https://josephausterweil.github.io/probintro/sampling/index.htmlThe question you can’t sum. Chibany wants an answer about a whole semester of bentos — and for the first time, no formula gives it. This Part’s move: when you can’t compute, wander and count. Chibany wanders the campus (Markov chains), the campus web (random walks and PageRank), their own memory (search as a censored walk), and finally learns to answer any question by sampling — Monte Carlo, particle filtering, and MCMC — closing with the question of whether minds sample too.Hugoen-usThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/sampling/markov-chains/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/sampling/markov-chains/index.htmlA Habit That Only Remembers Yesterday In Chapter 8 we learned to draw what we already know as a graph: nodes for variables, arrows for “depends on,” and a Markov factorization that let us read a joint distribution straight off the picture. That word — Markov — is about to come back, but pointed at something new. Until now our graphs were snapshots: a fixed set of variables, all at once. This chapter adds the one ingredient every story eventually needs. Time.https://josephausterweil.github.io/probintro/sampling/random-walks-networks/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/sampling/random-walks-networks/index.htmlWhen the States Are Connected In Chapter 13 a Markov chain was an abstract thing: a handful of states and a transition matrix saying how to hop between them. Chibany’s states were T and H; the three-state example’s were just “1, 2, 3.” But where do the states and their transitions come from? Very often, from a picture of how things are connected — a network. Chibany has been sketching again.https://josephausterweil.github.io/probintro/sampling/memory-search/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/sampling/memory-search/index.htmlThe Walk Inside Your Head The last two chapters built a machine: a Markov chain (Chapter 13), then a random walk on a network (Chapter 14). This chapter argues that the machine is not just a model of card decks and web pages — it is a model of you. Specifically, of how you search your own memory. Try it right now, the way Chibany’s class did. Name as many animals as you can, out loud, for thirty seconds. Go.https://josephausterweil.github.io/probintro/sampling/monte-carlo/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/sampling/monte-carlo/index.htmlA Question You Can’t Sum For three chapters now we have run Markov chains and watched where they settle. In Chapter 15 we ended on a promise: that running a chain to learn about a distribution is an idea with a name — Monte Carlo — and that the next part of the course turns it into a tool. This is that part. Here is the kind of question that motivates it. Chibany eats two bentos a day, all semester, and their weights wander — some days a light onigiri set, some days a heavy katsu-and-rice. Jamal is curious.https://josephausterweil.github.io/probintro/sampling/particle-filtering/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/sampling/particle-filtering/index.htmlWhy this chapter (and a note on the assignment) Chapter 16 estimated a fixed quantity — the average bento, a tail probability — from a batch of samples. But data often arrive one at a time, and the thing you’re estimating moves. This chapter adapts importance sampling to that streaming case. It is not needed for the Monte Carlo assignment, but it is the natural bridge between the Monte Carlo of Chapter 16 and the Markov chain Monte Carlo of Chapter 18 — and it is where sampling becomes a model of how a mind might track a changing world in real time.https://josephausterweil.github.io/probintro/sampling/mcmc/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/sampling/mcmc/index.htmlRunning Chapter 13 Backwards In Chapter 13 we were handed a transition matrix $P$ — Chibany’s tonkatsu/hamburger habit — and asked a question: what is its stationary distribution $\pi$? We answered it two ways, by power iteration (just run the chain) and as the eigenvalue-1 eigenvector, and both landed on the same 70/30. The chain came first; the distribution $\pi$ fell out of it. This chapter reverses the arrow. Jamal: “Last week we started with a chain and found where it settles. But for inference I have the opposite problem. I know the distribution I want — it’s the posterior — I just can’t sample from it.”https://josephausterweil.github.io/probintro/sampling/sampling-the-mind/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/sampling/sampling-the-mind/index.htmlA note on this chapter and the assignment This chapter teaches the full method — a working Gibbs-plus-Metropolis sampler for a hierarchical model — but on a different application (Chibany rating bento shops) and with different data and a different derivation than the Monte Carlo assignment. That is deliberate: you should finish here able to build such a sampler, then meet a fresh version of the problem on the assignment, so you are exercising the skill rather than copying an answer. Where the assignment goes somewhere this chapter does not, we’ll say so plainly.