https://josephausterweil.github.io/probintro/structure/index.htmlThe Bento Provenance Project. Solving the Mystery of the Two Peaks told Chibany what arrives — a 70/30 mixture of tonkatsu and hamburger. This Part is about the rules behind it: which concepts generalize from a few examples, what causes what, how surprising each day really is, and what each individual student’s true rate is. The Bento Journal becomes a research instrument. graph LR A[Generalization] --> B[Bayes Nets] B --> C[Conditional<br>Independence] C --> D[Causal<br>Bayes Nets] D --> E[Information<br>Theory] E --> F[Hierarchical<br>Bayes] Chapters Bayesian Generalization Bayesian Networks Conditional Independence and d-Separation Causal Bayes Nets and the Do-Operator Information Theory: Surprise, Uncertainty, and the Collider Hierarchical Bayes This project is generously funded by the Japanese Probabilistic Computing Consortium Association (JPCCA).Hugoen-usThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/structure/generalization/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/structure/generalization/index.htmlBayesian Generalization How do you learn a concept from a handful of examples? You see three numbers that fit a hidden rule — or a few bentos with a golden sticker — and somehow you know which other things fit too. This chapter shows that the same Bayes’ rule you already know becomes a model of human generalization once you make a single shift: a hypothesis is a set. Try it yourself A companion notebook builds the number game and the size principle interactively: 📓 Open in Colab: 07_generalization.ipynbhttps://josephausterweil.github.io/probintro/structure/bayes-nets/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/structure/bayes-nets/index.htmlDrawing What You Already Know Chibany is sitting at their desk, staring at the histogram from Chapter 5 — the one that started the whole mystery:https://josephausterweil.github.io/probintro/structure/conditional-independence/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/structure/conditional-independence/index.htmlReading Independence Off a Graph In Chapter 8, Chibany learned to draw a model as a graph and run it both forward (sampling) and backward (inferring a hidden cause). But a Bayes net does something subtler than store a distribution compactly: its shape tells you, at a glance, which variables carry information about which. This chapter is about that. We’ll find that a graph has only three basic wiring patterns — chain, fork, and collider — and that knowing how each one behaves lets you answer “does observing $A$ tell me anything about $B$?” without computing a single probability. One of the three patterns behaves backwards from what intuition expects, and that surprise — the collider — turns out to be the engine behind some of the most counterintuitive reasoning in all of probability.https://josephausterweil.github.io/probintro/structure/causal-bayes-nets/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/structure/causal-bayes-nets/index.htmlWhen the Arrows Mean Causes So far, the arrows in our Bayes nets have meant one specific thing: “carries information about.” An arrow $A \to B$ said the two were probabilistically linked, and Chapter 9 taught us to read exactly which variables inform which off the graph’s shape. But the arrows never claimed anything about causation. A fork $A \leftarrow B \to C$ makes $A$ and $C$ dependent without either one causing the other.https://josephausterweil.github.io/probintro/structure/information-theory/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/structure/information-theory/index.htmlMeasuring Surprise We’ve reached the last chapter of the Bayes-net spine, and it ties everything together with a single idea: information. So far we’ve talked about whether one variable tells you something about another — dependence, independence, d-separation. Information theory lets us say how much, in a precise, additive unit: the bit. Chibany has been keeping a journal of which bento they get each day. Some days they predict tonkatsu and they’re right; some days they’re caught off guard. Today they were sure it would be tonkatsu — and got a hamburger.https://josephausterweil.github.io/probintro/structure/hierarchical-bayes/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/structure/hierarchical-bayes/index.htmlTwo extremes that both feel wrong Chibany has been keeping a journal: for each student who brings them a bento, they record whether it was tonkatsu or hamburger. After a while the journal looks like this — each student with a tonkatsu count $k_i$ out of their total bento count $n_i$: Student Tonkatsu $k_i$ Total $n_i$ Raw fraction $k_i / n_i$ Alyssa 70 100 0.70 Ben 28 40 0.70 Carmen 6 10 0.60 Diego 3 5 0.60 Emi 2 2 1.00 Farid 0 1 0.00 Chibany wants, for each student, a believable estimate of $\theta_i$ — that student’s underlying probability of bringing tonkatsu. Two obvious strategies both fail: