Explorables

Most explanations of probabilistic programming ask you to read first and understand later. These pages invert that. Every figure here is a small machine: drag the numbers, click the data, step the algorithms, and watch the mathematics respond. You will have felt why a sampler stalls or a particle filter collapses before you meet the equation that says so. The genre is borrowed with gratitude from Bret Victor's explorable explanations and 3Blue1Brown's visual mathematics.

Two habits worth forming:

• Touch everything. If a number looks interesting, try dragging it — dashed underlines mark the ones that respond. Canvases are clickable more often than not.
• Mind the seed. Wherever randomness appears, a scrubbable seed appears with it. The same seed always replays the same run — which is not a gimmick, it is fugue's worldview: a recorded run is a trace, and traces can be replayed. You will meet this idea everywhere.

One color language runs through every page: prior × likelihood = posterior, with coral for the current sample and violet for momentum and structure. Once you know it, every canvas and every equation on this site reads at a glance.

The six machines

1. Anatomy of a Probabilistic Program — a coin, a prior you can bend, and data you can click into existence. Bayes' rule as something your hands learn first.
2. The Model Is a Score — five observations you can drag along an axis, a prior, and an exact posterior that follows your hand. Underneath, fugue's Model monad performed one effect at a time — then handed to a different performer. The page that explains fugue itself.
3. Random Walks in Posterior Space — a real regression: drag the data points and watch the posterior heatmap deform under the sampler's feet, while accepted samples paint fit-lines through your data. Live split-R̂ and ESS referee the whole thing.
4. Rolling, Not Guessing: Hamiltonian Monte Carlo — the same regression, but the sample gets momentum and rolls. Leapfrog trajectories, divergences, and a side-by-side race against the random walk at a matched budget — visible in the data, not just the parameters.
5. Particles That Tell Stories — Sequential Monte Carlo as a population: propagate, weight, resample. Watch lineages go extinct and understand degeneracy by witnessing it.
6. A Field Guide to Distributions — every distribution fugue ships, with real parameterizations, a sampler racing its own density, and the natural return types that make fugue's models type-safe.

Read them in order for a course, or jump to what you came for — each page stands alone and links onward. When a page convinces you, the matching tutorial turns the intuition into working Rust; the code on every explorable compiles against fugue-ppl 0.2.0 as written.

One more honesty note, and a point of pride: the samplers behind the MH, HMC, and SMC machines are not JavaScript imitations — they are the fugue crate itself, compiled to WebAssembly, driven one transition per animation frame. Same kernels, same diagnostics, same seeded reproducibility as production; only the drawing is JavaScript. (Browsers without the WASM package fall back to a faithful JS mirror.) When you want to type your own model at these kernels, the Playground is exactly that.