Synthetic parameter-recovery example

The synthetic single-date example is the recommended first calibration case. It generates observations from a known shifted-exponential LPM, adds controlled noise, estimates the same LPM family, and compares the posterior with the known truth.

The generation settings define:

Setting	Value
LPM	exp_shifted
True mu	28 years
True shift	4 years
Tracers	CFC-11, CFC-12, CFC-113, SF6
Relative observation error	4%
Random seed	12345

Run

From a source checkout with PyAges installed:

python -m examples.synthetic.lpm_recovery_single_date.run_lpm_recovery_single_date

The default runner reuses the reviewed, versioned synthetic observations, runs the single-date Metropolis-Hastings workflow, and writes truth-aware summaries. Its 5,000-step single chain is pedagogical; it is not a publication convergence protocol.

Run the multi-chain qualification

PyAges 2.0 includes a second configuration for multi-chain qualification:

pyages run examples/synthetic/lpm_recovery_single_date/lpm_recovery_single_date_multichain.yaml

This profile uses four bounds_stratified starts, 1,500 pilot transitions and 4,000 production transitions per chain, thinning: 1, master seed 20260831, and required R-hat < 1.01 plus bulk/tail ESS >= 300. It retains 2,999 rows per chain, or 11,996 rows after qualification and pooling. The current sequential runner performs 22,000 MH transitions.

The corresponding extensive test is:

python -m pytest -q --run-extensive tests/examples/test_synthetic_recovery_multichain_scientific.py

It verifies marginal recovery of mu=28 and shift=4, their two-dimensional joint ellipse, recovery of mu+shift=32, and all four fitted latent tracer responses. The fixed-seed run has maximum R-hat 1.003066, minimum bulk ESS 1540.59, and minimum tail ESS 1647.96.

The fitted response distribution is not posterior predictive: it varies the retained LPM parameters but does not draw new observation noise. The result qualifies this one versioned noisy realization, not repeated-noise coverage. Operational details and trace inspection are in Running and qualifying multi-chain MH.

Regeneration is deliberately explicit because a numerical-method change can alter the versioned reference data:

python -m examples.synthetic.lpm_recovery_single_date.run_lpm_recovery_single_date --regenerate

Review the resulting Git diff and the relevant scientific invariants before accepting regenerated values.

Interpret

Start with parameter_recovery_summary.txt. It records each true parameter, posterior mean and standard deviation, and the mean-minus-truth difference. Then compare:

• 01_data_model_space.png: noisy observations, reachable models, posterior samples, and the noise-free true model;

• 02_parameter_summary.png: marginal posterior distributions and true parameter values;

• 03_objective_summary.png: sampled objective landscape, posterior samples, and truth.

The known truth makes parameter recovery visible, but it does not eliminate finite-chain Monte Carlo error or structural non-identifiability. Continue with Bayesian Inference and Diagnostics before interpreting field results.

The source files and notebook remain in examples/synthetic/lpm_recovery_single_date/ on GitHub.