Add rrule for BridgeStan's log_density function

[nsiccha]

Adds a MooncakeBridgeStanExt package extension providing rrule!! for BridgeStan.log_density.

BridgeStan's log_density calls into compiled Stan C++ code, so Mooncake cannot differentiate through it generically. This extension uses the existing log_density_gradient! API — which Stan already computes analytically — for an exact, efficient reverse-mode rule.

• tangent_type(StanModel) = NoTangent: StanModel holds C pointers Mooncake must not recurse into
• Two rrule!! methods: positional log_density(sm, q) and kwarg log_density(sm, q; propto, jacobian)
• Tests cover both call forms across all propto/jacobian combinations

[github-actions]

Mooncake.jl documentation for PR #1006 is available at:
https://chalk-lab.github.io/Mooncake.jl/previews/PR1006/

[codecov]

Codecov Report

❌ Patch coverage is 0% with 20 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
ext/MooncakeBridgeStanExt.jl	0.00%	20 Missing ⚠️

📢 Thoughts on this report? Let us know!

[github-actions]

Performance Ratio:
Ratio of time to compute gradient and time to compute function.
Warning: results are very approximate! See here for more context.

┌────────────────────────────┬──────────┬──────────┬─────────────┬─────────┬─────────────┬────────┐
│                      Label │   Primal │ Mooncake │ MooncakeFwd │  Zygote │ ReverseDiff │ Enzyme │
│                     String │   String │   String │      String │  String │      String │ String │
├────────────────────────────┼──────────┼──────────┼─────────────┼─────────┼─────────────┼────────┤
│                   sum_1000 │ 110.0 ns │     1.91 │        1.92 │    1.09 │        5.46 │   10.1 │
│                  _sum_1000 │ 951.0 ns │     6.75 │        1.01 │  1620.0 │        33.5 │   1.09 │
│               sum_sin_1000 │  6.57 μs │     3.19 │        1.38 │    1.77 │        10.8 │   1.89 │
│              _sum_sin_1000 │   5.1 μs │     3.21 │        2.28 │   318.0 │        13.8 │   2.57 │
│                   kron_sum │ 436.0 μs │     6.96 │        2.93 │    4.45 │       173.0 │   10.8 │
│              kron_view_sum │ 427.0 μs │     8.19 │        4.17 │    17.5 │       175.0 │   5.31 │
│      naive_map_sin_cos_exp │  2.17 μs │     2.48 │        1.44 │ missing │        7.19 │   2.32 │
│            map_sin_cos_exp │  2.14 μs │     2.87 │        1.51 │    1.75 │        6.24 │   2.93 │
│      broadcast_sin_cos_exp │   2.3 μs │     2.51 │        1.43 │    4.57 │        1.46 │   2.21 │
│                 simple_mlp │ 183.0 μs │     6.56 │        2.84 │    1.71 │        13.6 │    3.1 │
│                     gp_lml │ 201.0 μs │     10.3 │        2.46 │    4.43 │     missing │   8.04 │
│ turing_broadcast_benchmark │  1.97 ms │     4.09 │        3.06 │ missing │        30.5 │   2.05 │
│         large_single_block │ 391.0 ns │     4.43 │         2.0 │  4500.0 │        30.5 │    2.2 │
└────────────────────────────┴──────────┴──────────┴─────────────┴─────────┴─────────────┴────────┘