Nested Archimedean copulas#367
Conversation
|
Codecov Report❌ Patch coverage is
Additional details and impacted files@@ Coverage Diff @@
## main #367 +/- ##
==========================================
+ Coverage 78.28% 79.21% +0.93%
==========================================
Files 84 88 +4
Lines 5106 5508 +402
==========================================
+ Hits 3997 4363 +366
- Misses 1109 1145 +36 ☔ View full report in Codecov by Sentry. 🚀 New features to boost your workflow:
|
Yes, we'd need |
Add the density kernels for nested (hierarchical) Archimedean copulas in
src/nested/. The nested-copula density is the mixed partial of the nested
CDF; differentiating the composition of generators is Faà di Bruno's
formula, carried here by partial Bell polynomials expressed through
truncated Taylor series:
* _composition_taylor — Taylor coefficients of ϕ⁻¹_outer ∘ ϕ_inner,
* _faa_di_bruno_coeffs — partial-Bell-polynomial step per child block,
* _cauchy_product — convolution of sibling contributions,
* _phi_deriv / _assemble_density — contraction against ϕ⁽ᵏ⁾ of the
outer generator,
* _process_node / _leaf_log_jacobian / _nested_logpdf — the recursion,
with censored leaves shifting the argument but not the differentiation
order (the per-variable survival likelihood).
Follows the algorithm of Yang & Li, arXiv:2605.23134 (2026).
Built only on the public Generator interface (ϕ, ϕ⁻¹, ϕ⁽ᵏ⁾, ϕ⁻¹⁽¹⁾); no
parallel generator system. Generic in the value type with a Float64
default; BigFloat / Double64 coordinates flow precision through the whole
recursion for adversarial high-d / deep-tail inputs.
FrankTaylorGenerator.jl adds Taylor1-compatible ϕ / ϕ⁻¹ methods for the
Frank generator (its default log1mexp form has no Taylor1 method), using
the equivalent expm1/log1p closed forms; the scalar ϕ path is untouched.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Add the public NestedArchimedeanCopula{d,TG} <: Copula{d} and wire it into
the module (include + export). Follows the algorithm of Yang & Li
(arXiv:2605.23134).
* Unified constructor NestedArchimedeanCopula(G; leaves=, children=):
children may be ArchimedeanCopula OR NestedArchimedeanCopula, nesting
to arbitrary depth; children are auto-placed on consecutive free dim
blocks or pinned with `child => dims`. A purely flat declaration
(leaves only) returns the native ArchimedeanCopula fast path. The legacy
positional form NestedArchimedeanCopula(G, children) still works.
* Distributions._logpdf — uncensored nested density (element-typed).
* logpdf(C, u; censored=δ) — optional per-variable right-censoring mask:
the mixed partial of the nested CDF over observed dims only (the
survival likelihood); default (omitted) is the plain density.
No nesting-validity check: the caller supplies a valid parameter
combination.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Add test/NestedArchimedeanCopula.jl (wired into runtests.jl) covering:
* flat declaration dispatches to the native ArchimedeanCopula with a
bit-for-bit identical logpdf;
* uncensored density vs an INDEPENDENT reference — the nested CDF
assembled from the generators and mixed-partial-differentiated by
nested ForwardDiff (no shared code path with the Faà di Bruno
recursion), same-family and heterogeneous trees;
* uncensored density vs an EXTERNAL acopula reference (log-likelihoods
committed under test/data/nested/), Clayton/Gumbel/Frank, d = 10, 20;
* censored likelihood (logpdf with a mask) vs the ForwardDiff reference,
incl. the bivariate Clayton closed form and the omitted-mask ==
plain-density default;
* arbitrary-depth nesting;
* constructor validation (bad dim tiling) and a fit/smoke usage.
Adds DelimitedFiles to the test target (with its [compat] entry, so Aqua's
dependency-compat check stays green) for reading the reference CSVs.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Add docs/src/manual/nested.md (linked in docs/make.jl) covering the definition, building trees (auto/pinned dims, root leaves, mixed families, arbitrary depth), the Float64/BigFloat precision story, and the optional per-variable censored / survival likelihood via the `censored` keyword of `logpdf`. Follows the algorithm of Yang & Li (arXiv:2605.23134). Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Move the per-variable right-censored survival likelihood to the SklarDist
level, where the margins and data live (a Copula{d} stays the pure
dependence on [0,1]^d). The copula-scale mixed partial remains the internal
kernel that SklarDist calls, and is exposed as the lower-level
`censored_logpdf`.
* logpdf(S::SklarDist, x; censored = δ, T = Float64): survival
log-likelihood = Σ_{i observed} log f_i(x_i) + log[ mixed partial of
the copula CDF over the OBSERVED coords at u = (F_i(x_i)) ]. Omitted /
all-false mask reduces to the plain joint density; all-true reduces to
log cdf(S, x). Does NOT route through Distributions.censored (that maps
the censoring atom to the copula boundary u=1 and returns -Inf).
* censored_logpdf(C::Copula, u, δ; T = Float64): copula-scale mixed
partial over observed coords, GENERAL over Copula{d}. Implemented for
flat ArchimedeanCopula (via ϕ⁽ᵏ⁾) and NestedArchimedeanCopula (via the
Faà di Bruno recursion); any copula supplying _censored_copula_logpdf
gains the SklarDist survival likelihood for free.
* NestedArchimedeanCopula loses its logpdf(...; censored) kwarg method
(kept pure); gains an analytic Distributions._cdf (ϕ_root ∘ Σ ϕ⁻¹) so
cdf no longer falls back to slow numerical integration.
Default working precision is Float64 (matching the uncensored path);
BigFloat is opt-in via T= for adversarial high-d / deep-tail inputs.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Tests (test/NestedArchimedeanCopula.jl, 45 pass):
* censored_logpdf (copula scale): bivariate Clayton closed form; FLAT
ArchimedeanCopula via ϕ⁽ᵏ⁾ (general non-nested path); nested across
sectors vs ForwardDiff reference; all-observed == density; all-censored
== log cdf.
* SklarDist survival likelihood: bivariate Clayton closed form on the data
scale; omitted mask == plain joint density; FINITE where the
Distributions.censored route returns -Inf; all-censored == log cdf(S, x);
nested factor == observed-margin densities + copula censored_logpdf.
Docs (docs/src/manual/nested.md): the survival likelihood section now uses
logpdf(SklarDist(C, margins), x; censored = δ), with the math, the
"don't use Distributions.censored" warning, and the note that it is general
over copulas; censored_logpdf documented as the copula-scale building block.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Replace the local `_phi_deriv` (generic Taylor-based ϕ⁽ᵏ⁾) with calls to upstream `ϕ⁽ᵏ⁾(G, k, t)` in `_assemble_density` and in the censored-likelihood helper. Rewrite `_composition_taylor` as a thin wrapper around upstream `taylor(f, x₀, d)`. No parallel derivative machinery; closed-form per-family ϕ⁽ᵏ⁾ overrides now win dispatch automatically. `FrankTaylorGenerator.jl` retained (still needed by `_composition_taylor` for nested-Frank trees, since `log1mexp` has no Taylor1 rule); docstring updated to reflect the narrower role. Tests: same 24/26 pass as before the refactor (the 2 unchanged errors are a pre-existing `rng` test-harness scoping issue, identical on upstream/main). Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
thisiscam
force-pushed
the
nested-archimedean
branch
from
3cf02b1 to
845ab0c
Compare
…poke CensoredLikelihood Make NestedArchimedeanCopula a first-class citizen of the conditioning/ subsetting framework so per-variable censoring is an emergent capability of the standard API rather than a bespoke function. New src/nested/NestedConditioning.jl: - SubsetCopula(::NestedArchimedeanCopula, dims): prune the generator tree to the observed marginal (drop 0-observed subtrees, collapse 1-observed subtrees to a bare leaf by single-coordinate marginalisation-invariance, keep >=2), re-index to 1:p; collapse to a flat ArchimedeanCopula when no sub-nesting survives. Its existing _logpdf gives the observed-marginal density via the Faa-di-Bruno walk. - NestedDistortion + DistortionFromCop(::NestedArchimedeanCopula, js, ujs, i): closed-form conditional marginal whose cdf is the mixed partial over the conditioned set / observed-marginal density, routed through the O(d^2) tree walk (not ForwardDiff). General p, so it also accelerates the per-coordinate distortions the generic ConditionalCopula builds. Per-variable censoring is now the "gist recipe" logpdf(subsetdims(X,O), x_O) + logcdf(condition(X,O,x_O), x_C). - Delete src/CensoredLikelihood.jl (bespoke censored_logpdf / logpdf(::SklarDist; censored=)); keep the internal _censored_copula_logpdf nested kernel as the numerator behind the distortion. - Fix _cdf(::NestedArchimedeanCopula) to flow Real/ForwardDiff.Dual eltypes (was forcing Float64, which crashed the generic multi-censored fallback). - Retarget the censoring tests to the gist recipe against the same independent references; fix the pre-existing rng-scoping bug. 50/50 pass; the JAX-CSV bit-identity regression is untouched. The multi-censored-dim conditional CDF (|unobserved|>=2) stays on upstream's generic ForwardDiff fallback (correct, slower) — ConditionalCopula stores the inner copula as a field, not a type param, so it cannot dispatch on the inner type; a fast multi-dim path is a documented follow-up. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
…walk The single-censored conditional and the observed-marginal density already used our nested kernel, but the MULTI-censored-dim conditional CDF (|censored| >= 2) went through SklarDist(ConditionalCopula, …), whose _cdf calls the generic _partial_cdf — i.e. upstream's nested ForwardDiff over the closed-form CDF. That nests one ForwardDiff.derivative per observed coordinate: Dual-of-Dual type explosion at compile time, O(2^k) at run time, infeasible in high dimension. Add a _partial_cdf(C::NestedArchimedeanCopula, is, js, uᵢₛ, uⱼₛ) specialisation that returns exp(_censored_copula_logpdf(C, assembled_u, cens, T)), the order-|js| mixed partial via our polynomial Faà di Bruno tree walk. ConditionalCopula stores its inner copula in an abstract field but _cdf calls _partial_cdf(CC.C, …), which dispatches on the runtime type (concretely nested), so the override is selected without ConditionalCopula carrying the inner type. Net: the gist recipe routes through our algorithm for ANY number of censored coordinates — zero ForwardDiff. - Tests 50/50 -> 58/58: tighten the multi-censored gist tolerance and add a direct-kernel-equality + CDF-contract (monotone, [0,1]) testset. JAX-CSV bit-identity regression untouched and green; full ConditionalDistribution.jl suite green (non-nested conditioning/sampling unaffected). - Docs/comments updated: multi-censored now uses the tree walk for any censoring count; precision caveat (Float64 high-order fast-tail -> BigFloat) noted. Caveat (documented): end-to-end BigFloat *through* condition() is not yet enabled — upstream's ConditionalCopula/DistortionFromCop store Float64 fields; BigFloat reaches the kernel via direct calls or the single-censored path. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
|
So you do need Then, you code should not use the taylor1() function but rely on the new Regarding Adding the two methods you proposed is fine as a temporary measure. I feel like those methods belong to the Frank generator file and not really their own file, maybe you could move them there ? Can you please add some examples of nested archimedean copulas to the GenericTests.jl file ? |
| @@ -0,0 +1,33 @@ | |||
| # ============================================================================= | |||
There was a problem hiding this comment.
The content of this file should be moved to the file of the frank generator
| function _composition_taylor(outer::Generator, inner::Generator, t₀::T, d::Int) where {T} | ||
| coefs = taylor(x -> ϕ⁻¹(outer, ϕ(inner, x)), t₀, d) | ||
| return T[coefs[k+1] for k in 1:d] | ||
| end |
There was a problem hiding this comment.
So, the derivatives of ϕ⁻¹(outer, ϕ(inner, x)) should IMHO be obtained by manually running the faa-di-bruno formula leveraging the methods we have for the derivatives of these two functions : computing these derivatives is sometimes challenging in term of precision, and the direct taylor() version does not use the potentially overwritten version of those.
There was a problem hiding this comment.
I deliberatedly chose the current version because 1) it's simple and clear on paper 2) NAC density most likely requires BigFloat in most cases, due to the convolution-of-the-taylor-polynomials step requiring a wide range of precision.
That said, a more robust version we had in our preprint and also the JAX implementation was based on implicit differentiation (in Appendix A.4), which avoids the high-order derivative of the inverse.
Do we want that implemented?
There was a problem hiding this comment.
No, its better to use the higher-order derivatives of the inverses as it allows implementers of generators to specify them if they want. Can you leverage the implementation of FdB/Bell that you already have ?
There was a problem hiding this comment.
1. I tried FdB method and found it is broken because of catastrophic cancellation at the FdB summation. On a Frank/Frank example, the derivative of the nesting becomes completely incorrect, even after feeding in highly-accurate high-order derivatives using BigFloat-512.
2. The direct jet and the implicit solver are both fine and equally accurate. Max rel.
error of the edge-link coefficients vs a 2048-bit reference:
precision | direct implicit fdb
Float64 | 2.3e-12 2.9e-12 1.2e+00
64-bit | 9.4e-16 4.2e-16 2.0e-04
96-bit | 2.1e-25 7.7e-25 1.3e-13
This is on Frank/Frank. From my experience with the JAX package, certain nesting
3. IMHO the productive fix for this class of numerical issue is to let the user override the composition's Taylor calculation directly (composition_taylor). Then, per generator-pair, they can pick the naive taylor method, the implicit method, or a taylor1 over a hand-simplified composition ----- maximal flexibility to retain precision. I have just updated this PR to include this hook --- and we have all three methods: direct, implicit, FdB implemented as user-specifiable options.
Repro (self-contained, plain Copulas.jl): https://gist.github.com/thisiscam/8fa9a4ea2b2dffa474cd4ea131efdc21
There was a problem hiding this comment.
Well done here.
Thats looks great. In the documentation where you explain the problem and the possibility to overwrite the default choice, can you add a proper example while you make the argument, starting with the case of someone just trying to compute density of a NAC and obtaining 1) good results for e.g. clayton/clayton, then 2) switching to frank and obtaining bad results (show diff with bigfloats) and 3) using the hook to fix it, and 4) even proposing to implement it yourself through the third option.
That would help clarify things for a user.
EDIT: The faa-di-bruno version should be removed then since its performing so poor.
| _kid_dims(ch::NestedArchimedeanCopula) = ch.dims | ||
|
|
||
| function SubsetCopula(C::NestedArchimedeanCopula{d, TG}, dims::NTuple{p, Int}) where {d, TG, p} | ||
| # `subsetdims` already short-circuits p==1 (Uniform) and p==d (C unchanged), |
There was a problem hiding this comment.
subsetdims indeed short-circuits p==1 (Uniform) and dims == 1:d (unchanged) but not every p==d (reordering)
SubsetCopula(::NestedArchimedeanCopula, dims) relabelled the pruned marginal by SORTED surviving dims, ignoring the requested order. For a non-exchangeable nested structure this returned the wrong marginal under a reordered subset (e.g. subsetdims(C, (3,1)) gave the (1,3) copula). Relabel by the requested `dims` order instead. (Regression test lands with the nested test-suite update.) Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Relocate the Taylor1-specialised ϕ/ϕ⁻¹ methods for FrankGenerator (needed by the direct edge-composition path because Frank's θ>0 scalar forms use log1mexp, which has no Taylor1 method) from the standalone src/nested/FrankTaylorGenerator.jl into src/Generator/FrankGenerator.jl, and drop the now-unused include. No behaviour change. (A future TaylorSeries⊕LogExpFunctions extension giving log1mexp(::Taylor1) would retire these; the implicit edge-composition method already avoids them.) Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
…ethod) The per-edge Taylor expansion of ϕ⁻¹_outer ∘ ϕ_inner now goes through an overloadable hook composition_taylor(outer, inner, t₀, d), defaulting to _composition_taylor_direct (the current jet-over-composition, byte-identical). Ships a second building block, _composition_taylor_implicit (the implicit- equation method of Yang & Li App. A.4 / acopula): it solves ϕ_outer(h)=ϕ_inner order-by-order using only forward derivatives ϕ⁽ᵏ⁾ and a scalar ϕ⁻¹_outer, never differentiating the inverse and never composing generators on a single jet. Users select a method or register a closed form per generator pair the same way they override ϕ⁽ᵏ⁾ — by dispatch, no keyword. Ships the Clayton/Clayton closed form as the worked example (more accurate than the direct jet: 4.9e-77 vs 8.1e-77 rel-err vs a 512-bit reference). Default path is bit-identical; implicit matches direct to ~1e-15 (Float64) / ~1e-30 (BigFloat) across families and on full nested densities. Docs section shows how to switch methods and add custom compositions. Also carries the nested test-suite update (edge-composition tests + the subsetdims-reordering regression test): 58 -> 92 assertions, all passing; the JAX-CSV bit-identity regression is untouched. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
- fit(CopulaModel, C0, U) / fit(C0, U): MLE of the generator parameters holding the tree structure fixed (unconstrained reparam + LBFGS); coef/coefnames/dof read from the fitted tree so AIC/BIC are correct. - Distributions._rand! via inverse-Rosenblatt; _logpdf promotes the working type so ForwardDiff generator params flow through Float64 data. - Two nested Clayton entries added to the GenericTests bestiary; fit-recovery / subsetdims-order / rand / coef tests. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
- ϕ⁻¹⁽ᵏ⁾(G, k, t): overridable kth derivative of the inverse generator (generic Taylor-jet default; exact Clayton closed form). - _composition_taylor_fdb: a third selectable edge composition behind the composition_taylor hook, assembling ϕ⁻¹_outer∘ϕ_inner from ϕ⁻¹⁽ᵏ⁾(outer) and ϕ⁽ᵏ⁾(inner). Opt-in only; the default remains the direct jet. - Note: catastrophically ill-conditioned for fast-tail generators (Frank/Joe) at high d — kept as a selectable option, not the default. See PR discussion. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
| function _composition_taylor(outer::Generator, inner::Generator, t₀::T, d::Int) where {T} | ||
| coefs = taylor(x -> ϕ⁻¹(outer, ϕ(inner, x)), t₀, d) | ||
| return T[coefs[k+1] for k in 1:d] | ||
| end |
There was a problem hiding this comment.
Well done here.
Thats looks great. In the documentation where you explain the problem and the possibility to overwrite the default choice, can you add a proper example while you make the argument, starting with the case of someone just trying to compute density of a NAC and obtaining 1) good results for e.g. clayton/clayton, then 2) switching to frank and obtaining bad results (show diff with bigfloats) and 3) using the hook to fix it, and 4) even proposing to implement it yourself through the third option.
That would help clarify things for a user.
EDIT: The faa-di-bruno version should be removed then since its performing so poor.
| k < d && (gpow = _poly_mul_inc(gpow, g, d)) # ĝᵏ⁺¹, truncated to order d | ||
| end | ||
| return h | ||
| end |
There was a problem hiding this comment.
So this shoudl be removed then, thanks for trying out
| h[k] = (θ_in_pow / θ_out) * binom * base^(r - k) | ||
| end | ||
| return h # [h₁,…,h_d], length d, no constant | ||
| end |
There was a problem hiding this comment.
this function should move to the ClaytonGenerator file, with correct comments on top explaining what it should compute
| # dispatches on a TEMPLATE INSTANCE `C0::NestedArchimedeanCopula` that carries the | ||
| # full tree — neither `fit(NestedArchimedeanCopula, U)` (a bare type) nor | ||
| # `fit(typeof(C0), U)` can rebuild the tree, since the type parameters | ||
| # `{d, root-TG}` encode only the dimension and the root generator family. |
There was a problem hiding this comment.
I understand why you did this, otherwise MLE is not possible. I kinda agree with you, but this shoul dbe in the docs too : maybe you could add an example of fitting a NAC on some dataset ? inside the example parts of the documentation
fit() optimises through a parametrisation `α -> NestedArchimedeanCopula` decoupled from the generator objects: default (per-generator, unchanged); enforce_nesting=true (same-family θ-ordered trees parametrise each child's θ as a non-negative increment over its parent, so every optimiser step is a valid nesting); or a fully custom reparam=(α->C), init=α₀ (share parameters, fit on another scale, encode any constraint) — ForwardDiff differentiates straight through it. dof now reads the free-parameter count (nparams) so AIC/BIC stay correct even when a custom reparam shares parameters. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
…ride to its generator file Remove _composition_taylor_fdb + _poly_mul_inc and the ϕ⁻¹⁽ᵏ⁾ interface (generic + Clayton closed form): the FdB route is catastrophically ill-conditioned (per-PR discussion) and was the only consumer of ϕ⁻¹⁽ᵏ⁾. The direct (default) and implicit methods remain behind the composition_taylor hook. Move the closed-form Clayton/Clayton composition_taylor override into Generator/ClaytonGenerator.jl, with a header comment explaining what it computes. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Add a Fitting section (MLE on a fixed tree + the three parametrisation modes: default / enforce_nesting / custom reparam) and rewrite the edge-composition section around the three selectable methods (direct default / implicit / closed-form per pair). Method choice is about availability and speed; BigFloat is the precision fix for deep-tail / high-d inputs. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
…ied reparam The decoupled parametrisation already lets the caller encode any constraint via `reparam`; a built-in enforce_nesting hard-coded one constraint and re-introduced a per-family whitelist into fit(). Remove it and its helpers; the docs and a test now demonstrate nesting as a user-written reparam (child θ = parent θ + softplus(δ)). Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
A custom parametrisation no longer needs a dummy template copula: `fit(CopulaModel, reparam, init, U)` takes the map and its initial point directly and infers the dimension from `reparam(init)`. Distinguished from the template form `fit(CopulaModel, C0, U)` by arity; `fit(C0, U)` remains a quick shim. (No untyped `fit(reparam, init, U)` shim — that would be type piracy on Distributions.fit.) Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
…ote, update fit examples The underbrace texttt labels broke MathJax (subscripts + thin-spaces inside texttt); use plain text labels and give the code mapping in prose. Remove the Distributions.censored warning. Update the fit examples to the template-free fit(CopulaModel, reparam, init, U) form. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Rename _composition_taylor_direct/_implicit to composition_taylor_direct/composition_taylor_implicit (users select them via the composition_taylor hook), and give the hook + both methods docstrings — including the DIY-with-taylor recipe. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
…upports p==d via lrnv#369) Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
2 participants
Adds support for hierarchical Archimedean copulas of arbitrary tree depth with per-subtree generator choices. Density via Faà di Bruno over the tree (Bell-polynomial recursion + Cauchy product per node); closed-form CDF.
Reuses upstream
ϕ⁽ᵏ⁾,taylor(), and the existing generator interface throughout — no parallel derivative machinery. Tests check the tree-density against an independent nested-ForwardDiffreference and against frozen CSVs from a JAX implementation.Three coordination questions before un-drafting:
src/nested/FrankTaylorGenerator.jlis 4 lines addingϕ/ϕ⁻¹ofFrankGeneratoron::Taylor1(works aroundlog1mexpnot having a Taylor1 rule). Happy to split as a tinymain-targeted PR first and rebase.src/CensoredLikelihood.jlcurrently has our per-variable censored likelihood. Once Missinglogpdf(::Copulas.DistortedDist{Copulas.ArchimedeanDistortion{Copulas.ClaytonGenerator{Float64}, Float64, 1}, Normal{Float64}}, ::Float64)#345 lands, the natural form is the decomposition viacondition/subsetdims. Prefer (a) wait for Missinglogpdf(::Copulas.DistortedDist{Copulas.ArchimedeanDistortion{Copulas.ClaytonGenerator{Float64}, Float64, 1}, Normal{Float64}}, ::Float64)#345 and rebase onto it, (b) drop censoring from this PR and follow up separately, or (c) keep here for now and remove on rebase?condition/subsetdimsaren't specialised forNestedArchimedeanCopulayet — the generic AD-on-_cdffallback works (we provide a closed-form_cdf) but is slow. Want a tree-walking specialisation in this PR or as a follow-up?Reference: Yang & Li, "Archimedean Copula Inference via Taylor-Mode AD," arXiv:2605.23134 (Algorithm 1 = the nested-AC density).