Uncertainty quantificationEngineering methods

Bayesian inference & UQ

Not a single answer — a distribution over answers, with calibrated confidence. Markov-Chain Monte Carlo fits a model to noisy data and quantifies how sure you should be: posteriors that match the closed form where one exists, credible bands on predictions, model comparison, and a coverage check.

DjiniousLabModelling & simulation

Posterior distributions of two parameters with their correlated joint density and a well-mixed MCMC trace

MCMC vs analytic posterior (mean diff): 0.002MCMC vs analytic posterior (mean diff)
Gelman–Rubin R̂ (converged): 1.001Gelman–Rubin R̂ (converged)
coverage of the 90% intervals: 86%coverage of the 90% intervals
credible intervals contain the truth: ✓credible intervals contain the truth
design notebooks: 10design notebooks
requirements PASS: 5 / 5requirements PASS

The honest answer is a distribution, not a number.

Every other model in this portfolio gives one answer. But fit a model to noisy data and the right answer isn't "the rate is 0.7" — it's "0.7, and here's exactly how sure I am." Bayesian inference produces that: a full posterior distribution over the parameters, and calibrated uncertainty on every prediction. Markov-Chain Monte Carlo is the engine. This program builds it from the conjugate case you can check by hand up through calibrating a dynamical model and comparing competing ones — and it complements every other flagship, because uncertainty can wrap any of them.

Start where you can check the answer exactly.

For some models the posterior is analytic — a Beta prior and binomial data give a Beta posterior, in closed form. The program starts there, watching the posterior sharpen onto the true value as data arrives, because it's the case where a sampler can be validated to the last digit. Get the machinery right where you can verify it, then trust it where you can't.

A Beta-Binomial posterior distribution sharpening onto the true value as more data is observed — The correctness anchor: with a conjugate prior the posterior is analytic, and it visibly sharpens onto the truth as data accumulates. The one case where the answer is exact — the benchmark every sampler in the program must reproduce.

MCMC posterior marginals for two parameters, their joint density, and a well-mixed trace — The hero — Metropolis–Hastings MCMC: a hand-rolled sampler recovers a model's parameters as full posterior distributions (top), their correlated joint density (bottom-left), from a well-mixed chain (bottom-right). Each marginal is centred on the true value, and the histogram lands exactly on the analytic posterior. The engine of modern Bayesian computation, built from scratch.

A fit you can't doubt is a fit you can't trust.

Least-squares draws one line through the data and stops. Bayesian regression draws the whole posterior over lines — and the spread of that fan, narrow where data is dense and flaring where it's sparse, *is* the uncertainty. That's the difference between a prediction and a prediction you can act on: not just the best guess, but how far it might be off, everywhere.

A fan of posterior-sample regression lines through noisy data, widening where data is sparse — The fan is the uncertainty: instead of one best-fit line, Bayesian regression gives a posterior over lines. Where the data pins the model the lines bunch tight; where it doesn't, they splay. A single least-squares line throws all of that away.

Calibrate a dynamical model — with error bars.

The same machinery infers the parameters of an ODE. Given noisy observations of a decaying signal, MCMC recovers the decay rate as a posterior (centred on the truth), and pushing that posterior forward through the model produces a posterior-predictive band — a forecast that carries its own uncertainty. And the program checks that the uncertainty is honest: across many replications, the 90% credible intervals contain the truth about 90% of the time. Calibration, verified.

An ODE decay trajectory calibrated to noisy data with a posterior-predictive credible band — Model calibration with uncertainty: MCMC infers a decaying ODE's rate constant from noisy observations and propagates the posterior forward into a 90% predictive band around the trajectory. A forecast that knows how uncertain it is — and whose intervals are calibrated to contain the truth.

Let the data pick the model — and punish complexity.

Given competing models — a straight line versus a parabola — which does the data support? A more complex model always fits the training data better, so the honest comparison penalizes complexity. The Bayesian information criterion does exactly that, and over a sweep of polynomial degrees it bottoms out at the true degree: enough flexibility to fit the signal, no more. Occam's razor, made quantitative.

A model-comparison criterion minimized at the true polynomial degree — Occam's razor, quantified: across polynomial degrees the information criterion is minimized at the degree that generated the data — complex enough to capture the signal, penalized for going further. The data picks its own model, and over-fitting is scored against.

Every number re-derived at sign-off.

The V&V notebook rebuilds each sampler from scratch with fixed seeds and re-derives the requirements against theory, printing a PASS/FAIL board.

Result

MCMC vs analytic posterior: mean diff 0.002
Parameter recovery (CI ∋ truth): 1.85 / 0.53
Multi-chain R̂: 1.001
Predictive-interval coverage: 86%
Model comparison (BIC): true degree

Requirement

MCMC vs analytic posterior: R-01
Parameter recovery (CI ∋ truth): R-02
Multi-chain R̂: R-03 ≈ 1
Predictive-interval coverage: R-04 ≈ 90%
Model comparison (BIC): R-05

The sampling paradigm, anchored to theory.

The deliverable is ten notebooks and the dossier — MCMC samplers and posteriors, not an acausal network, so there's no custom block or canvas. The sampler is random-walk Metropolis–Hastings (not gradient-based HMC/NUTS), likelihoods are Gaussian, chains are modest length, the models are low-dimensional, model comparison uses an information criterion rather than a full marginal likelihood, and nothing is calibrated to a real-world dataset. What the program proves is the Bayesian paradigm itself — a distribution over answers with calibrated uncertainty — validated against closed-form posteriors where they exist, and reaching dynamical-model calibration, uncertainty propagation, and model comparison where they don't. It is the uncertainty layer that can wrap any other flagship.