https://josephausterweil.github.io/probintro/learning/index.htmlThe Mystery of the Two Peaks. Early summer, and something has changed: this semester’s bentos arrive wrapped in furoshiki — opaque. Chibany, refusing to peek, starts weighing them, and the weights refuse to behave: the average is 441 g, yet no bento ever weighs 441 g. Solving that mystery takes this whole Part: continuous quantities need densities instead of counts, bumps need Gaussians, learning needs priors that become posteriors, and the two peaks themselves need mixture models.Hugoen-usThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/learning/mystery-bentos/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/learning/mystery-bentos/index.htmlThe Weight of the Matter It’s a new semester at the university, and Chibany’s students have been bringing them bentos again. But this time, something is different. Last semester, the bento boxes were transparent. So Chibany could see whether there was tonkatsu or hamburger inside. But this semester, the bento boxes are opaque. Thus, Chibany can’t see what is inside when they receive the bento. They want to know IMMEDIATELY what the meal is, but it would be extremely rude to open the bento box as the students are watching. Thankfully, Chibany is curious, and they’re a probabilist.https://josephausterweil.github.io/probintro/learning/continuous/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/learning/continuous/index.htmlFrom Counting to Measuring Chibany stares at their histogram. They understand expected value now. The 455g average makes sense as a mixture of 500g tonkatsu and 350g hamburger bentos. But something still bothers them. Look at these actual measurements from their first week: Monday: 520g (tonkatsu) Tuesday: 348g (hamburger) Wednesday: 505g (tonkatsu) Thursday: 362g (hamburger) Friday: 488g (tonkatsu) The weights aren’t exactly 500g and 350g! They vary. And here’s the deeper question: What’s the probability that a bento weighs exactly 505.000000… grams?https://josephausterweil.github.io/probintro/learning/gaussian/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/learning/gaussian/index.htmlThe Bell Curve After learning about the uniform distribution in Chapter 2, Chibany realizes something: real measurements rarely spread evenly across a range. When they measure 1000 tonkatsu bentos carefully, the weights don’t spread uniformly between 495g and 505g. Instead, most cluster near 500g, with fewer and fewer measurements appearing as you move away from that center value. This pattern appears everywhere in nature: Heights of people Measurement errors Test scores Daily temperatures And yes, bento weights! This is the Gaussian distribution (also called the Normal distribution), and it’s arguably the most important probability distribution in statistics.https://josephausterweil.github.io/probintro/learning/bayesian-learning/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/learning/bayesian-learning/index.htmlThe Learning Problem Chibany has a new challenge. They receive shipments from a new supplier, but don’t know the mean weight of their tonkatsu bentos. They believe the supplier is trying to hit 500g (like their usual supplier), but they’re not certain. Maybe the supplier aims for 495g? Or 505g? The question: How can they learn the true mean weight from observations? This is Bayesian learning: starting with a prior belief, observing data, and updating to a posterior belief.https://josephausterweil.github.io/probintro/learning/mixture-models/index.htmlThu, 02 Jul 2026 00:00:00 +0000https://josephausterweil.github.io/probintro/learning/mixture-models/index.htmlReturning to the Mystery Remember Chibany’s original puzzle from Chapter 1? They had mystery bentos with two peaks in their weight distribution, but the average fell in a valley where no individual bento existed. We now have all the tools to solve this completely: Chapter 1: Expected value paradox in mixtures Chapter 2: Continuous probability (PDFs, CDFs) Chapter 3: Gaussian distributions Chapter 4: Bayesian learning for parameters Now we combine them: What if we have multiple Gaussian distributions mixed together, and we need to figure out both which component each observation belongs to AND the parameters of each component?