Modular numerical controls for sommer::mmes

sommer development team

Purpose

For a clickable decision tree with searchable controls, defaults, examples, and compatibility checks, open the interactive mmes_control guide.

mmes() separates statistical model specification from numerical policy. Formulas, data, weights, family, constraints, and known covariance parameters remain mmes() arguments. Numerical choices belong in one validated object:

fit <- mmes(
  fixed = trait ~ environment,
  random = ~ genotype,
  rcov = ~ units,
  data = trial_data,
  control = mmes_control()
)

This vignette documents the modular API, defaults, mathematical ownership, eligibility constraints, and migration from the removed numerical formals. It is explanation only: code is displayed, not run. Building this vignette does not fit models, benchmark methods, or change hardware settings.

Public interfaces

The current public numerical interface is:

mmes(fixed, random, rcov, data, W, weights = NULL,
     naMethodX = "exclude", naMethodY = "exclude",
     naMethodRandom = "exclude", naMethodR = "exclude",
     contrasts = NULL, family = stats::gaussian(), vcc = NULL,
     covPar = NULL, control = mmes_control(), .internal = NULL)

mmes_control(
  formulation = "henderson", task = list(), representation = list(),
  solve = list(), logdet = list(), score = list(), curvature = list(),
  update = list(), proposal = list(), convergence = list(),
  output = list(), execution = list(), diagnostics = list()
)

fixed, random, rcov, data, and W remain model inputs. weights declares independent blocks in a supplied W; it does not create weights. family, vcc, covPar, missing-data policies, and contrasts retain their statistical or data-preparation roles. .internal is reserved for package recursion and is not a user control.

mmes_control() validates top-level and nested module names. Unknown fields, duplicate names, and malformed named lists error. Removed options are not accepted through ..., aliases, or a legacy engine argument. Controls record requests; they do not fit a model or establish model-specific eligibility. Fitting validation remains authoritative.

Ownership and defaults

The default control retains the established numerical bundle: Henderson formulation, REML fitting, exact AI curvature, EM recovery, no extrapolation, 30 inner iterations, full covariance information, coefficient-matrix retention, and inherited solver/resource defaults. These defaults preserve the prior numerical bundle; they do not authorize new combinations.

Module Mathematical ownership Fields and defaults
formulation Equation domain "henderson"; "henderson" or "direct".
task Fit versus known-covariance solve; objective; outer PQL mode="fit", objective="reml", pql=list(maxit=20L, tol=1e-5, estimateTheta=FALSE, warmStart=TRUE).
representation Equation/covariance representation compression="off", factorScores="none", fixedRank="error", equations="normal", sparseFactors="off", matrixFree="auto"; modes can inherit global options, and equations defaults to "saddle-point" if that mode is globally forced.
solve Linear solves, preconditioning, reuse, elimination method="auto"; PCG fields below; blockSchur, blockBanded, factorSchurMultiFactor, and dFirstML default to "auto".
logdet Determinant approximation scaling="none", lanczosSteps=20L.
score Covariance derivatives and trace estimation traceProbes=8L, estimator="hutchinson", rank=16L, covarianceChecks="auto", traceBatch="off".
curvature AI metric and reduced-direction assembly method="exact-ai", reduced="off".
update Covariance-coordinate update and recovery method="ai", recovery="em", blendWeight=NULL, commonScale="off".
proposal Proposal scaling and extrapolation stepWeight=NULL, extrapolation="none".
convergence Inner iterations and stochastic accuracy maxIters=30L, likelihoodTol=1e-4, parameterTol=1e-4, accuracy="fixed", maxTraceProbes=NULL, maxLanczosSteps=NULL, logDetTol=1e-3, scoreTol=1e-3.
output Coefficient and covariance uncertainty; setup output inverse=0L, covariance="full", coefficientMatrix=TRUE, setup=FALSE, selectedInverse="auto", panelSize=64L.
execution Memory, hardware, threads, precision, seed denseMemoryMB=1000, memoryPolicy="speed", gpu=FALSE, gpuMinSize=256L, blasThreads=NULL, inverseTol=1e-6, seed=0L.
diagnostics Messages, package-age warning, profiling verbose=TRUE, dateWarning=TRUE, profile="off".

The PCG fields under solve default to tol=1e-8, maxIters=0L, preconditioner="diagonal", nystromRank=32L, warmStart=FALSE, recycleRank=0L, recycleMethod="coarse", recycleState=NULL, exportRecycle=FALSE, rhsMethod="independent", blockSize=16L, and residualRefresh=0L. maxIters=0L selects the solver’s automatic limit. PCG-specific requests matter only when an eligible PCG path uses them.

output$inverse controls final coefficient-inverse work and fit-time PEVs. Mode 0L computes neither. Mode 1L computes random-effect PEVs from the selected inverse without forming full Ci. Mode 2L forms full Ci and computes PEVs. To defer this work, retain the default mode 0L and output$coefficientMatrix=TRUE, then call postPEV(fit, mode=1L) for PEVs only or postPEV(fit, mode=2L) for PEVs and full Ci.

Options may supply defaults for experimental modes, memory budget/policy, and GPU settings. For example, a non-NULL sommer.mme.denseMemoryMB option replaces the 1000 MB default. Explicit control values override option-derived mode defaults. Use them for reproducible intent. Hardware and library availability can still affect execution.

Constructors and valid values:

The complete field reference and mapping from old names are in ?mmes_control.

Decision graph and constraints

Rank-deficient fixed effects

By default, mmes() stops with diagnostics when the fixed design contains zero columns or linear dependencies. For Gaussian identity-link fits, both the Henderson and direct engines can instead use a shared sparse C++ QR basis selected once after observation filtering:

fit <- mmes(
  response ~ locale + cell,
  data = observations,
  control = mmes_control(representation = list(fixedRank = "drop"))
)
fit$fixedRank[c("rank", "retained", "omitted")]
summary(fit)$betas

The original coefficient layout is retained. Omitted coefficients are constrained to zero and explicitly marked aliased, with no inferential standard errors or tests. Retained coefficients are conditional on those constraints. This preserves the fitted-mean space but need not reproduce ASReml’s particular coefficient constraints. Different fixed bases can shift REML likelihoods by a constant.

fit$fixedRank$relation is a sparse map from the retained design to the original columns. Direct-engine VarBeta stays in retained coordinates; it is not expanded to a potentially huge original-column covariance. Henderson prediction contrasts are checked against the original model’s estimability relations. Non-estimable rows are rejected. Wald tests, likelihood-ratio comparisons, Satterthwaite/Kenward-Roger inference, and emmeans require an explicitly full-rank refit. Existing direct-engine prediction limitations are unchanged.

Character classify predictions for rank-reduced fits stop before a Cartesian level grid is constructed. To explicitly request observed combinations:

cells <- predict(fit, D = "locale:cell", observed = TRUE)
cells$observed

This path averages the fixed predictor over retained records within each observed group and excludes random contributions by default. It is not an equally weighted balanced-grid marginal mean. Random contributions can be requested explicitly through Dtable, without changing its fixed rows:

rules <- fit$Dtable
rules$include[rules$type == "random"] <- TRUE
conditional <- predict(fit, D = "locale:cell", observed = TRUE, Dtable = rules)

For random terms, include alone uses empirical within-group design averages; include plus average uses equally weighted co-occurring indicator levels; average alone uses equally weighted selected fitted coefficient levels. Random level restrictions are given through entries of levels named by the fitted random term. The returned Dtable contains only term-level rules; selections are returned separately in levels and can be reused with those rules. A levels entry named exactly as D may select and order observed labels. Unobserved requests produce an error. Numeric classify variables and custom fixed-effect rules require explicit estimable contrasts instead. See ?predict.mmes for random-level filtering and weighting details.

Observed combinations can still produce a large dense covariance. maxMemoryMB defaults to the fitted dense memory budget (1000 MB otherwise) and checks estimated prediction workspace before constructing grouped contrasts or solving. Restrict levels or use smaller explicit contrast batches if necessary. PEV = FALSE does not remove the mandatory vcov.

This option is not supported with PQL, solve mode, conditional FA, factor scores, observation rotation or compression; these combinations fail explicitly rather than silently ignore the rank policy.

Conditional FA: reusable posterior moments with AI or EM

The default ordinary AI workflow is unchanged. The opt-in representation$equations="conditional" selects sparse latent FA equations using the existing formula preparation, observation map, covariance descriptors, model methods, controls, and execution-plan reporting. The optimizer is a separate block: update$method="ai" (the default) uses exact REML scores and average-information curvature on these equations; update$method="em" uses conditional-moment EM. The earlier update$method="conditional-fa" API retains its EM behavior even without an explicit representation request.

long <- stackTraits(wide, traits = trait_names, keep = "id")
Ainv <- solve(A)
attr(Ainv, "inverse") <- TRUE

fit <- mmes(
  value ~ trait,
  random = ~ vsm(fam(trait, k = 2), ism(id), Gu = Ainv),
  rcov = ~ vsm(fam(trait, k = 1), ism(record)),
  data = long,
  control = mmes_control(
    representation = list(equations = "conditional"),
    update = list(method = "ai"),
    convergence = list(maxIters = 300),
    output = list(covariance = "none", coefficientMatrix = FALSE,
                  inverse = 0, panelSize = 32)
  )
)
fit$convergence
fit$conditionalFA$likelihood
fit$conditionalFA$updates  # accepted AI steps and any EM recovery
fit$conditionalFA$ai       # final score norm, curvature rank and workspace
fit$executionPlan$admission$conditionalFA
covmatrix_mmes(fit, term = 1, se = FALSE)

Missing responses are excluded by the usual observation map, not imputed. Repeated individuals and incomplete residual pairing blocks are supported; record must identify the actual experimental unit. The genetic incidence must select exactly one individual and one trait per observed value. The fixed design may contain covariates and interactions but must have full column rank. Each trait must retain observations. Known positive-definite relationship precision matrices retain the usual Gu contract. Each covariance source must be entirely free or entirely fixed. FA specifics can be heterogeneous or homogeneous; diagonal and homogeneous residuals are also supported.

Mathematical blocks

Write the latent model as

$$ y=Xb+Z_g(L_g f_g+s_g)+L_e f_e+\epsilon,\quad \operatorname{Var}(s_g)=\Psi_g\otimes A,\quad \operatorname{Var}(\epsilon)=\operatorname{diag}(\psi_{e,t(i)}). $$

Residual factors are indexed by pairing key, genetic factors by individual. The prepared sparse design is denoted by \(D\) and its random-effect precision by \(Q\) (with zeros on fixed-effect coordinates). At every iteration:

  1. Assemble \(H=D^\mathsf{T}R_0^{-1}D+Q\) and factor it numerically.
  2. Compute conditional means and the entries of \(H^{-1}\) needed by the complete-data quadratics, including fixed-effect uncertainty.
  3. Select the update block: exact REML score/AI contractions, or loading regressions and expected quadratic forms for EM. Do not treat predicted factors as observed data.
  4. Evaluate the observed restricted log likelihood and reject a material decrease. EM convergence requires likelihood and parameter stabilization; AI requires a small diagonally scaled REML score.

On the original response scale, omitting the Gaussian normalizing constant, the restricted objective uses

$$ -2\ell_R=\log|R_0|-\log|Q_{\mathrm{random}}|+\log|H| +(y-D\hat a)^\mathsf{T}R_0^{-1}(y-D\hat a) +\hat a^\mathsf{T}Q\hat a . $$

Integrating the fixed coefficients with a flat measure gives the restricted objective; their conditional second moments are essential to the EM update. Specific variances have a machine-precision positive floor to keep the conditional distribution well-defined during EM. AI uses log-specific coordinates and, for shared factors, logit allocations.

Exact score and AI blocks

For covariance coordinate \(\theta_i\), let \(V_i=\partial V/\partial\theta_i\), \(P=V^{-1}-V^{-1}X(X^\mathsf{T}V^{-1}X)^{-1}X^\mathsf{T}V^{-1}\), and \(z=Py\). The score and average information are

$$ s_i=\tfrac12(z^\mathsf{T}V_i z-\operatorname{tr}(PV_i)),\qquad \mathcal A_{ij}=\tfrac12(V_i z)^\mathsf{T}P(V_j z). $$

The implementation differentiates the sparse latent design, residual diagonal and random precision. Selected posterior moments give the exact score traces; relationship-factor solves apply random prior covariances without forming a dense relationship inverse. The already factored \(H\) projects derivative right-hand sides in bounded panels. Neither \(V\) nor \(P\) is constructed. Shared loading derivatives affect both covariance sources in the same coordinate.

Independent loading matrices retain rotationally redundant coordinates. A diagonally scaled, rank-revealing curvature eigensolve drops numerically null directions rather than inverting a singular information matrix. Steps have a bounded coordinate displacement and a backtracking observed-likelihood Armijo test. If no AI proposal is accepted, an EM recovery step uses the same posterior moments; conditionalFA$updates records "ai" or "em-recovery" explicitly. This changes optimization, not the REML objective. conditionalFA$ai$scoreNorm is the diagonally scaled score checked against convergence$parameterTol; a small displacement alone cannot certify AI convergence.

Near-zero conditional noise can make the latent equations ill-conditioned even when the marginal covariance remains well-conditioned. Conditional AI therefore has explicit numerical bounds: writing \(\tau=\max(10^{-8},\texttt{execution\\)inverseTol})$, free physical specifics must be at least \(\tau\operatorname{var}(y)\) and shared allocations lie in \([\tau,1-\tau]\) (inverseTol must be below 0.5). Initial values must satisfy these bounds. An active-set solve excludes outward directions at a bound; convergence uses the projected KKT score, not an unconstrained zero-gradient claim. The actual varianceFloor, allocationFloor, and activeBounds are reported in conditionalFA$ai. These physical-coordinate constraints are not identical to the ordinary engine’s descriptor-coordinate bounds. EM recovery stays within the AI feasible set. Explicit conditional EM retains its earlier machine-precision floor, so optima at numerical boundaries need not match exactly between optimizers.

conditionalFA$rawLikelihood retains this original-scale objective. For compatibility with the ordinary solver’s standardized-response reporting, llik and conditionalFA$likelihood add \((n-\operatorname{ncol}(X))\log(\operatorname{sd}(y))\). This constant does not affect updates or convergence. AIC and BIC retain the existing solver’s fixed-design-column penalty convention; the covariance-model parameter count is recorded separately rather than silently changing those established fields.

Reuse and memory policy

The native conditional-moment block reuses symbolic LDLT analysis only while the sparse pattern is identical. Numeric factors are recomputed at every covariance update. Fixed sparse quadratic contractions are compiled once; identical factor incidence/missingness patterns share their Gram contractions. Selected inverse entries are obtained in bounded panels. No dense observation covariance or full posterior inverse is constructed. EM does not assemble an AI matrix; conditional AI does assemble a dense parameter-by-parameter matrix and observation-by-parameter derivative RHS.

output$panelSize controls inverse/projection-panel width; execution$denseMemoryMB admits their dense workspace and, for AI, the retained RHS/curvature/eigensolve workspace (not a process-memory cap). Oversized AI requests fail explicitly rather than switching optimizers. Sparse factor fill, latent genetic specifics, and the number of exact inverse solves can still be expensive, especially with dense relationships. This first engine prioritizes exactness and bounded dense intermediates; it is not a claim of MegaLMM-like thousand-trait runtime. EM can converge slowly near boundaries. Compare converged objectives and predictions, not equal iteration counts.

Shared-loading covariance model

For common trait axes, use matching tags:

random = ~ vsm(fam(trait, k = 2, shared = "axes"), ism(id), Gu = Ainv)
rcov   = ~ vsm(fam(trait, k = 2, shared = "axes"), ism(record))

This changes the covariance family to \(G=L D_g L^\mathsf{T}+\Psi_g\) and \(R=L D_e L^\mathsf{T}+\Psi_e\). Both sources must be free with equal rank. Diagonal factor variances are updated from conditional quadratics and normalized so \(D_g+D_e=I\); the compensating scale is absorbed into \(L\). Starting loading columns must be positive scalar multiples between sources. Defaults satisfy this. The common basis is in fit$conditionalFA$state$loading, with allocations weightG and weightE; sign and ordering are not uniquely identified. loadings_mmes() instead reports triangular marginal loadings for each source. Other engines reject shared tags rather than ignoring the restriction.

Output and compatibility

Fits remain mmes objects. fitted(), residuals(), coef(), covariance parameter reports, and explicit covmatrix_mmes(..., se=FALSE) remain available. theta has named empty entries rather than dense trait matrices. The covariance descriptors and conditional state are the authoritative factorized covariance representation. Covariance-parameter SEs, retained public C, coefficient PEVs, and uncertainty-dependent methods are not available on this path; they fail explicitly. No implicit handoff to another estimator is made. For shared models, the usual marginal parameter reports are redundant representations; the independent parameter count is reported in fit$conditionalFA$parameterCount. The public llik retains its usual one-row history shape. monitor contains the final marginal covariance parameters only, not every intermediate EM state; the likelihood history is retained in conditionalFA$likelihood.

Weights, non-Gaussian families, ML, multiple random covariance sources, rotation, user vcc, random-regression/overlapping incidence, rrm(), spatial residuals, and combinations with AI acceleration/augmentation are not admitted. Such models continue to use the unchanged ordinary engine unless the conditional engine is explicitly requested.

Validation and measured limits

The posterior kernel is checked against dense means, inverse entries and log determinants, including symbolic reuse/invalidation. End-to-end tests compare incomplete-record REML, GLS and genetic BLUPs with dense reference equations, check shared and independent EM monotonicity, and verify convergence against a separately optimized one-dimensional restricted likelihood with a fixed genetic covariance. Unequal genetic/residual ranks and homogeneous specifics have separate coverage. Conditional AI scores and curvature are also checked against independently finite-differenced marginal covariance matrices, including rank-two shared loadings and homogeneous specifics. Convergence is compared with a separately optimized restricted likelihood; numerical-bound KKT convergence, EM recovery and legacy-EM parity have separate tests.

Baseline-versus-candidate checks used the enhancer split-plot, compound-symmetry and diagonal GxE, multivariate, random-regression, field row/column, and genomic relationship models. Covariance parameters, BLUPs, likelihood histories and convergence flags matched exactly for all seven. This is regression evidence, not an independent certification of a model optimum. The earlier conditional EM selector and the explicit conditional EM control also produce identical histories, parameters and predictions. Measured R allocations changed by less than 0.03%; elapsed times were noisy on the shared machine, so unchanged ordinary paths are not claimed as a speedup.

A separate single-thread, synthetic score benchmark with 2,000 traits, rank 5 and 200 individuals compared the new regression score routine with U %*% solve(Sigma, L): maximum absolute difference was below \(5\,10^{-14}\), median elapsed time was 0.003 versus 0.146 seconds (seven repetitions), and profiled R allocations were 3.84 versus 30.67 MiB. This measures score calculation only, not model fitting or process peak memory.

Conditional EM is not uniformly competitive with AI. In a four-trait, 24-individual, four-replicate incomplete-data experiment with a near-boundary genetic specific variance, AI converged in about 5 seconds; conditional EM had not met both convergence tolerances after 1,500 updates and about 123 seconds. The engine reports nonconvergence explicitly. These measurements motivate future acceleration/reduced-latent-system work; they do not establish thousand-trait fitting scalability.

On the same near-boundary fixture, conditional AI with active numerical bounds converged in 16 updates rather than the 1,500 unfinished EM updates. Conditional AI took about 1.5 seconds versus 0.6 seconds for ordinary AI. These shared-machine timings do not establish a general speed advantage; the numerical bounds and stopping rules must also be considered when comparing optima.

Ordinary fitting paths

The sequence below shows dependencies, not a sequence of arbitrary method switches. Several paths dispatch early, and later choices remain gated by the model, task, solver, and output request.

Model formulas, covariance descriptors, data and weights
|
+-- task$mode = "solve"
|   +-- Known covariance parameters; no covariance estimation
|   +-- Gaussian identity and restricted solve/output requests
|
+-- task$mode = "fit"
    |
    +-- Non-Gaussian family
    |   +-- Outer PQL uses task$pql
    |   +-- Inner fits remain subject to Gaussian-path restrictions
    |
    +-- Gaussian identity / inner Gaussian fit
        |
        +-- formulation = "direct"
        |   +-- Observation-space fitting
        |   +-- No Henderson solver selection
        |
        +-- formulation = "henderson"
            +-- solve$method = "pcg"
            |   +-- REML estimation; iterative solves and stochastic
            |       determinant/score evaluation
            +-- solve$method = "auto", "ldlt", or "cholmod"
                +-- Deterministic sparse-factorization paths
                +-- Exact determinant/score paths when eligible

Representation, evaluation, update, proposal, convergence, output,
execution, and diagnostics requests apply within those branches only
when their model-dependent admission checks pass.

Model-independent combinations are checked by one compatibility table when mmes_control() builds the object and again in mmes(). A supplied setting that conflicts with another, or that the selected path would ignore, stops the call. The error names every offending setting and suggests a fix. Restating a default is always accepted. Rules that depend on the data, family, vcc or rotation are checked at fit time. See ?mmes_control (section Compatibility) for the rule list.

mmes_control(solve = list(method = "ldlt"), score = list(traceProbes = 32L))
#> Error: Incompatible mmes_control() settings:
#> * PCG preconditioning, recycling, stochastic log-determinant and
#>   trace-estimation settings only apply when solve$method='pcg' ...
#>   Offending: score$traceProbes. Remove them or set solve$method='pcg'.

Examples of coupled prerequisites:

For the descriptive capability inventory and conceptual decision graph, inspect the internal catalog:

capabilities <- sommer:::.mmes_capabilities()
capabilities$modules
capabilities$controls
capabilities$decisions
capabilities$edges

This inventory does not inspect a fitted model, resolve eligibility, or predict the selected execution path. It complements, but does not replace, control validation and fit-time diagnostics.

Migration from removed numerical formals

Move every numerical choice into control; do not pass old formals to mmes(). Use one control object rather than conflicting duplicated values.

Former argument or path Current setting
henderson=TRUE/FALSE formulation="henderson"/"direct"
solver solve$method
nIters convergence$maxIters
tolParConvLL convergence$likelihoodTol
tolParConvNorm convergence$parameterTol
tolParInv execution$inverseTol
REML=TRUE/FALSE task$objective="reml"/"ml"
solveOnly=TRUE/FALSE task$mode="solve"/"fit"
pqlControl task$pql
computeCi output$inverse
getPEV No separate replacement; output$inverse controls fit-time PEVs. Use postPEV() to compute PEVs after fitting.
returnParam output$setup
blend, blendWeight update$recovery, update$blendWeight
stepWeight proposal$stepWeight
acceleration="aitken" proposal$extrapolation="aitken"
acceleration="hybrid-ai" curvature$method="hybrid-ai"
compression representation$compression
factorScoreAugmentation representation$factorScores
experimental$sparseFactors, matrixFreePcg, saddlePoint representation$sparseFactors, representation$matrixFree, representation$equations
experimental$blockSchur, blockBanded, factorSchurMultiFactor, dFirstML Same-named solve fields
experimental$pdCertificates, traceBatch score$covarianceChecks, score$traceBatch
experimental$reducedAI, commonScale curvature$reduced, update$commonScale
experimental$cholmodFinalSelected, panelCi2 output$selectedInverse, output$panelSize
experimental$profile diagnostics$profile
covarianceInformation output$covariance
retainCoefficientMatrix output$coefficientMatrix
PCG tol, maxIters, preconditioner, reuse solve fields
PCG determinant scaling and Lanczos depth logdet$scaling, logdet$lanczosSteps
PCG trace probes, estimator, rank score$traceProbes, score$estimator, score$rank
PCG covariance checks and trace batching score$covarianceChecks, score$traceBatch
PCG accuracy budgets convergence fields
PCG seed execution$seed
Experimental elimination requests Corresponding solve fields
Experimental sparse factors, matrix-free, saddle point Corresponding representation fields
Experimental trace checks/batching score$covarianceChecks, score$traceBatch
Experimental reduced AI curvature$reduced
Experimental common scale update$commonScale
Experimental inverse selection/panel size output$selectedInverse, output$panelSize
Experimental profiling diagnostics$profile
denseMemoryMB, memoryPolicy, gpu, gpuMinSize, blasThreads Same-named execution fields
verbose, dateWarning Same-named diagnostics fields
recordDiagnostics, verboseDiagnostics Removed flags; use diagnostics$verbose for fit output and diagnostics$profile for profiling
engine Pass the complete object as control
... legacy/pass-through options No replacement; set documented module fields only

For example, former settings spread across top-level formals and engine lists now have one canonical owner:

control <- mmes_control(
  formulation = "henderson",
  task = list(objective = "reml"),
  solve = list(method = "pcg", tol = 1e-8, maxIters = 0L),
  score = list(traceProbes = 16L),
  convergence = list(accuracy = "adaptive", maxIters = 40L),
  output = list(inverse = 0L, covariance = "full")
)

The statistical model is unchanged by choosing a numerical representation or solver only when that method is mathematically equivalent and eligible. Do not interpret the module structure as permission to combine unsupported methods.