Add FullyBayesianGP: HMC-marginalized hyperparameters + Bayesian model averaging

[blankjul]

What

A new surrogate backend, FullyBayesianGP (in pysurrogate.models), that marginalizes the GP length-scales and nugget by Hamiltonian Monte Carlo instead of point-estimating them by maximum likelihood, then predicts by Bayesian model averaging:

mean = <mu_s>        var = <sigma2_s> + Var_s(mu_s)

The extra between-sample term is the hyperparameter uncertainty an MLE fit drops — which is why an MLE variance is overconfident on scarce/noisy data (one length-scale guess, no acknowledgement it was a guess).

Why it's cheap to add

No new gradient code. The HMC potential is the DACE profile likelihood the Kriging search already descends (log-lik = -(n/2)·log(obj)), so _GPPosterior just wraps DaceProblem through a log transform plus a log-space prior. Each posterior sample becomes a frozen Dace fit (optimizer=None); prediction averages their Predictions. Builds directly on the Prior foundation from #9.

New files

• dace/hmc.py — generic HMC with dual-averaging step-size adaptation and trajectory-length jitter (breaks the fixed-leapfrog resonance that was under-sampling the widest posterior direction).
• models/bayesian.py — FullyBayesianGP + the _GPPosterior wrapper.
• tests/models/test_bayesian.py — 10 tests.

Verification

• HMC recovers a known 3D Gaussian (mean + full covariance).
• Log-posterior gradient matches finite differences to ~1e-9.
• Deterministic given the seed.
• DACE golden stays 19/19 byte-identical (purely additive).
• pyclawd check green; fast tier 531 passed.

Honest scope

Regime	Verdict
Noisy data (calibration)	✅ Clear, reliable win. d4/n40: mean NLPD 0.2 vs 0.9 (4× better calibrated) at comparable RMSE, and never catastrophically overconfident — MLE-Kriging is on ~half the seeds. Exactly what a BO acquisition needs.
Sparse high-D (SAAS)	❌ Not shipped. Couldn't demonstrate a win in tractable regimes (MLE-ARD is either already perfect or everything fails), so the headline is calibration + noise-robustness, not sparsity.

The calibration test asserts the aggregate NLPD win (robust) rather than per-seed (which MLE occasionally wins when it happens to be calibrated) — marked @slow.

🤖 Generated with Claude Code