Using Odds instead of Probabilities with Bayes' rule

Grant Sanderson (3Blue1Brown) recently released a video on Bayes' rule that advocates for talking in terms of odds rather than in terms of probabilities, simply because that will make it much easier to quickly estimate the Bayes factor of your prior.

The idea is that instead of using the standard formulation that you'll find in the EA concepts map, you can instead reduce the calculation to a simple multiplication of the prior odds by the Bayes factor (i.e., sensitivity / false positive rate). Since this is just straight multiplication, you can do quick estimates in your head that would have taken actual calculation had you instead used probabilities.

Instead of: $P\left(H|E\right)=\frac{P\left(H\right)\times P\left(E|H\right)}{\left(P\right(E|H)\times P(H\left)\right)+\left(P\right(E|\neg H)\ast P(\neg H)}$

Sanderson proposes: $O\left(H|E\right)=O\left(H\right)\times P\left(E|H\right)P\left(E|\neg H\right)$

Note that "odds" appear in the form $1:2$ instead of the equivalent probability  $1/3$.

This is not a new proposal. For example, Arbital's Bayes' rule: Odds form mentions that it is the more convenient form of Bayes' rule, and Gregory Lewis points out that there are especially good reasons for Bayesians to prefer odds to probabilities. Nevertheless, 3Blue1Brown's video really hit home to me as to just how much dramatically easier it is to use odds instead of probabilities when using Bayes' rule — and especially how much less prone it is to misunderstandings.

Beyond just making calculations easier, there's something I really like about attaching a number to test accuracy that doesn't even look like a probability. If you hear that a test has a 9% false positive rate, that's just such a disastrously ambiguous phrase. It's so easy to misinterpret it to mean there's a 9% chance that your positive test result is false.

But imagine if instead the number that we heard tacked on to test results was that the Bayes factor for a positive test is 10. There's no room to confuse that for your probability of having the disease.

— Grant Sanderson

While this may not be news to others, I found this to be a worthwhile approach to how we can talk about Bayes' rule in a way that will reduce misunderstandings with others. Most EAs who mention Bayes' rule on the EA Forum continue to speak in terms of probabilities, not odds (paradoxically even in cases where they use the term "Bayes factor") so I think that there are still others in the EA community that may also find 3Blue1Brown's video helpful.

Note that the video is aimed at an audience that isn't already familiar with Bayes' rule. Depending on your existing knowledge, you may find it useful to skip ahead to 11:01 to hear specifically about how using odds rather than probabilities can make Bayes' rule easier and less prone to misunderstandings.

I will also suggest, if the community finds it helpful, that this video be added to the Bayes' rule EA concepts map entry.