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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).

Thus our prior can affect how we interpret the evidence. The first prior is an inductive prior - things that happened before are predicted to happen again with greater probability. The second prior is anti-inductive - the more red balls we see, the fewer we expect to see in the future.

In both cases, you started out believing something about the barrel - presumably because someone else told you, or because you saw it with your own eyes. But then their words, or even your own eyesight, was evidence, and you must have had prior beliefs about probabilities and likelihoods in order to interpret the evidence. So it seems that an ideal Bayesian would need some sort of inductive prior at the very moment they were born. Where an ideal Bayesian would get this prior, has occasionally been a matter of considerable controversy in the philosophy of probability.

A real life example may come in hand to better understand the consequences of the priors on the reasoning about any subject. Consider two leaders from different political parties - each one has his own beliefs about social organization and the roles of people and government in the society. This differences can be attributed to a wide range of factors, from genetic variability to education influence in their personalities and condition the politics and laws they want to implement. However, both leaders - and, most importantly, the voters - should note that neither has reason to belief his reasoning is better than the other, unless he can demonstrate that his priors lead to a better political model and improvements in the society.

Prior probability

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This specific term usually refers to a prior already based on considerable evidence - for example when we estimate the number of red balls after doing 100 similar experiments or hearing about how the box was created.

As a complementary example, suppose there are a hundred boxes, one of which contains a diamond - and this is all you know about the boxes. Then your prior probability that a box contains a diamond is 1%, or prior odds of 1:99.

Later you may run a diamond-detector over a box, which is 88% likely to beep when a box contains a diamond, and 8% likely to beep (false positive) when a box doesn't contain a diamond. If the detector beeps, then represents the introduction of evidence, with a likelihood ratio of 11:1 in favor of a diamond, which sends the prior odds of 1:99 to posterior odds of 11:99 = 1:9. But if someone asks you "What was your prior probability?" you would still say "My prior probability was 1%, but I saw evidence which raised the posterior probability to 10%."

Your prior probability in this case was actually a prior belief based on a certain amount of information - i.e., someone told you that one out of a hundred boxes contained a diamond. Indeed, someone told you how the detector worked - what sort of evidence a beep represented. In conclusion, the term prior probability is more likely to refer to a single summary judgment of some variable's prior probability, versus the above Bayesian's general notion of priors.

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