Probability
2011-07-29 — 2025-11-03
Wherein an account is presented of probability as models for event regularities, founded on conditional probability and Rényi and Cox formulations, and an introductory animated lecture is noted.
A placeholder where I mention things I occasionally refer people to, or that others have referred to me, but whose exact reference I sometimes forget.
1 What even is probability theory about?
It’s about models for regularities in how events occur. In Bayesian terms, probability tells us how to update our beliefs about the world when we observe new events, given a model of how things happen. In frequentist terms, it’s about the distribution of outcomes from experiments that exhibit some regularity (usually conditional dependence). There are other models too, but I know them less well.
Neither of those is “directly” “about” the world, but we can do very well by using either as a model, so maybe they’re pretty good models. That gap between the world and our models leads some to claim that probability does not “exist” (Nau 2001; Spiegelhalter 2024), which is technically true under a sufficiently restrictive definition of “exist”, but not one I use in day-to-day life.
By the way, I wrote and animated an introductory lecture about this way of explaining probability.
2 Probability hacks
Terry Tao, and his Neat Probability Hacks for Dummies Like Me
Dominic Yeo
3 Rényi axioms
A different formulation based on the algebra of conditional probability. See Rényi probability; c.f. Cox probability.
4 Cox probability
…is also founded on conditional probability like the Rényi flavour, but it’s more subjectivist AFAICT. See Cox probability.
5 Categorical probability
See Jacobs (2025).