SBBRMI: no embedded Laplace (Stan laplace_marginal) for @plate cell latents, blocked upstream on stan-dev/math#3438 #3

Summary

BRM has no spelling that integrates a @plate cell latent out with Stan's embedded Laplace approximation (laplace_marginal, Stan 2.37 or later), so cell latents are always sampled. An opt-in SBBRMI lowering was investigated and is on hold: on a synthetic model it does not run with the Stan that BRM compiles against, and it could not be sampled even with the pending upstream fix.

Model

Synthetic: 18 cells with 8 to 16 Gaussian latents each, log_T ~ Normal(location, scale), where location and scale depend on 56 population parameters. The latents are observed through T = exp(log_T): by noisy counts on about 70 % of the entries, and by the cell total with a 0.2 % relative standard deviation.

Findings

With BridgeStan 2.9.0 (Stan 2.39, Stan Math 5.3.0):

• laplace_marginal returns NaN for 17 of the 18 cells at the true parameter values, with solvers 1, 2 and 3 and tolerances from 1e-3 to 1e-8.
• The cause is upstream. The likelihood is not log-concave (its negative Hessian is indefinite even at the mode in 12 of 18 cells). Solvers 1 and 2 need a positive-definite matrix and fail from the default start, so Stan falls through to solver 3. Solver 3 reaches the mode, but its compute_log_determinant takes log of the signed LU diagonal. That is NaN whenever partial pivoting leaves a negative pivot, although det(B) > 0 in every cell.
• Fix: Return the final Newton iterate as the Laplace latent mean, and fix the solver-3 log determinant stan-dev/math#3438 (open).
• Started at the exact conditional mode, all three solvers return the correct Laplace value.

With #3438 applied to a local copy of Stan Math:

• Values are finite, but one gradient costs about 440 times the joint model's.
• The marginal is not smooth at small scales: directional finite differences agree with the gradient at step 1e-2 and diverge below 1e-4.
• NUTS (4 chains) did not finish warm-up in 30 minutes. The joint model sampled 4 × 1,000 draws in about a minute.

Resume condition

Revisit when a BridgeStan release ships a Stan Math with #3438, and its marginal is smooth on this shape. Re-measure on this model before adding a lowering. Any such lowering must be opt-in. A per-unit Laplace for the RK backend (RKBRMI) is being developed separately.