Probability
July 29, 2011 — January 18, 2025
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A placeholder where I mention some things I occasionally refer people to or have been referred to, but forget the reference for.
1 What even is probability theory about?
It’s about models for regularities in how events occur. In Bayesian terms, probability is about how we should update our beliefs about the world when we see new things happen, given some model for how things happen. In frequentist terms, it’s about the distribution of outcomes of experiments with some kind of regularity of distribution (usually conditional dependence). There are other models too, but I know them less well.
Neither of those is “directly” “about” the world, but it seems we can do very well by using either of them as a model for the world, so maybe they are pretty good models for the world. This gap between the world and this model of it 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 that I use in day-to-day life.
BTW 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
Founded on conditional probability (Mečíř 2020; Taraldsen 2019) which “feels” more natural to me, but YMMV.
4 Cox probability
Also founded on conditional probability, but more Bayesey I think. See Cox probability