block tensor grid. TRUE (defaults) or a list(max_cells =, f0 =, skew_max =) refines a few high-weight, mutually non-adjacent interior cells, replacing each with a small curvature-aware CCD node cloud so a coarse base grid resolves the sharply-peaked directions without the k^d tensor blow-up. The local curvature is a diagonal finite difference of the outer log-marginal over the cell's own grid neighbours (no mode-find; only the off-centre nodes are new solves), warm-started from the cell's inner mode; each refined cell's sub-nodes carry partition-of-unity design weights so the total integration weight is conserved (no double-count). max_cells (8L) caps the refined cells; f0 (1.1) is the CCD radius. The design scale is shrunk per cell so the cloud fits the cell's Voronoi box (the local-Gaussian mass beyond it belongs to the neighbouring cells). A cell keeps its cloud only while the nodes' own log-marginals stay within skew_max (the gamma3_ok band, 0.5) of the quadratic the cloud was placed from, measured as a standardized cubic magnitude; above it the cell is put back as its own mass atom, which on a skewed outer target is measurably closer than the design. Engages only on the tensor path (the curvature stencil needs axis neighbours), at >= 4 transformable latent axes, with no active phi_grid; otherwise it is a no-op. The applied refinement is summarised on the result as $local_ccd_info. Also driven automatically by k_refine = "ccd".
adaptive_grid_cutoff (10), adaptive_grid_stride (2L), adaptive_grid_max_frac (0.75), adaptive_grid_min_cells (48) – tuning for integration = "grid_adaptive". adaptive_grid_cutoff is the log-density keep / expand radius from the peak (larger keeps more cells, closer to the dense tensor); adaptive_grid_stride the coarse-seed subsample stride per axis; adaptive_grid_max_frac the kept-fraction ceiling past which the builder declines back to the dense tensor; adaptive_grid_min_cells the smallest dense tensor worth the adaptive machinery – below it the builder declines BEFORE any inner solve (on a small tensor the coarse seed is already most of the grid, so there is no tail to skip). Ignored by the other integrators. The kept-cell / dense / solve counts are returned as $adaptive_grid_info.
inner_refresh (1L) – inner-Newton Cholesky factor reuse interval (Shamanskii / chord method). For a non-quadratic positive arm (e.g. a beta cover arm) the latent Hessian changes every inner iteration, so the default re-factorizes the sparse Cholesky on each step – the dominant per-grid-cell cost. inner_refresh = m > 1 re-factorizes only every m-th inner step and reuses the cached factor in between (refreshing early whenever a reused solve fails). The gradient is exact on every step and each step is line-search safeguarded, so the converged mode is unchanged and the final mode-pass Hessian (log_det, SEs) is always fresh; only the path to the mode uses a stale curvature, which may cost a few extra inner iterations. Applies to the sparse joint path with the default control$hessian = "lm" curvature; the dense small-n_x path re-factorizes a cheap Hessian and ignores it. 2L-4L is a good range for a slow beta arm.
k_quality ("report") – the reliability intent for the outer Pareto-$\hat{k}$, a single statement of how reliable the fit should be. "report" (default) computes the diagnostic and reports the achieved band. "ok" / "good" additionally name a TARGET band (the $\hat{k}$ confidently usable, resp. good) and raise the default k_samples (to 800L / 2000L, unless you set it) so the bootstrap CI can resolve it. "none" disables the diagnostic. The fit carries an honest verdict – k_quality_requested, k_quality_reached, k_quality_best, k_quality_miss, k_quality_reason, k_quality_rounds, k_quality_k_trace – and never silently downgrades: if the requested band is not confidently met it reports the band actually reached and why. For "ok" / "good", when the first fit does not reach the band the engine escalates by REFINING THE INTEGRATION GRID (see k_refine), widening / densifying it where the posterior mass escapes its current bounds and re-diagnosing, up to k_max_rounds rounds. Refinement is the first lever on either kind of miss because it lowers $\hat{k}$ itself rather than the uncertainty around it. Once refinement is exhausted – no boundary mass left to chase – what remains depends on the miss (k_quality_miss): a "resolution" miss ($\hat{k}$ confidently outside the band) ends the chase, while a "precision" miss (the bootstrap CI straddling a boundary, so $\hat{k}$ may already be inside the band) spends its remaining rounds doubling the importance-draw budget, the one lever refinement does not touch.
k_refine ("grid") – the integration-refinement rung for k_quality "ok" / "good". "grid" (default) re-fits with adaptive grid refinement (adaptive_grid) each escalation round, driven by the bad $\hat{k}$, so a too-coarse / too-narrow grid is widened / densified where the importance weight concentrates until the band is reached or the budget is spent. "ccd" instead refines a few high-weight, mutually non-adjacent interior cells with local curvature-aware CCD node clouds (see local_ccd), the right rung when the grid is too coarse to resolve a sharply-peaked direction rather than too narrow; it forces a tensor base grid (the curvature stencil needs axis neighbours) and engages only on the multi-block path at >= 4 transformable latent axes. "none" disables refinement: the band is reported but not chased.
k_max_rounds (2L) – the escalation round budget for k_quality "ok" / "good": the maximum number of re-fit rounds after the first fit. A refinement round allows one more pass than the last refinement round did; a round taken after refinement is exhausted leaves the grid where it stands and doubles the importance-draw budget instead. 0L disables escalation (single-shot, the band is reported but not chased).
diagnose_k (TRUE), k_samples (500L) – compute the outer Pareto-$\hat{k}$ accuracy diagnostic by importance-sampling the joint hyperparameter posterior against the proposal the integrator fits (mixed per-axis transforms: log for positive scales, logit for the BYM2 mixing weight, identity for the copy coefficient $\alpha$). k_samples is the number of importance draws, each one an extra inner joint solve, and is where more tail information comes from: a tighter k-hat needs MORE actual tail ratios, so increase k_samples (not k_bootstrap). It is not a pure precision knob, though. Under the automatic PSIS tail rule (min(S/5, 3 sqrt(S))) the fitted tail FRACTION shrinks as 3 / sqrt(S), so a larger budget describes a DEEPER quantile of the weight distribution and the reported k-hat can move materially – measured on a synthetic heavy-tailed outer target, 0.57 at 500 draws and 7.94 at 50000 (gcol33/tulpa#631). Set k_tail_points in proportion to k_samples to hold the fraction and get a strictly sharper estimate of the same number; the k_quality escalation's draw rung does exactly that. The draws are RNG-restored so the fit's modes / draws are unchanged. A fit carrying an axis whose support is not safely known (CAR_proper's rho_car) declines to the quadrature-ESS fallback (pareto_k = NA). FALSE skips the diagnostic. "by_arm" additionally computes a k-hat restricted to each arm's hyperparameter axes (the other arms held at their posterior mean), reported in pareto_k_by_arm, to localise which arm drives a tail-heavy joint k; the joint k itself is unchanged. Per-arm k is defined for the multi-block layout with two or more arms and declines for the single-block shared-field layout.
k_threads (NULL) – outer-thread width for the diagnostic's importance batch. The k_samples re-solves are independent and run after the grid (every core free), each solved single-threaded once the batch saturates the pool, so widening it is a bit-identical wall-clock speedup (the k-hat is unchanged). NULL follows the fit's own thread grant – the larger of n_threads_outer and the inner n_threads – so a serial fit keeps a serial diagnostic while a threaded fit gets a free parallel one. "auto" uses the physical performance-core count (capped at 2 under R CMD check); an integer pins the width (1L forces serial). Always capped at k_samples.
k_bootstrap (1000L) – bootstrap replicates for the outer Pareto-$\hat{k}$ uncertainty. The k-hat is a single fixed number for a fit + proposal; its sampling uncertainty GIVEN the proposal is estimated by resampling the diagnostic's raw importance log-ratios with replacement and re-fitting the GPD tail k_bootstrap times (no new inner solves). Reports pareto_k_se_boot (bootstrap SE), pareto_k_ci_low / pareto_k_ci_high (the 2.5\ closed-form GPD-shape MLE asymptotic SE $(1 + k)/\sqrt{M}$, a cross-check), and pareto_k_band_confident (TRUE iff the bootstrap CI lies within one reliability band). The bootstrap measures how UNSTABLE the current tail estimate is; it cannot create tail information. Increase k_samples, not k_bootstrap, to obtain more tail information. 0L skips it (point k-hat only). The per-arm k carries the same fields.
k_tail_points (NULL) – number of upper-tail order statistics for the GPD fit. NULL uses the automatic PSIS rule $\lceil\min(0.2 N, 3\sqrt{N})\rceil$. An explicit value is an EXPERT tail-threshold control, capped at the 20\ so the fit stays an extreme tail; it is NOT a precision knob (a request that drags body ratios into the tail lowers variance but biases the k-hat). The used and requested counts are reported in pareto_k_tail_points / pareto_k_tail_points_requested.
k_conf_bands (NULL) – the reliability-band boundaries the bootstrap CI is tested against for pareto_k_band_confident. NULL (default) uses the sample-size-dependent boundaries $c(0.5, \min(1 - 1/\log_{10} S, 0.7))$ at the realised draw count $S$ (Vehtari et al. 2024): the good cut is 0.5 and the usable cut tightens below 0.7 for small $S$ (about 0.565 at $S = 200$, reaching 0.7 only past $S \approx 2154$). Supply a strictly-increasing numeric vector to fix the boundaries instead, e.g. c(0.5, 0.7) for the size-independent good / ok / unreliable split.
diagnose_skew (TRUE), skew_idx (NULL) – compute the inner-Laplace skewness diagnostic ($inner_skew, gamma_3, Rue Martino & Chopin 2009 Sec 3.2.3) at the fitted MAP grid cell: one extra Newton solve via the same kernel_fn the outer diagnostic reuses, scoring every arm's fixed-effects coefficients by default (pass skew_idx, 1-based indices in the joint [arm1_beta | arm1_re | arm2_beta | ... | blocks] latent layout – see $arm_layout – to probe additional indices). This is the complementary layer to diagnose_k: that scores the outer hyperparameter-grid integration around a FIXED inner Laplace, this scores whether that inner Gaussian approximation is itself a good fit. A genuinely coupled arm (cell_coupling != "separable" on that arm, e.g. tulpaObs's occu_cover) has no per-obs likelihood for the separable formula to score; it is scored instead by the contraction of the cell third-derivative tensor, which differences the cross-arm Hessian the coupling spec already returns. A fit that can carry neither reports NaN, not a silently wrong 0, and $inner_skew_declined says WHY – "coupled_arm" marks arms the inner layer could not score at all, distinct from a diagnostic that was simply switched off – while $inner_skew_arms_declined names the arms with no oracle, including on a partially scored fit. See diagnostics() for the combined verdict. Wired for both the single-block backends (icar/bym2/car_proper) and the multi-block path (a per-group RE, a trend field, or an arm-specific field block).
within_cell ("box_uniform") – the WITHIN-CELL construction the reported per-axis hyperparameter intervals are read with. The outer grid's weights say how much mass each cell holds; they do not say how it is spread inside the cell, and a quantile needs both. "box_uniform" puts the cumulative FULL mass at each cell EDGE and interpolates between edges; "chord" puts the cumulative MID-mass at each cell coordinate and interpolates between coordinates – the same masses over the same boxes with the knots moved half a cell, which measures as a whole order of convergence (2.00 against 1.04 on a fixture with a closed-form posterior). THE DEFAULT IS "box_uniform" since 0.0.188, decided on FIXED-TRUTH coverage at the placement the engine ships, with auto_recenter = "resolve" as the default. Summed |coverage - nominal| over nominal 0.95 / 0.80 / 0.50, chord against box-uniform: 0.2900 / 0.1233 on the pre-registered fixed-truth instrument, 0.2004 / 0.0361 over 4680 truth-swept fits of the same fixture, and 0.2467 / 0.1572 over nine (config, axis) rows spanning seven families, at 0.69 to 1.08x the width. The conditional-coverage swing that held the default back reads 0.110 at the shipped placement against 0.415 on the coarse pinned grid it was measured on, and at nominal 0.50 it is the same on both reads. outer_grid_h_over_sd is how wide a cell is on each axis, and theta_within_cell is what each axis was actually read with. Only a "density" support admits it – a CCD design, a locally refined grid and a posterior sample are not cell partitions that tile – and an axis it declines on reports "chord" with a reason rather than erroring. Nothing else moves: point estimates, moments, draws and weights are untouched, and "chord" restores the previous report exactly.
skew_correct (TRUE) – consume the inner-Laplace expansion instead of only grading it: report Cornish-Fisher marginal quantiles at each coefficient's own gamma_3, about the centre gamma_1 + gamma_3 / 2, from summary() / confint() wherever the combined inner band (gamma_3 and the inner importance k-hat) says the leading-order expansion is in its regime, and the Gaussian quantiles everywhere else. It is post-processing on the reported quantiles: draws, modes and weights are untouched, so a fit run with it off is bit for bit the fit it was before. The band that bounded the relocation itself (centre_unreliable) is off (Inf): every finite cutoff was measured to decline the coefficients the correction helps most. $skew_correction records the per-coefficient gamma_3, gamma_1 and the centre they form, the band, the k-hat, the combined band, the eligibility and the reason behind it; the skew_applied attribute on summary() / confint() records what was actually used at the requested level. A FULLY COUPLED fit declines it: the location term's contraction against a covariance block is not reachable from the cell third-derivative oracle, and an absent gamma_1 is never read as zero. Such a fit therefore reports what it reported before the default moved, to the bit – a declined coefficient keeps the grid-mixture read. See tulpa_nested_laplace() for the measurement behind the default.
subspace_debias (FALSE) – correct only the latent directions the inner-layer diagnostics flagged, by exact Metropolis along the Gaussian-conditional-mean surface through each cell's mode, and leave the rest at their Gaussian conditional. Settings and semantics are the ones documented on tulpa_nested_laplace(). On a fully coupled fit gamma_3 is NaN for every arm, so the selector rests on the derivative-free inner Pareto-k-hat; where that also bands the coordinate reliable, idx pins the set explicitly. An EMPTY selection leaves the fit bit-for-bit identical to the plain path. A fit whose inner solve took the s2z rank-1 or the PSD eigen-clamp path carries no usable factor to build the surface from and is left uncorrected, the same two paths diagnose_skew declines on.
cila (FALSE) – corrected integrated Laplace, the second inner-layer debias (after Lai, Margossian and Sheldon, arXiv:2605.20345). Where subspace_debias selects coordinates and runs exact Metropolis on them, this selects nothing: at every outer cell it draws n_points points from the whole inner Gaussian, weights each by the exact joint density it came from, and reports the weighted particles. The cell marginals and the latent posterior both converge to the exact ones as the effort grows, so n_points is the only dial. TRUE takes the defaults; a list overrides n_points (1024L), variant ("qmc", a Sobol net; "is" for iid draws, "rqmc" for the net under n_shift random shifts), n_shift (8L), n_draws (the reported draw count) and seed (the auxiliary stream, engine-owned so requesting the correction leaves every other posterior draw unchanged). Below 512 points a cell's particle set is too coarse to be a marginal at all and the request is refused. The correction reports DRAWS, so every coefficient-facing method reads them instead of the grid's Gaussian mixture, and the corrected per-cell masses become the fit's own weights / log_marginal with weights_source reporting "cila"; the pre-correction pair is kept as $cila$laplace and $cila$retained_mass is the original share of the cells that produced a usable particle set. $cila also carries the variant actually run and the PSIS grade (pareto_k, rel_ess) of the correction's own weights. A cell whose inner solve factorized SPARSELY draws through the CHOLMOD factor's own triangular and permutation solves; an LDL' factor carries no square root to draw with and is declined with "sparse_factor_not_ll".
auto_recenter (TRUE) – re-centre a default outer grid axis on its posterior mode and refit when the fit rails against a boundary node. FALSE integrates over the grid exactly as given, whatever it is, and records outer_grid_recenter_declined = "auto_recenter_disabled". The joint rescues trigger on the whole grid's collapsed-edge regime rather than on a per-axis rail, so the per-axis policy names tulpa_nested_laplace() takes ("rail", "resolve", "always") are refused here with an error rather than accepted and ignored.
recenter_pilot (FALSE) – detect the placement above on a THINNED grid rather than on the full one. Placement reads two things, the argmax cell and an FD curvature stencil at it, and reads both off log_marginal; neither is read off the integration the detecting pass paid for, so every inner Newton solve of that pass is discarded the moment a placement fires. With the pilot on, the detecting grid keeps each axis's endpoints and at most TRUE's three nodes between them (an integer names its own resolution), and the full grid is solved once, at the placed axes.

Off by default because the trade is a property of the WORKLOAD. With F full cells, P pilot cells and a firing rate p, the un-piloted cost is F + p (S + F) against the pilot's P + p S + F, so the pilot pays exactly when P < p F. Measured on a 3-axis (sigma, alpha, phi) donor + copy fixture over 48 paired seeds: 0.78x the cells solved on the configuration that places on 10 of 12 seeds, and 1.29x on the one that never places.

A coarse detector is not the same detector. The collapse trigger is ess_grid < 2, which falls with the cell count, so a pilot fires somewhat more readily than the full grid: on that same sweep the two disagreed on 9 of 48 pairs, every one of them the pilot placing where the full grid did not, and 7 of the 9 reached a HIGHER maximum inner log-marginal after placing (+0.35 to +2.14 nats) while 2 lost 0.46 and 0.72. It cannot go the other way: a declined pilot is followed by the full fit, which then gets its own detection, so the pilot only ever adds a placement. A fit that declines both is bit-identical to the same fit with the knob off. What the pilot detected on is recorded in outer_grid_pilot.
max_grid_cells (2048L) – cell-count ceiling on a multi-block tensor outer grid, refused with an error above it. Each cell is one inner Newton solve, so the default catches per-block grids that multiplied out to a run nobody asked for; a deliberate converged tensor reference grid (4 axes at 7 levels is 2401 cells) raises it here, which integration = "ccd" cannot serve since a CCD is a different integration design.
checkpoint (NULL) – grid-cell checkpoint/resume. Set list(path = "fit.ckpt", resume = TRUE) to make a killed or interrupted fit resumable: each completed outer-grid cell is appended to path, and a later call with the same responses + grid + control loads the finished cells and solves only the rest. EVA-scale joint fits run for hours, so a wrapper teardown, reboot, or OOM near the end otherwise loses the whole run. resume = TRUE (the default when a path is given) continues from an existing file; resume = FALSE removes it first and starts over. Adaptive-grid refinement cells are checkpointed under their own coordinate keys, so resume covers them too. A file written for different data or solver settings is rejected (fingerprint mismatch) rather than resumed onto a stale result.

Value

A list of class c("tulpa_nested_laplace_joint", "tulpa_nested_laplace", "list") with:

theta_grid, theta_names – outer-grid hyperparameter values (includes the alpha axis when copy is set).
log_marginal, weights – per-grid-point log-marginal and integration weights (sum to 1).
theta_mean, theta_sd – posterior moments per hyperparameter, including alpha when copy is set.
theta_median, theta_ci_lo, theta_ci_hi – weighted-quantile median and 2.5/97.5 empirical CI per hyperparameter axis (same names as theta_mean). Recommended summary for right-skewed scale-like axes (alpha at small n_pos, sigma/range/phi near a boundary), where the posterior mean is pulled by the right tail away from the bulk and mean +/- 1.96 sd mis-states the uncertainty.
modes – [n_grid x n_x] matrix of inner modes.
n_iter – inner Newton iterations per grid point.
arm_layout – list with per-arm beta_start, re_start, spatial offset(s) and n_x for decoding modes.
prior, responses, copy – echoed inputs.
timing – named numeric of wall-clock seconds: total plus the setup (validation / encoding / grid construction), grid (inner Laplace solves, including adaptive-refinement and consistency passes), postproc (weight / moment / marginal assembly) and diagnostics (outer Pareto-$\hat{k}$) phases. The grid phase is the one that scales with grid size and core count. Surfaced one-line in print.
pareto_k, pareto_k_is_ess, pareto_k_scope – outer Pareto-$\hat{k}$ accuracy diagnostic and its importance-sampling ESS (both NA when control$diagnose_k = FALSE or the fit declines; see the diagnose_k control knob). pareto_k < 0.7 indicates the nested integration is reliable; >= 0.7 that the (skewed / heavy- tailed) hyperparameter posterior is misfit by the Gaussian grid and the fit should escalate to an exact debias.
pareto_k_declined – when pareto_k is NA, WHY: "not_requested" (control$diagnose_k = FALSE; nothing is wrong), "unguessable_axis: <axis>" (a support the engine will not guess, e.g. car_proper's rho_car – a PERMANENT limitation of that family, so read the quadrature ESS instead), "draws_too_few", "grid_too_small", "no_varying_axis", "degenerate_proposal" (the outer mode curvature came back non-finite – a signal about the fit), "not_applicable", or "internal_inconsistency" (an engine bug worth reporting). NA on a fit whose $\hat{k}$ WAS computed.
pareto_k_proposal_source – how the outer importance proposal the $\hat{k}$ scores was built: "mode_hessian" from the Laplace curvature at the hyperparameter mode (the CCD design's, or a finite-difference Hessian when a sharp posterior collapses the grid), "grid_moment" from the grid-weighted covariance, "moment_matched" when the moment-matching refinement improved on either, "grid_mixture" when the faithful mixture-over-grid-cells proposal won, or "skew_normal" when the skew-normal rescue did. NA when the diagnostic is off or declines. The mode-Hessian source keeps the $\hat{k}$ meaningful when the grid concentrates on ~1 cell.
pareto_k_regime – whether the outer grid actually integrated hyperparameter uncertainty: "spread", or "collapsed_interior" / "collapsed_edge" when the quadrature effective sample size falls below two cells so the integration has degenerated to a point evaluation at the modal hyperparameter. An interior collapse is benign (the grid bracketed the mode; the fit is empirical Bayes at that mode and only the integrated hyperparameter uncertainty is missing); an edge collapse means the mode sits at an extreme node, so the grid may be too narrow. Attached from the stored weights, so it is present even with control$diagnose_k = FALSE.
pareto_k_grid_edge_axes, pareto_k_grid_edge_sides – for an edge collapse, which axes the dominant cell sits against and on which side ("lower" / "upper"); widen those axes and refit to confirm the mode is bracketed. Axes the grid pins to one value are excluded (pinned, not at a boundary).
axis_span – per outer axis, the span the fit worked over and the two terms that separate it from the nodes it was given: nodes (the range of the axis's initial continuum nodes), declared (their support, i.e. those nodes plus the half node step the outermost cells own – for k equally spaced nodes that is k / (k - 1) times the node range on the axis's integration coordinate, so 2x at two nodes and 1.125x at nine), integrated (the support after refinement, the same interval axis_support reports), refine (the mode refinement ran under: "none" / "densify" / "extend") and n_nodes (initial and final continuum node counts).
outer_grid_placement – "fixed" (the default sigma_grid axis was used as-is) or "auto_recentered" when a collapsed_edge on sigma triggered the mode-Hessian recenter-and-refit (see the prior argument above). outer_grid_recenter_attempts (integer) and outer_grid_prior_added (logical: whether the light default PC(U=3, alpha=0.01) sigma prior was applied) are set alongside it, plus outer_grid_prior_declined = "prior_pinned" when a second attempt ran but the caller's pinned prior_sigma held that prior back.
outer_grid_recenter_declined – on a "fixed" placement, why the recenter did not run: "grid_not_collapsed" (the grid already brackets the mode – the common case, no refit needed), "axis_pinned" (the caller pinned sigma_grid; mark it with auto_grid() if it is a default rather than a choice), or "no_usable_curvature" (the mode-Hessian the recenter needs was unavailable or degenerate, e.g. a car_proper grid whose rho_car axis has unguessable support). Absent when the fit WAS recentred.
outer_grid_pilot – present only when control$recenter_pilot ran: the pilot's resolution (n_pilot), its cell count (cells), the axes it thinned (axes) and those it could not (axes_kept), and what the placement was DECIDED from – the pilot's own regime, ess_grid and edge_axes. The reported grid is never the grid the trigger was read on, so a two-pass fit is legible from the object rather than only from the progress log. outer_grid_pilot_declined = "pilot_placement_declined" records a pilot whose detection placed nothing and which was therefore followed by the full fit and the full fit's own detection.
outer_grid_recenter_sd_clamp, outer_grid_recenter_sd_raw, outer_grid_recenter_sd_used – on a recentred fit, one entry per moved axis: which mode-SD bound the placement hit ("none" / "floor" / "ceiling"), the SD the finite-difference stencil MEASURED, and the SD the axis was actually laid from. A clamped axis was laid from a substituted spread rather than a measured one, and the reported interval is read off that span; the two used to be indistinguishable on the fit.
pareto_k_outer_skew – per-axis skewness of the hyperparameter marginal in the proposal's whitened coordinate, present only when a $\hat{k}$ above the good band triggered the skew-normal rescue pass. It is the EXPLANATION for an inflated $\hat{k}$: a symmetric Gaussian proposal against a right-skewed variance-component marginal has a heavy importance-ratio tail whatever the integration's quality. NULL when the Gaussian proposal already fit.
pareto_k_se_boot, pareto_k_ci_low, pareto_k_ci_high, pareto_k_se_formula, pareto_k_tail_points, pareto_k_tail_points_requested, pareto_k_band_confident – the outer Pareto-$\hat{k}$ uncertainty, present whenever the diagnostic ran. pareto_k_se_boot is the bootstrap SE of the k-hat and pareto_k_ci_low / pareto_k_ci_high its 2.5\ quantiles – the estimator's sampling spread GIVEN the proposal, not a posterior credible interval; pareto_k_se_formula is the closed-form GPD-shape MLE asymptotic SE cross-check. pareto_k_tail_points is the GPD tail size used and pareto_k_tail_points_requested the k_tail_points request (NA when automatic). pareto_k_band_confident is TRUE iff the bootstrap CI lies within one reliability band, NA when the bootstrap was off or could not fit. diagnose_draws and diagnose_cost_ratio (the diagnostic's draw budget and its wall-clock cost relative to the fit) are attached at the top level.
pareto_k_by_arm, pareto_k_by_arm_is_ess, pareto_k_by_arm_scope – present only with control$diagnose_k = "by_arm". Named (by arm) outer Pareto-$\hat{k}$ restricted to each arm's hyperparameter axes, the other arms held at their posterior mean, so a tail-heavy joint k can be localised to one arm. A per-arm entry is NA when that arm carries no varying axis. Each arm carries its own bootstrap uncertainty: pareto_k_by_arm_se_boot, pareto_k_by_arm_ci_low / pareto_k_by_arm_ci_high, pareto_k_by_arm_se_formula, pareto_k_by_arm_tail_points and pareto_k_by_arm_band_confident.
k_quality_requested, k_quality_reached, k_quality_best, k_quality_miss, k_quality_reason, k_quality_rounds, k_quality_k_trace – the reliability verdict for the control$k_quality intent. k_quality_requested echoes the intent; k_quality_best is the band THIS fit reached ("good" / "ok" / "unreliable", or "uncertain" when the bootstrap CI crosses a boundary) – the band of the returned fit, not the best band seen while escalating, which k_quality_k_trace carries; k_quality_reached is TRUE/FALSE for an "ok" / "good" target (NA for "report" / "none"), never silently downgraded; k_quality_miss classifies a miss as "resolution" (the $\hat{k}$ confidently outside the band – a grid deficiency) or "precision" (the bootstrap CI straddling a boundary – the estimator's variance), NA when there is no miss to classify, and is what the escalation reads once refinement is exhausted; k_quality_reason records why it stopped; k_quality_rounds is the number of escalation re-fits performed (0 when the first fit sufficed or escalation was off); and k_quality_k_trace is the outer $\hat{k}$ round by round, starting at the first fit's, so a chase that did not descend monotonically is visible. The refinement rungs rebuild the proposal from the grid they just changed, so a rising trace is a re-founded proposal on a strictly larger grid rather than a worse fit, which is why the LAST round is returned rather than the lowest-$\hat{k}$ one.
adaptive_grid_info – when adaptive_grid = TRUE, a list with triggered_axes (character) and n_points_added (integer) describing the refinement passes. NULL otherwise.
local_ccd_info – when local_ccd engaged (or k_refine = "ccd"), a list with n_cells_refined, n_nodes_added, the refined cells, the n_cells_before / n_cells_after grid sizes, n_design_nodes (the cells that carry an in-cell design weight rather than their own mass), design_mass (the share of the integration weight sitting on them) and cell_share (the share each refined cell held on the BASE grid, before any node was placed). The two shares differ: the replacement nodes sit nearer the peak than the cell's own coordinate did, so refining raises the share the refined region holds, and reading both separates how concentrated the base grid already was from how much the refinement concentrated it. Three per-cell readings come with it, on three orthogonal axes, none of them gating anything. Shape: misfit (the standardized cubic magnitude of the part of the cell the design cannot represent, the one reading skew_max gates on, with misfit_declined / cells_declined for the cells put back as their own mass atoms). Centring: offset (the norm of the whitened gradient, i.e. how far the cell's own peak sat from the cell's coordinate in units of its marginal spread) and mode_gain (the same displacement in the cell's own curvature units, 0.5 * g' (-H)^-1 g in nats, read off the same fit's quadratic coefficients and comparable across cells whose curvature differs; NA where -H is not positive definite, a cell with no interior peak to be displaced from). Mass: log_mass_ratio (the cell's refined mass over the coarse atom it replaced, log_mass_refined - log_mass_coarse, both carried on the log scale with the cell's outer design weight in them) and max_node_weight (the share the single largest node takes of its own cell's refined mass). Each carries a _declined twin for the cells the gate put back, whose nodes were evaluated before the score was read. misfit certifies that the design can represent the cell's shape and nothing more: a cell can be exactly quadratic and still be read at a point that is not representative of it (offset / mode_gain), and refining it can still move how much mass it competes for against its unrefined neighbours (log_mass_ratio, the embedded-rule local error indicator of adaptive cubature). NULL when local CCD was off or declined (single-block, < 4 axes, an active phi_grid, or no peaked interior cell).
integration_requested, integration_declined – what integration asked for, and why the CCD did not run. $integration names the integrator that RAN, and .nl_node_support() keys the interval construction off it, so a caller who asked for a moment rule and received a density grid reads the reason here rather than inferring it. integration_declined is NA_character_ when nothing was declined, and otherwise one of "axis_count" (fewer transformable latent axes than the requested mode's threshold), "unguessable_axis" (a CAR_proper rho_car or a non-BYM2 rho), "degenerate_axis" (a single-valued axis), "modefind_ridge" / "modefind_boundary" / "modefind_degenerate" / "modefind_failed", "hessian_singular", "hessian_not_pd", "copy_atom_components" (more copy atoms than the design splits into, see copy_atom_mass), "copy_atom_mass" (a declared atom mass outside [0, 1)) or "placement_budget" (placing the design would cost more inner solves than integrating the tensor grid it replaces; see the ccd_budget control knob). Multi-block fits only.
ccd_modefind_evals, ccd_modefind_budget, ccd_modefind_rounds, ccd_modefind_seconds – what PLACING the CCD cost: inner Newton solves spent by the mode-find (seed, step calibration and every round of every mixture component), the ceiling they were spent against, the rounds taken and the wall-clock seconds. Present on any multi-block fit that attempted a placement, including one that declined – a decline reports what it had already paid – and absent when no placement was attempted, which is not the same as a placement that cost nothing. Read alongside $integration to judge whether the CCD earned its design on this model rather than arguing it from the round caps.
weight_kind – one entry per outer-grid cell, "mass" or "design". A tensor cell holds the mass of its own cell and a CCD node holds a design weight, so a fit integrated by one rule reports one value throughout; a locally refined grid carries both, and this says which per cell instead of leaving it to be read off integration.
theta_interval_read, theta_interval_design_mass – what the reported per-axis theta_median / theta_ci_lo / theta_ci_hi were read off, and how much of the integration weight sits on nodes whose cumulative sum is not a CDF. "density" is a weighted quantile over cells that discretize the posterior (design_mass 0), "moment_rule" an interval from the moments a central-composite design delivers (design_mass 1), and "mixed" the locally CCD-refined grid, which carries both kinds at once. A mixed support still reports the weighted quantile, which measured best against a converged reference in both design_mass regimes; the share is the regime variable, since on that part of the support the quantile is bounded by the refined cells' own grid neighbourhoods rather than by the posterior. Every nested path stamps the pair, not only the multi-block driver.
within_cell_requested, theta_within_cell, theta_within_cell_declined – the WITHIN-CELL construction the same intervals were read with. The kind above says what the integrator left; this says how each cell's mass was spread inside its own box when the grid was read back. "box_uniform" is the default and puts the cumulative full mass at each cell edge; "chord" (control$within_cell) puts the cumulative mid-mass at each cell coordinate, the same masses over the same boxes with the knots moved half a cell. The construction is recorded per axis, and an axis whose cell partition could not be built falls back to "chord" on its own with the reason in theta_within_cell_declined ("support_<kind>", "single_node", "boxes_do_not_tile", "no_usable_node").
theta_cell_edge_coord, theta_cell_edge_declined – per axis, the COORDINATE that axis's outer cell edges were mirrored in, and why the support the axis declares did not produce them. A declared support is authoritative: its mirrored edge is used when finite and inside it, and otherwise the axis reports its extreme grid coordinate – which the containment test has already placed inside the support – rather than falling through to a coordinate guessed from the node values, which is what put a lower bound of 0 on a positive axis and a bound above 1 on a unit one. An axis that declares no support at all keeps the guess by design, but its mirrored edge is checked for being a representable double that brackets the coordinates, and falls back to the extreme grid coordinate with "mirrored_edge_not_representable" when it is not. theta_cell_edge_declined is NA where nothing was declined.
outer_grid_cell_width, outer_grid_axis_sd, outer_grid_h_over_sd – per axis, the cell width and the posterior SD in that axis's own coordinate, and their ratio. A within-cell reconstruction resolves an interval endpoint to within one cell, so how much of the reported width and of its realized coverage is a property of where the grid fell rather than of the posterior is governed by this ratio; below .nl_diag("grid_resolved") the cells are narrower than the posterior they discretize and the two constructions converge to each other. NA on an axis whose own marginal is maximal at an endpoint, which is the placement question outer_grid_railed_axes reports.
outer_grid_resolution_declined – per axis, why that axis carries no ratio, from the closed vocabulary too_few_nodes, mode_at_edge, coord_not_finite, spacing_not_finite, stencil_degenerate, curvature_not_negative; NA on an axis that scored. The reasons are not interchangeable: mode_at_edge says the grid does not contain that axis's own posterior mode, which is a stronger statement about the fit than any ratio, and the whole-grid resolved verdict is withheld while one is present rather than being read off the axes that did score.
prune_cheap_log_marginal, prune_mask, prune_n_pruned, prune_tol, prune_screen_iters – present only when prune = TRUE and the safety gate did not fall back. Cheap-pass log-marginals at every cell, a logical mask of pruned cells, the pruned-cell count, the threshold actually applied, and the screening depth the driver actually ran at. Pruned cells have log_marginal = -Inf so they get zero weight under .nl_normalise_weights_safe.
prune_log_gap_cut, prune_cheap_lm_spread, prune_min_keep, prune_n_floor_restored – the screen read on the scale it operates on: how many nats below the best screened cell the tolerance cut at, how many nats the screened surface spans, the kept-cell floor applied, and how many cells that floor put back. A cut that is a sliver of the spread is a tolerance with no resolution on this grid.
prune_fallback_triggered, prune_fallback_reason – present only when a fallback to the full grid fired. The returned fit is the full-grid (unpruned) result; the reason string records whether the cheap-pass safety gate tripped or the screened grid left the placement pass no curvature.
n_threads_outer_requested, n_threads_outer_realised – the outer-grid width asked for and the width the grid ran at. The driver clamps the request to the team the OpenMP environment hands out, so a fit launched under OMP_NUM_THREADS=1 with n_threads_outer = 10 runs serial; the pair is what a timing has to be recorded against.

(sigma, alpha) parameterization

Each arm's linear predictor reads $$\eta_{arm} = X_{arm} \beta_{arm} + \sigma_{arm} \cdot z_s,$$ where $z_s$ is a unit-precision latent (ICAR(tau=1) for ICAR/BYM2, or the BYM2 mix; CAR_proper uses the structure of $D - \rho_{car} W$). All arms share the donor amplitude $\sigma$ from the outer-grid sigma_grid axis. The copy arm's amplitude is $\sigma_{arm} = \alpha \cdot \sigma$, where $\alpha$ is a direct outer-grid axis taken from copy$alpha_grid. The Cartesian product is over (sigma, [rho/rho_car,] alpha[, phi]). Direct $\alpha$ as an outer-grid axis (rather than a post-hoc ratio $\alpha = \sigma_{pos} / \sigma_{occ}$) avoids plug-in bias on the weakly-identified ratio at small n_pos and lets a regularizing hyperprior land on $\alpha$ directly.

References

Rue, Martino & Chopin (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. JRSS-B 71(2):319-392.

Examples