I’d like some feedback on a possible article or series.
I am thinking about writing and/or recording videos on the measure theoretic foundations of probability.
The idea is: empirical probability (probabilities of coin flips, dice rolls, and finite sequences) is fairly well taught and approachable. However, theoretical probability (the type of measure theory needed to prove theorems or justify calculations on real numbers and infinite sequences) is, in my opinion, not well taught outside of specialist courses. I’d be interested in working through the intuition of measure theory based probability, lightly touching on history, and using key counter-examples. An issue is: this treatment only increases one’s ability to read and reason, not one’s ability to calculate or build models.
Some points I would consider are:
- The main style and emphasis differences between integral calculus and measure theory.
- Why probability is fundamentally a measure on sets, not of elements.
- Why we can’t use a theory that assumes all sets are measurable.
- Why we have to accept a theory where sometimes probability zero events happen.
- Why we assume countable additivity, instead of finite additivity.
- Why we can assume a uniform distribution on an interval of real numbers, but not of an interval of rational numbers.
- Why we can’t reliably reason about conditional probabilities of the form
P[A | B]when
P[B]is zero. This is a remaining flaw in the current theory.
A summary is: modern probability theory has the above non-intuitive features, but when shown with properly worked examples we can show these are small compromises compared to what other axiom systems give us.
One problem for this in teaching is: materials that span the introduction of the Kolmogorov axioms (such as the books of de Finetti, von Mises, and maybe even Savage) de-emphasize the counterexamples that chased the world into accepting the Kolmogorov axioms. I think if we treated these issue directly the trade-offs in working with modern measure theoretic probability would look a bit more favorable.
What are people’s opinions? Of interest? Not of interest? Work as articles? Or work as video lectures?
Data Scientist and trainer at Win Vector LLC. One of the authors of Practical Data Science with R.