Our ability to talk past each other is impressive :)
would have been an easier way to illustrate your point). I think this is actually the assumption you’re making. [Which is a horrible assumption, because if it were true, you would already be perfectly calibrated].
Yes this is almost the assumption I am making, the general point of this post is to assume that all your predictions follow a Normal distribution, with μ as “guessed” and with a σ that is different from what you guessed, and then use X2 to get a point estimate for the counterfactual σ you should have used. And as you point out if (counterfactual) σ=1 then the point estimate suggests you are well calibrated.
Our ability to talk past each other is impressive :)
Yes this is almost the assumption I am making, the general point of this post is to assume that all your predictions follow a Normal distribution, with μ as “guessed” and with a σ that is different from what you guessed, and then use X2 to get a point estimate for the counterfactual σ you should have used. And as you point out if (counterfactual) σ=1 then the point estimate suggests you are well calibrated.
In the post counter factual σ is ^σz