Reimplement particle MCMC (SMC and PG) natively, removing AdvancedPS

[yebai]

Opened for discussion.

Reimplements SMC and PG/CSMC directly on Libtask + DynamicPPL and drops the AdvancedPS
dependency. A rewrite based on #2848 (thanks @charlesknipp; commits co-authored).

Each observe is one filtering step: under ParticleMCMCContext, every likelihood term
becomes a Libtask.produce, so a particle is a suspended model execution we advance, weight,
and resample. The conditional-SMC reference is reproduced by replaying a per-particle
TracedRNG (no stored trajectory); a fork is reseeded to branch off. Particle state lives in
the task's taped globals (no task_local_storage), and @addlogprob! reweights via the
likelihood accumulator.

Posteriors, log-evidence, and exact reference reproduction match the old implementation;
gibbs.jl, Aqua, and log-density-metadata tests pass; performance is on par.

Fix #2781
Fix TuringLang/AdvancedPS.jl#110
Fix TuringLang/AdvancedPS.jl#39
Fix TuringLang/AdvancedPS.jl#6

[github-actions]

Turing.jl documentation for PR #2853 is available at:
https://TuringLang.github.io/Turing.jl/previews/PR2853/

[codecov]

Codecov Report

❌ Patch coverage is 96.98276% with 7 lines in your changes missing coverage. Please review.
✅ Project coverage is 86.25%. Comparing base (11f081b) to head (bf9357d).

Files with missing lines	Patch %	Lines
src/mcmc/particle_mcmc.jl	96.98%	7 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #2853      +/-   ##
==========================================
+ Coverage   85.09%   86.25%   +1.16%     
==========================================
  Files          23       23              
  Lines        1516     1586      +70     
==========================================
+ Hits         1290     1368      +78     
+ Misses        226      218       -8     

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[charlesknipp]

Overall, I think this is a very nice revision of #2848. Although, there are a few minor choices I intentionally made when drafting that should be reconsidered here.

1. Passing AbstractMCMCEnsemble through to reweight! (only for SMC) for consistency
2. Producing the log score upon tilde_observe (or accloglikelihood!! for @addlogprob!) to ensure log score is up to date with log likelihood accumulation.

[yebai]

Thanks, this was a valuable review. What changed, and where I took a different route:

Score ↔ accumulator consistency. The produce now fires from tilde_observe!! and an accloglikelihood!! overload after the accumulator is updated, so the produced score can't lag the accumulated log-likelihood and @addlogprob! terms reach the accumulator too. It emits the accumulator's delta rather than a stashed increment, so the accumulator stays the single source of truth.

SMC output. AbstractMCMC.sample now runs the sweep and bundles the whole population in one shot — SMCState and the step-loop are gone, and discard_initial/thinning no longer have anything to apply to. I took it a little further: a final resampling step returns an equal-weight sample (so mean(chain[…]) needs no weighting), and chains carry log_normalizing_constant (renamed from logevidence, the standard SMC term) plus a per-observation ess_per_step diagnostic.

Parallelism. I diverged here. MCMCThreads/MCMCDistributed parallelise independent chains; within-sweep parallelism spreads one sweep's particles, which resample all-to-all every step and are live Libtask tasks — so a distributed reweight would be communication-bound and effectively infeasible. Rather than dispatch reweight! on the ensemble, I kept the two as separate knobs (chain-level = sample's ensemble argument; within-sweep = a threads-only multithreaded sampler option) and fixed the HISTORY line that blurred them.

PG state Done — the retained Particle already carries the varinfo and rng needed to resume, so PGState is gone.

Naming / parametrization SMCContext; and Particle{RT<:TracedRNG,WT<:Real} so logweight follows LogProbType.

"Is it always InitFromPrior?". Yes — verified a value is never already present at tilde_assume!! (particle varinfos start empty, each variable is assumed once, and the CSMC reference replays RNG seeds rather than reusing stored values), so an InitFromParams branch would be dead code. I documented the reasoning in place instead of adding it.