Inverse UQ

The scripts in apps/iuq/ perform parameter inference using Markov chain Monte Carlo (MCMC) sampling with a Polynomial Chaos (PC) surrogate model constructed by the sibling Forward UQ workflow. Besides conventional Bayesian inference, embedded model-error inference is also supported.

The workflow can run out-of-the-box as a demo, but is intended to be adapted to specific problems — there are many tuning knobs that typically need adjusting.

The folder contains:

• workflow_iuq.x — an example end-to-end workflow script.

• run_infer.py — the main inference script.

• create_data_truepc.py — synthetic data generation from a PC model.

• prep_data.x — an alternative example of manual data preparation.

• postp_infer.py — post-processing and visualisation.

Workflow overview

A typical IUQ session follows three steps (after the forward surrogate has been built with the UQPC workflow):

0. Prepare data — Create or supply observational data (ydata.txt) and the corresponding noise variance (ydatavar.txt). Use create_data_truepc.py for synthetic data from the PC surrogate, or prep_data.x for manual data setup.

1. Run inference — Execute run_infer.py to perform MCMC-based Bayesian calibration using the PC surrogate as the forward model. Use -m for embedded model-error inference.

2. Post-process — Run postp_infer.py to generate posterior diagnostics: MCMC chain traces, marginal PDFs, prior-vs-posterior comparisons, and posterior predictive checks.

The example workflow workflow_iuq.x demonstrates all three steps.

Prerequisites

All scripts expect results.pk (produced by uq_pc.py in the Forward UQ workflow) to be present in the working directory.

create_data_truepc.py

Create synthetic observational data for inference testing.

This script loads a previously built PC surrogate from results.pk, evaluates it at a true parameter vector, adds Gaussian noise, and writes out synthetic data and variance files for use by the inference workflow.

Outputs:

• p_true.txt — True parameter vector used for data generation.

• ydata.txt — Synthetic observational data of shape (outdim, neach).

• ydatavar.txt — Data-noise variance array of shape (outdim,).

Example:

python create_data_truepc.py -s 0.5 -e 7
python create_data_truepc.py -s 0.1 -e 3 -c p_true_input.txt

Arguments:

Flag	Default	Description
-s, --sig	0.5	Noise standard deviation added to the PC model output.
-e, --each	1	Number of replicate samples per output location.
-c, --cfile	None	True-coefficients file. If omitted, random values are drawn.

prep_data.x

An alternative Bash script for manually preparing ydata.txt and ydatavar.txt. Users can edit this script to hard-code their own observations and noise variances.

Example:

bash prep_data.x

run_infer.py

Perform MCMC-based Bayesian inference using the PC surrogate.

This script loads the PC surrogate from results.pk, configures the likelihood, prior, and adaptive MCMC sampler, and runs Bayesian calibration. Parameters with total sensitivity below 1 % are automatically fixed at their nominal values.

Expected inputs:

• results.pk — PC surrogate from the UQPC workflow.

• ydata.txt — Observational data.

• ydatavar.txt — Data-noise variance.

Outputs:

• calib_results.pk — Pickled calibration dictionary containing the MCMC chain, MAP parameters, posterior object, and diagnostics.

• chain.txt — MCMC chain (plain text).

• mapparams.txt — MAP parameter estimate.

• surr_error_var.txt — Surrogate approximation error variance.

Example:

# Classical inference (no model error)
python run_infer.py

# Inference with embedded model error
python run_infer.py -m

Arguments:

Flag	Default	Description
-m, --merr	False	Enable embedded model-error inference. When set, a Gaussian marginal likelihood (gausmarg) is used with additional stochastic output-order terms.

Note

Key inference settings (MCMC length, burn-in, adaptation schedule, prior type, likelihood type, etc.) are configured inside the script. Users should edit the model_infer_config dictionary to match their problem.

postp_infer.py

Post-process and visualise Bayesian inference results.

This script loads calib_results.pk and results.pk, then generates:

• Chain diagnostics — MCMC trace plots for each inferred dimension (chain.png).

• Posterior predictive — Model predictions from posterior samples evaluated through the surrogate.

• Input posterior PDFs — Triangular pair-plot of parameter posteriors (pdf_tri_post_inputs.png).

• Output posterior PDFs — Triangular pair-plot of output posteriors with observed data overlaid (pdf_tri_post_outputs.png).

• Prior vs posterior — Kernel-density and scatter comparisons between training-set (prior) and posterior parameter samples (pdf_prior_vs_post.png).

• Posterior predictive envelopes — Output predictions at MAP, with data and error bars (postpred_map.png).

Expected inputs:

• results.pk — PC surrogate results.

• calib_results.pk — Calibration results from run_infer.py.

• ydata.txt — Observational data.

• ydatavar.txt — Data-noise variance.

• surr_error_var.txt — Surrogate error variance.

Example:

# Without known true parameters
python postp_infer.py

# With known true parameters overlaid on posteriors
python postp_infer.py -c p_true.txt

Arguments:

Flag	Default	Description
-c, --cfile	None	True-parameters file. When provided, the true values are overlaid on posterior PDF plots for comparison.

Note

Parameter and output names are read from pnames.txt and outnames.txt when present; otherwise generic names (Par_, Out_) are used.

workflow_iuq.x

An annotated Bash script that demonstrates the complete inverse-UQ workflow: data preparation, inference (with or without model error), and post-processing.

Run it from the apps/iuq/ directory (after having run the UQPC forward workflow to produce results.pk):

bash workflow_iuq.x