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Naso (Oct 24 2022 at 14:08):I've seen people say that probability is dual to statistics.
Example from Quora:
"Suppose that you have a bag of red and blue marbles, and you reach in and grab some. In probability, you want to answer questions about what’s likely to be in your hand given the contents of the bag. In statistics, you want to answer questions about what’s likely to be in the bag given the contents of your hand. "
Is there a way to see this as a formal/categorical duality?
Tobias Fritz (Oct 24 2022 at 17:06):That's a really cool way to put it! At least in the example that you're quoting, the "duality" is known as Bayesian inversion: simply apply Bayes theorem in order to determine a conditional probability of the form from .
In categorical terms, this duality can indeed be formalized as a type of categorical duality, namely a dagger. I showed this in categorical probability quite generally as Remark 13.10 in this paper. (I only gave a sketch of the proof there, but will be happy to answer questions.)