Comment by mitthrowaway2
4 hours ago
But again this seems like a TypeError.
If you're asking me for my forecast and my confidence in my forecast, I would say this:
Forecast: It will rain tomorrow. Confidence: 0.7
If you say "No no, I mean what is your confidence in the 0.7 number" then I have no idea what you're talking about. 0.7 is my confidence. It's not valid to attach a confidence to that number. When the next day comes, we simply find out whether it rains or does not rain, and whoever put the highest probability on the correct outcome wins.
If you're doing this with intermediate steps then I have to ask: where did the 0.7 probability come from as an intermediate step, and what is the epistemic meaning of the 0.8 attached to it? If you only felt 80% sure that you were 70% sure that it would rain tomorrow, then what is the proposition that you are 70% confident about?
Addendum: The closest thing I can think of that matches the GP's description is conditional probabilities.
For example, if it rains tonight then I am 70% sure it will rain tomorrow too. I am also 80% sure it will rain tonight. And I am 100% sure that if it doesn't rain tonight, it also will not rain tomorrow. Then I can chain these together: The chance of rain tomorrow is P(Rain tomorrow) = P(Rain tomorrow | rain tonight) * P(Rain tonight) = 0.8*0.7 = 0.56.
You can assign conditional probabilites to the correctness of your model itself. Then the phrasing is more like "I am 70% confident that it will rain tomorrow, conditional on my understanding of meteorology being correct" and "I am 80% sure that I understand meteorology correctly". Then you also need to add in a term for P(Rain | not understanding correctly).