Use AbstractPPL AD interface

[yebai]

Migrates DynamicPPL's AD plumbing to the new AbstractPPL.prepare / value_and_gradient!! interface introduced in AbstractPPL 0.15, retiring DynamicPPL's direct dependency on DifferentiationInterface.

The LogDensityFunction constructor now hands AD preparation to AbstractPPL.prepare(adtype, logdensity_internal, x; context=..., check_dims=false) and LogDensityProblems.logdensity_and_gradient simply forwards to AbstractPPL.value_and_gradient!!. The internal _use_closure heuristic that picked between Base.Fix1 closures and DI.Constant arguments per backend is removed — that decision now lives in AbstractPPL. logdensity_at is renamed to logdensity_internal (with a const alias for compatibility), and LogDensityAt becomes a deprecation shim that returns an AbstractPPL.Evaluators.VectorEvaluator.

Replacing #1354. Tests currently fail due to #1364

Notes: The primal and gradient times are identical to main except for tiny models, where this branch beats main for logdensity evaluation while gradient remains comparable.

Details

The apparent AD regression in the tiny linked row is mostly caused by a much faster
primal denominator, not by slower raw gradients.

Reported row:

main:    primal = 23.6 ns, Mooncake ratio = 7.16   -> raw grad ~= 169 ns
branch:  primal = 4.63 ns, Mooncake ratio = 33.43 -> raw grad ~= 155 ns

So Mooncake is slightly faster in raw time for this row. The ratio looks worse because
t(logdensity) became tiny.

Root cause

On Julia 1.11, old main's optimised primal still contains a call boundary:

invoke DynamicPPL.tilde_assume!!(...)

Current branch inlines through the same path, so the one-dimensional linked model reduces
to loading params[1] and evaluating two scalar Normal logpdfs.

Local Julia 1.11.9 checks:

old main e2d4b5da:
  primal ~= 9.4 ns

old main + monkey-patched @inline tilde_assume!!:
  primal ~= 3.2 ns

current branch:
  primal ~= 2.5 ns

Adding @inline only to LogDensityProblems.logdensity did not materially change old
main. The important boundary is the assume/init path.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 81.70%. Comparing base (d2052a1) to head (8d27685).

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1363      +/-   ##
==========================================
- Coverage   82.30%   81.70%   -0.61%     
==========================================
  Files          50       50              
  Lines        3543     3536       -7     
==========================================
- Hits         2916     2889      -27     
- Misses        627      647      +20     

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[github-actions]

DynamicPPL.jl documentation for PR #1363 is available at:
https://TuringLang.github.io/DynamicPPL.jl/previews/PR1363/

[shravanngoswamii]

@yebai few suggestions and improvement comments.

[shravanngoswamii]

Mooncake and Enzyme are faster on this PR. So the 10× regression reported in comments are no longer valid.

main branch d2052a1:
Model                      dim    linked     primal    FwdDiff    RvsDiff    Mooncake    Enzyme
-----------------------------------------------------------------------------------------------
Simple assume observe*       1     false    3.09 ns      12.60    1542.87       40.05      8.81
Simple assume observe*       1      true    3.09 ns      12.88    1690.22       43.55      9.27

This PR 3bdeb74:
Model                      dim    linked     primal    FwdDiff    RvsDiff    Mooncake    Enzyme
-----------------------------------------------------------------------------------------------
Simple assume observe*       1     false    3.26 ns      12.76    2101.00       37.30      8.23
Simple assume observe*       1      true    3.02 ns      13.37    1832.40       38.87      8.85

My system:

julia> versioninfo()
Julia Version 1.12.6
Commit 15346901f00 (2026-04-09 19:20 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 16 × 12th Gen Intel(R) Core(TM) i5-12500H
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, alderlake)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 16 virtual cores)

julia> 

[yebai]

Mooncake and Enzyme are faster on this PR. So the 10× regression reported in comments are no longer valid.

To clarify, these metrics were never valid; they arose from a misunderstanding. For some models, log-density evaluations occur on the order of nanoseconds, making the gradient-to-primal ratios highly noisy across runs and machines. For these tiny models, the primal and gradient timings themselves are more reliable. More generally, these benchmarks are intended primarily to catch obvious allocation or type-stability regressions and are meant for DynamicPPL developers only.