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Cyan comments on What's the Value of Information? - Less Wrong
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OK, so this is as good a place as any to whinge about my pet peeve.
You can't shift your prior! You can update your probability, after which it's your posterior probability. The terms "prior" and "posterior" are only defined relative to some piece of evidence. Of course, the posterior relative to one piece of evidence can be the prior relative to the next, but usually people are not talking about this sort of sequential setup.
Nothing personal, johnlawrenceaspden. It's typical for folks around here to write things like "update my prior" when they really mean "update my probability", and it's like nails on a chalkboard every time for me .
Agreed that it is loose talk. I think the reason is that the posterior becomes the prior for the next inference, so you can think of your beliefs sloshing around and changing in response to information. After all, even the very first prior will likely have come from somewhere, and be the posterior of some other process.
Aren't they (at least implicitly)?
Sure, I suppose. But usually there's only one piece of evidence being discussed explicitly, and I think it makes little sense to use the word "prior" to refer to the probability that results from updating on it.