TL;DR: A stand-alone HTML file contains an interactive app that implements the 3Blue1Brown demonstration of Bayes’ Theorem as the geometry of changing beliefs. This app complements an existing alternative app. One of my favorite YouTube channels is Grant Sanderson’s 3Blue1Brown. The channel’s intellectually inspiring videos have set the standard for visualizing mathematics. Of course, my attention has been drawn in…

TL;DR: Check out Henrik Godmann’s “Probability Playground” with over 5 million simple exercises on probability theory! Every year in September and October, Dora Matzke and I teach a course on Bayesian inference. Part of the course involves mastering Bayes’ theorem and the associated rules of probability — the law of total probability, conditional probability, the odds form and the probability…

It has long been known that the frequentist one-sided p-value from Null-Hypothesis Statistical Testing can be interpreted as a poor person’s Bayesian posterior probability. Specifically, the one-sided p-value is usually a good approximation to the posterior probability that the effect is in the direction opposite to that stipulated by the alternative hypothesis (e.g., Casella & Berger, 1987; Jeffreys, 1939; Marsman…

The LLM revolution fundamentally changes how many researchers do their work. The disruption extends even to mathematics, the purest science of them all (xkcd, “purity”, #435): Armed with an LLM, even researchers without much mathematical training can quickly solve complex mathematical problems. Now that sophisticated mathematical assistance has become so readily available, does it still make sense for students to…

A few years ago, my colleagues and I decided to test the Diaconis-Holmes-Montgomery (DHM) hypothesis that a fair coin, when flipped in the air and caught in the hand, tends to land on its starting side slightly more often than 50% (Diaconis, Holmes, & Montgomery, 2007). In fact, DHM suggested that the effect would be about 1%, and they indicated…

Before proceeding, I should say something about Richard Feynman. In his physics work, Feyman was clearly a genius. And as a public persona, Feynman was certainly also charismatic (see for instance his interviews on YouTube). Moreover, Feynman helped popularize science by writing a series of books meant for a general readership. However, the available evidence suggests that Feynman had some…

LT;DR In the current era of AI, there is no longer any excuse for manuscripts that contain typographical errors, grammatical errors, unclear formulations, errors in equations, incorrect references, missing references, and a demonstrated lack of awareness for related work. A recent example of my own writing demonstrates that an AI (i.e., ChatGPT Pro) can offer high-quality feedback. Incorporating two rounds…

Recently I was hunting for an epigraph to start a manuscript on the topic of forgetting. Claude helpfully  suggested several options including one from 1821: “For a more literary and philosophical register, Thomas De Quincey offers something that resonates with the persistence of memory traces: “There is no such thing as ultimate forgetting; traces once impressed upon the memory are…