…we can set P(h/b) equal to any probability we like by choosing P(h) appropriately.

Not in any useful way. The only way you can set P(h|b) to .999… for example (and still respect background evidence) is if you define h so vaguely that it is predictive of nothing (e.g. h = “Hercules is a name in some book somewhere” vs. “Hercules existed as a real historical person and conquered the Peloponnesus in the year 1803 B.C.E.”). All non-trivial theories cannot be dinked this way. So I’m not sure what you are talking about here.

Bayesian reasoning actually solves Humean induction by removing the necessity of time from the equation. The laws of probability hold for sets of evidence even atemporally, therefore one never has to infer from past to future to get the same results, and since the same laws apply across both axes (space and time), Hume’s problem is avoided as long as you only make proper statistical statements about the future and not statements of objective certainty.

Of course, you could walk this all the way back to Cartesian skepticism (“maybe we’ve all been deluded about everything!”) but that’s a defect of all epistemologies, and therefore is not any greater a defect of Bayesian epistemology.

As for the setting of priors to 50/50 at zero empirical knowledge for simple binary hypotheses (hence you are correct for non-binary problems ramification becomes more problematic, but “x exists” is almost always binary), if subjective Bayesians deny this as you claim (I’m not sure they do, you may be over-generalizing from certain cases to all, e.g. non-binary to binary hypotheses), then they are refuted on this point by the arguments in Proving History, pp. 83-88, 110-14. Be aware that we can only be talking about artificial a priori priors, i.e. we analytically create their a priori status by subtracting information (thus we can manipulate the information however we need to develop them). It’s not like there’s any such thing as an actual a priori prior (maybe for Hal 9000 the second he is first turned on, but that’s certainly not us).

Anyway, I am fairly certain none of this was what Antony was thinking.

But it is also a problem if the evidence is itself in doubt – the standard bayesian methods do not allow for conditionalization on uncertain evidence (Jeffrey conditionalization allows this, but isn’t widely accepted). With historical claims, I imagine there are lots of cases where different experts disagree on what counts as evidence in the first place, and the standard bayesian machinery has a hard time dealing with this.

I treat this issue several times in Proving History as well, most pertinently I usually exclude all significantly uncertain existential claims from being evidence, classifying them as hypotheses instead, unless we can condition on the uncertainty, e.g. we can calculate “P conditional on e having a 2% chance of being true,” if we really wanted to. In probability theory that’s not difficult at all. The math is just tedious. I advise historians to avoid that and only put in e and b what all parties can agree should go there (all parties, that is, who agree to certain axioms defined in chapter two, and actually abide by them…thus lunatics can be excluded from “all,” for example, as can irrational dogmatists, liars, and so on; in practice, it amounts to determining what the Bayesian conclusion is within a population that is rational, neither insane nor unyieldingly dogmatic, and sincerely committed to certain basic axioms that non-controversially define history as a knowledge-seeking profession, since we don’t care what the conclusion is in other populations, e.g. liars or lunatics).

if e is a part of the background b, then P(e/b) is one, and P(h/e.b) reduces to P(h/b), and evidence e no longer ‘confirms’ hypothesis h.

There seems to be possible confusion here. P(e|b) = [P(h|b) x P(e|h)] + [P(~h|b) x P(e|~h)] and thus is always conditional on h and ~h. So it can never be 1 simply by being subsumed under b, because P(e|b) isn’t the probability that the evidence exists, it’s the probability that it would be produced by h and by ~h. Those latter probabilities never change. And the effect of them is represented in the revised prior, hence the probabilities are calculated into the effect e has on the prior now that it’s in b.

Your subsequent comment (which I addressed above) suggests that you are here confusing the term P(e|b) in BT with what I identify as P(e|o) in my article. Which gets us back to exactly what’s wrong with Antony’s objection. If, again, that’s what she was saying.

But yes, regarding the moon thing, you’re spot on. Except it wouldn’t be 50/50 if we had none of that evidence, as if every other object we encountered in the universe is made of cheese. But then, that effect of background evidence is more generally your point.

You would falsify BI by falsifying BT by using the same mathematics that proved BT, i.e. the theorem could have been proved false (or unprovable a la Gödel), but instead it just so happens Bayes’ Theorem was proven formally valid. So anyone who accepts any common axiomatic mathematics must accept BT. That could still turn out to be a mistake, but it’s extremely unlikely at this point (that all logical certainties could turn out only mistakenly so is a point I make in Proving History, pp. 24-25, and here that point entails falsifiability).

(Have you donated to the atheist hurricane relief fund like my second paragraph asks?)

This comes out in a special way on the comments section of his blog. Like you, Richard, he vets posts before making them public. Unlike you, he doesn’t post critical comments–unless he has something biting and witty to reply with. I once posted correcting his assessment of your credentials. I was brief, and courteous. It didn’t make it through his filter. That pretty much lost him my respect, right there.

Ted Bunn also picked up on the story, and offered perhaps the best response to anyone who thinks this way: challenge them to a game of poker, and insist that they stick to their principles. Either they have been dealt a winning hand, or they have not. It has already happened so no amount reasoning can allow a probability to be assigned. In which case it makes no difference whether or not they look at their cards, so they should play blind.

So you could hypothetically start at a state of zero empirical knowledge, where b contains only a priori knowledge (e.g. logic and mathematics) and your priors are therefore 50/50 (when there is literally no evidence yet to favor h over ~h, or vice versa, then that logically entails P(h|b) = P(~h|b) = 0.5), and then start adding evidence into e one bit at a time

If someone does not want to believe you, you could point out that this principle is actually widely used in robotics and machine learning. It is kind of hard to argue against real world technical applications.

An example: You want to do robot localization. Your robot can sense features of the environment and can move. Immediately after the robot is turned on, it knows nothing about the environment (all locations have the same probability). This are the priors. The robot then gets the first sensory input and computes the posterior probabilities. The process can be repeated using the posteriors as new priors and the algorithm converges pretty quickly – regardless of the priors unless you set the initial priors to zero.

Good priors obviously help to speed up the convergence but don’t really matter. If someone is interested I can recommend the book: “Probabilistic Robotics” that addresses the technical side of this.