# Priors

A Bayesian uses Bayes's Theorem to update beliefs based on the evidence. This requires that, even in advance of seeing the evidence, you have beliefs about what the evidence means - how likely you are to see the evidence, *if* various hypotheses are true - and how likely those hypotheses were, in *advance* of seeing the evidence. To calculate a posterior probability using Bayes's Theorem, you need a prior probability and likelihood distribution.

Suppose you had a barrel containing some number of red and white balls. If you start with the belief that each ball was independently assigned red color (vs. white color) at some fixed probability between 0 and 1, and you start out ignorant of this fixed probability (the parameter could anywhere between 0 and 1), then each red ball you see makes it *more* likely that the next ball will be red. (By Laplace's Rule of Succession.)

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; and where an ideal Bayesian would get this prior, has occasionally been a matter of considerable controversy in the philosophy of probability.

## Prior probability

This phrase usually refers to a point estimate already based on considerable evidence - for example when we estimate the number of women who start out with breast cancer at age 40, in advance of performing any mammographies.

The probability that you start with before seeing the evidence. One of the inputs into Bayes's Theorem.

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 this is 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 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. For a more complicated notion of prior beliefs, including prior beliefs about the meaning of observations, see "priors". ("Prior probability" is more likely to refer to a single summary judgment of some variable's prior probability, versus a Bayesian's general "priors".)

## Blog posts

- Priors as Mathematical Objects
- "Inductive Bias"
- Probability is Subjectively Objective
- Bead Jar Guesses by Alicorn - Applied scenario about forming priors.