Wait, what? Bayesians never assign 0 probability to anything, because it means the probability will always remain 0 regardless of future updates.
Yes. This name for this is Cromwell's rule.
And "prior probability", by definition, means that we throw out all previous evidence.
Not quite. The prior probability is the probability of the hypothesis and the background information, independent from the evidence we are updating on. This includes previous evidence. We usually write the "prior probability" as P(H), but it should really be written as P(H.B), where "H" is hypothesis and "B" is background information.
For example, let's say I am asking you to update your belief that Julius Caesar existed given a recently discovered, apparently first-hand account of Caesar's crossing the Rubicon. Your prior probability should NOT exclude all previous evidence on whether Caesar actually existed - e.g. official Roman documents and coins with his face. Ideally, your prior probability should be your posterior probability from your most recent update.
Right. What I want to do is calculate the probability that a random conscious entity would find itself living in a world where someone satisfying the definition of Julius Caesar had existed. And then calculate the conditional probability given the evidence, which is everything I've ever observed about the world including the newly discovered account.
Obviously that's not what you do in real life, but the point remains that everything after the original prior (based on Kolmogorov complexity or something) is just conditioning. If we're going to talk about how and why we should formulate priors, rather than what Bayes' rule says, this is what we're interested in.
Recently I've been struck with a belief in Aliens being present on this Earth. It happened after I watched this documenary (and subsequently several others). My feeling of belief is not particular interesting in itself - I could be lunatic or otherwise psychological dysfunctional. What I'm interested in knowing is to what extend other people, who consider themselves rationalists, feel belief in the existence of aliens on this earth, after watching this documentary. Is anyone willing to try and watch it and then report back?
Another question arising in this matter is how to treat evidence of extraordinary things. Should one require 'extraordinary evidence for extraordinary claims'? I somehow feel that this notion is misguided - it discriminates evidence prior to observation. That is not the right time to start discriminating. At most we should ascribe a prior probability of zero and then do some Bayesian updating to get a posterior. Hmm, if no one has seen a black swan and some bayesian thinking person then sees a black swan a) in the distance or b) up front, what will his a posterior probability of the existence of black swans then be?