The point that odds ratios are a more intuitive view of Bayes’ rule is not remotely original, but I thought I could improve on the existing explanations I’ve found.
The usual formulation: P(Hypothesis|Event) = P(E|H)*P(H)/P(E)
The odds ratio formulation: P(Hypothesis|Event) : P(Counterhypothesis|Event) :: P(E|H)*P(H) : P(E|C)*P(C).
Say that your acquaintance who usually comes to your events declines an invitation to dinner, saying they’re booked that night. You made a great fool of yourself around them last week, so you wonder if they’re free but are pretending not to be. Due to their repulsion.
You know the friend is busy every third night on average. This provides your prior belief – 1/3 chance that they have a real conflict, 2/3 chance that they’re free. Now you apply the knowledge to generate your new belief about whether they had a real conflict or not.
- Prior belief they had a real calendar conflict: 0.33
- How likely is it they’d decline if they had a conflict? Very likely. But not 100%. You’ve known them to ditch events when better opportunities came up. Dinner at my house is a middling event, and they’ve probably cooled on me, so there’s only a 4% chance they’d cancel preexisting plans for it.
So, in the 33% of universes (gray bar) where they have a conflict, 96% of the universes would look like the one I’m in now (blue bar). Which makes for a universe-share of 0.33 x 0.96 = 0.32.
- Prior belief they did not have a conflict: 0.67
- How likely is it they’d decline even if they didn’t have a conflict? Maybe 0.3. I was a great honking fool last week.
So, in the 67% of the universes where they didn’t have a conflict (chartreuse bar), 30% of those universes would look like the one I’m in now (orange bar). Which makes for a universe-share of 0.67 x 0.3 = 0.2.
You know you don’t live in the universes where they accepted the invite, so it’s a question of whether you’re in one of the 32 blue universes or in one of the 20 orange universes.
I find this slightly nicer than trying to jump straight to the figure 20/(20+32), which Bayes’ theorem is designed to do.