In Bayesian jargon, people talk about having a "flat prior" when they mean being very uncertain about a question. But more often you just give a probability that you think a question is true. We don't normally draw probability distributions to share with each other.

I'm wondering if there's a natural way to talk not just about the probability you think something is true, but an estimate of your confidence, in some quantitative way? And what would it mean to be "well-calibrated" in your uncertainty?

For example, I might be quite confident that a particular coin is fair and will come up heads 50% of the time (because I've gathered a lot of data), while being much less confident about another 50% bet even though I think it's as likely as not.

Aleatory and epistemic uncertainty often get wrapped up together so these estimates are not always proper probabilities nor measuresof confidence. You're separating them, good for you!