NO. 38 · Computational & Data

Bayesian Inference in Practice

The machinery of belief under evidence. You learn what an interval actually promises, then build posteriors and earn them: conjugate updates, Gibbs and Hamiltonian samplers, model weights, predictive checks. The path ends underground, where Bayesian PINNs deliver a velocity model as a posterior you can defend.

You can state exactly what a confidence or credible interval claims, derive a conjugate posterior by hand, sample an intractable one with Metropolis, Gibbs, or HMC, weigh models with Bayes factors and WAIC, check them against their own predictions, summarise draws into densities and quantiles, and carry the whole apparatus into uncertainty-aware seismic inversion.

15 competencies · 5 interactive widget challenges · 9 to 14 hours of guided study

For researchers who need posteriors they can defend, from lab data to velocity models

Draws on: Physics-Informed Neural Networks for Exploration Seismology Statistics and Data Science for Researchers

Uncertainty, honestly

What an interval actually claims

An interval is a promise about a procedure or a statement of belief; confuse the two and every error bar you publish says something you did not mean.

Reads: Exact vs asymptotic CIs Prediction intervals vs confidence intervals

Calibration and honest reporting

A 95 percent interval that covers 80 percent of the time is a quiet lie; calibration is how you catch it, and honest graphics are how you stop repeating it.

Reads: Calibration: when 95% really means 95% Communicating uncertainty without lying

Bootstrap intervals, hands onwidget challenge

Percentile, basic, BCa: three intervals from the same resamples, and they disagree exactly when the data are awkward. Knowing which to trust is a working skill, not trivia.

Reads: Bootstrap CIs (percentile, BCa, basic)

Bayes in earnest

Prior to posterior, by hand

Prior times likelihood, normalised: the whole doctrine in one line. Work the mechanics until a conjugate update feels like arithmetic, because everything after this is that line at scale.

Reads: Prior, likelihood, posterior, the mechanics Conjugate priors and analytic posteriors

Metropolis and Hamiltonian samplers

When the posterior has no formula you walk it instead; Metropolis steps and HMC glides, and knowing why each works is what separates a user of MCMC from an operator.

Reads: Metropolis-Hastings by hand Hamiltonian Monte Carlo intuition

Gibbs sampling, driven yourselfwidget challenge

Gibbs trades one hard joint distribution for easy conditionals, one axis at a time; drive it once and correlated parameters stop being a mystery.

Reads: Gibbs sampling on a 2D example

Bayes factors and WAICwidget challenge

Two models can both fit and still deserve different degrees of belief; Bayes factors and WAIC put a number on that, and teach you the number's own fragility.

Reads: Model comparison: Bayes factors and WAIC

Computation and checking

Posterior-predictive checks

A posterior that cannot regenerate the data that built it is confessing misfit; the predictive check is the cheapest honesty test in the Bayesian toolkit.

Reads: Posterior-predictive checks

Bootstrap, deeper, and cross-validation

The parametric bootstrap is predictive simulation in frequentist clothes, and cross-validation is the out-of-sample twin of WAIC; the two camps' checking tools are closer than either admits.

Reads: Bootstrap, deeper: parametric vs nonparametric Cross-validation done right

From draws to densities

A sampler hands you ten thousand draws, not a curve; kernel density estimation is how draws become the posterior you plot, and bandwidth is where that picture can quietly lie.

Reads: Kernel density estimation

Quantiles and distribution-free intervalswidget challenge

Credible intervals are read straight off the quantiles of your draws, and when nothing is Gaussian, quantile methods still keep their promise.

Reads: Quantile regression and distribution-free CIs

Bayes underground

PINN roles in classical inversionwidget challenge

Before Bayes goes underground you need the map of hybrid machinery: PINN as regulariser, initialiser, or solver, welded to classical FWI. This is the workflow your posterior will ride on.

Reads: PINN as regulariser, initialiser, or solver PINN-augmented classical FWI

Learned and generative priors

A learned regulariser is a prior you trained rather than chose, and a generative prior says what a plausible velocity model looks like before any wave arrives. This is the prior, industrialised.

Reads: Learned regularisers and ML priors Generative priors for velocity models

Bayesian and ensemble PINNs

Put a distribution over the network and the inversion inherits one; Bayesian and ensemble PINNs are the samplers of Part 7 sent underground.

Reads: Bayesian PINNs and ensemble PINNs

Uncertainty-aware velocity inversion

The destination: a velocity model delivered as a posterior, with the calibration standards of Stage 1 still binding kilometres down. What you learned about honest intervals is now a statement about the earth.

Reads: Uncertainty-aware velocity inversion