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.
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.
But that’s not what I’m talking about. I was specifically responding to your claim that:
“prior probability”, by definition, means that we throw out all previous evidence.
So far as I can tell, that’s not part of the accepted definition. For example, Jaynes’ work on prior probabilities explicitly invokes prior information:
in two problems where we have the same prior information, we should assign the same prior probabilities.
I don’t mean to come off as a dick for nit-picking about definitions. But rigorous mathematical definitions are really important, especially if you are claiming to argue something is true by definition—and you were.
Yes. This name for this is Cromwell’s rule.
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.
But that’s not what I’m talking about. I was specifically responding to your claim that:
So far as I can tell, that’s not part of the accepted definition. For example, Jaynes’ work on prior probabilities explicitly invokes prior information:
I don’t mean to come off as a dick for nit-picking about definitions. But rigorous mathematical definitions are really important, especially if you are claiming to argue something is true by definition—and you were.
Yes, I was wrong. I was explaining why I got so focused on the blank-slate version of the prior.
Oh, gotcha.