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In the context of Bayes's Theorem, Priors refer generically to the beliefs an agent holds regarding a fact, hypothesis or consequence, before being presented with evidence. More technically, in order for this agent to calculate a posterior probability using Bayes's Theorem, this referred prior probability and likelihood distribution are needed.

As a concrete example, suppose you had a barrel containing some number of red and white balls. You start with the belief that each ball was independently assigned red color (vs. white color) at some fixed probability. Furthermore, you start out ignorant of this fixed probability (the parameter could be anywhere between 0 and 1). Each red ball you see then makes it more likely that the next ball will be red (following a Laplacian Rule of Sucession).

On the other hand, if you start out with the prior belief that the barrel contains exactly 10 red balls and 10 white balls, then each red ball you see makes it less likely that the next ball will be red (because there are fewer red balls remaining)....

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