Exercise 1000.UQ.1 — Bayesian inference of a reduced climate model¶

Reference and scope¶

This exercise is fully detailed in the lecture notes, in section 10.14:

“An end-to-end uncertainty quantification workflow: a reduced climate model”

Students are expected to read this section carefully before starting the exercise. All equations, model structure, and physical assumptions are defined there and are not re-derived here.

The Python script provided in this repository is a direct implementation of the workflow described in the notes.

Script and data¶

🧪 Script
bayesian_climate.py

🧪 Datasets

Historical temperature data

Historical CO₂ data

Historical CH₄ data

These datasets correspond to those shown in the lecture notes (e.g. figures of historical data and posterior predictive checks).

Learning objectives¶

By completing this exercise, you will learn to:

Implement an end-to-end Bayesian uncertainty quantification workflow
Calibrate a reduced-order dynamical climate model using historical data
Distinguish clearly between:
- prior assumptions,
- likelihood (data constraints),
- posterior uncertainty
Interpret MCMC output beyond point estimates
Perform posterior predictive checks
Analyze time-dependent parameter importance using variance-based sensitivity analysis
Understand why uncertainty grows in future projections even when historical fit is good

What the script does (high-level workflow)¶

The script follows exactly the steps described in the lecture notes:

Define the forward model
- A coupled system of ODEs describing:
  - CO₂ concentration
  - CH₄ concentration
  - radiative forcing terms
  - planetary albedo
  - global mean temperature anomaly
  - aerosol forcing
Load historical observations
- Temperature anomaly
- CO₂ concentration
- CH₄ concentration
Specify uncertainty
- Bounded uniform priors for all uncertain parameters
- Gaussian observational error model
Bayesian inference
- Compute the log-posterior
- Find a MAP estimate (optimization)
- Use MCMC (ensemble sampler) to explore the posterior distribution
Posterior analysis
- Marginal and joint posterior distributions
- Parameter correlations
Posterior predictive uncertainty
- Forward propagation of posterior samples
- Probabilistic future projections under different emission scenarios
Global sensitivity analysis
- Time-dependent Sobol-style variance contributions
- Time-integrated parameter importance

How to run¶

From the repository root:

python chapters/1000_UQ/scripts/bayesian_climate.py

Practical notes¶

This script can be computationally expensive depending on:
- number of MCMC walkers,
- number of MCMC steps,
- ODE solver tolerances,
- use of multiprocessing.
For a quick test run, you may:
- reduce the number of walkers,
- reduce the number of MCMC steps,
- shorten the historical time window,
- disable multiprocessing.
A full run reproduces figures and behaviors discussed in the lecture notes. Expect runtimes from several minutes to longer, depending on settings.
Always verify that chains are well mixed before interpreting results.

Guided questions¶

1) Prior vs data: who constrains what?¶

Which parameters are strongly constrained by the historical data?
Which parameters remain close to their prior bounds?
How can this be inferred from marginal posterior distributions?

Relate your answer to the discussion of weak and informative priors in the lecture notes.

2) Parameter correlations and identifiability¶

Which parameters exhibit strong posterior correlations?
What physical or structural reasons explain these correlations?
Why is joint inference essential in this model?

Support your answer using joint posterior plots.

3) MAP estimate vs posterior distribution¶

Does the MAP parameter set provide a good fit to historical data?
Are there distinct parameter combinations that fit the data almost equally well?
What does this imply about relying on a single “best-fit” solution?

4) Posterior predictive checks¶

Does the posterior predictive interval capture the historical observations?
Are there systematic deviations during specific time periods?
Should these deviations be interpreted as parameter uncertainty or model limitations?

5) Uncertainty growth in future projections¶

Why does uncertainty grow with time even after calibration?
Why does uncertainty differ between business-as-usual and mitigation scenarios?

Relate your answer to feedback mechanisms discussed in the notes.

6) Time-dependent sensitivity¶

Which parameters dominate uncertainty in the early historical period?
Which parameters dominate long-term projections?
Why does parameter importance change with time?

Use the time-resolved variance contribution results to justify your answer.

Student tasks¶

Task 1 — Baseline calibration and posterior summary¶

Run the script as provided and deliver:

one plot comparing historical temperature data with:
- MAP prediction,
- posterior predictive uncertainty band;
one marginal or joint posterior plot (or a subset of a corner plot).

Write 8–12 lines discussing:

quality of the fit,
dominant sources of uncertainty,
whether the model appears overconfident or underconfident.

Task 2 — Robustness to inference choices¶

Modify one modeling or inference assumption, for example:

widen or narrow selected parameter priors,
change the assumed observational noise level,
change the number of MCMC walkers or steps.

Deliver:

one figure illustrating the impact on posterior or predictive uncertainty,
6–10 lines explaining what changed and why.

Task 3 — Identifiability analysis¶

Select two strongly correlated parameters and:

plot their joint posterior distribution,
explain the physical or mathematical origin of the correlation,
discuss what additional data or modeling changes could reduce this degeneracy.

Task 4 — Sensitivity interpretation¶

Using the sensitivity-analysis outputs:

identify the most important parameters by time-integrated contribution,
identify at least one parameter whose importance changes with time,
explain what this implies for long-term projections.

Write 8–12 lines.

Limitations (important)¶

This exercise illustrates a complete Bayesian UQ workflow, but it has important limitations:

The climate model is highly reduced and omits many physical processes.
Structural model discrepancy is not explicitly represented.
Observational uncertainty is treated in a simplified manner.
No rigorous convergence diagnostics are enforced.
Results are conditional on the chosen model structure and data.

All results should be interpreted as:

uncertainty conditioned on a specific model class,
not as definitive predictions of the real climate system.

Key takeaway¶

This exercise shows that:

good agreement with historical data does not imply predictive certainty;
uncertainty redistributes across parameters and time horizons;
long-term projections are dominated by feedback-driven uncertainty.

Replacing single trajectories with probability distributions is essential for responsible engineering analysis and decision-making.