Difference between revisions of "Priors"

In the Bayes's Theorem context, Priors refers 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 between 0 and 1. Furthermore, you start out ignorant of this fixed probability (the parameter could 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.