As a mathematical object, a Bayesian "prior" is a probability distribution over sequences of observations. That is, the prior assigns a probability to every possible sequence of observations. In principle, you could then use the prior to compute the probability of any event by summing the probabilities of all observation-sequences in which that event occurs. Formally, the prior is just a giant look-up table. However, an actual Bayesian reasoner wouldn't literally implement a giant look-up table. Nonetheless, the formal definition of a prior is sometimes convenient. For example, if you are uncertain about which distribution to use, you can just use a weighted sum of distributions, which directly gives another distribution.
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This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was Marginally Zero-Sum Efforts, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.
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