It seems like a consistent element of everyone's explanations so far has been that the bayesian has reasonable priors and is not naive. I'll respond to everyone at once here (hope that's okay) to keep things organized. cc @michaeldickens, @cole-wyeth, @gavin-runeblade, @thomas-castriensis (thanks everyone for the explanations so far).
edit: IDK how to mention/tag people :(
One thing I notice, the Bayesian needs a prior for 'unknown scam' but shouldn't they also consider the inverse? Like an unknown bank error in your favor type situation. I suppose that if the prior for that is small enough it doesn't matter so it's ignored as insignificant. Is the significance a matter of the ratio between priors of unknowns? Like it's way more likely to be an unknown scam than an unknown bank error, so we ignore the bank error.
Something bigger doesn't quite sit right with me: I don't think having priors and not being naive is our default state for knowledge frontiers (hence the initial framing and restrictions on knowledge of scams). Rather, the default is that we are missing priors (or they're wildly inaccurate) and we are naive to the ways in which we are naive (there is a lot that we don't know we don't know).
Here's a concrete example: consider a Bayesian in 1901 (before general relativity). Almost all measurements of planetary motion agreed with Newton's universal gravity (UG) and Mercury's orbit was our main (only?) counterexample. We had multiple (incorrect) theories about how Mercury's orbit might be perturbed by this body or that. The most recently discovered planet at the time (Neptune) was predicted via UG (0.125 of all known planets).
My understanding is that the Bayesian will, upon seeing measurements of all the planets' orbits, update based on successful predictions. Most of the predictions are correct for UG alone, and all of them are correct for UG + New Mystery Planet. Let's call these hypotheses
Now, based on the reasoning used (if I understand it correctly), the Bayesian already has reasonable priors (at least they believe so), and the general relativity hypothesis (playing the role of the scam hypothesis) is already accounted for in the Bayesian's calculations and updates. It doesn't matter that the Bayesian is unaware of how it works or not. [Unsure] The probability of GR being true is independent of whether the Bayesian knows about it or not
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. Let's call all the other hypotheses (excluding any UG hypotheses) just
Okay so in case of observing something uncontroversial, like Mars data,
Then, when observing Mercury, we see
Have I gone wrong somewhere? I'm not confident that I'm at the right midstate here. If there isn't a problem with my logic then this result feels intuitively wrong because we're getting more confident in
Also it seems like
that's how it seems it must be to me. I can't quite put my finger on it, but it feels wrong that learning about an idea independently of all observations of data somehow updates things retroactively. ↩︎
Can Bayesianism deal with the mail-order prophet scam? If so, how?
This scam uses a segmented mailing list and a large initial population to advertise predictions so that a 'winning' portion of recipients receive an apparently improbable series of correct predictions. Most recipients (the 'losers') receive an incorrect prediction at some point.
For clarity, here is the situation:
A Bayesian with no prior knowledge of the scam receives a letter advertising stock predictions via a new proprietary quantitative model. The letter predicts that by the end of the month, stock XYZ will be up. It also states that the firm will prove their legitimacy by sending 6 more predictions (one per month) that will correctly predict whether XYZ is up or down. Sure enough, the Bayesian receives each letter, and the predictions all end up correct!
Unbeknownst to the Bayesian, 63 other Bayesians also received the first letter, 31 others received the second, 15 the third, 7 the fourth, 3 the fifth, and 1 other received the sixth letter. Our unlucky Bayesian was the only one to receive the seventh.
The question is, assuming no knowledge of the scam or communication with the "losers" (for whom a stock prediction was wrong), should the Bayesian -- strictly adhering to Bayesian Epistemology -- believe that the firm's stock predictions are legit?
I don't know how to approach this from a Bayesian perspective and would appreciate any guidance from someone who's more familiar. My intuition is that the Bayesian should believe that the predictions are legit (which we observers know to be incorrect).
Also I feel I should disclose that I'm not a Bayesian (rather a Critical Fallibilist), and I'm looking for the best Bayesian answers to this problem (links to prior material welcome). It's often quite hard to find the best answers to a problem from a particular school, so please bear with me if I've missed something obvious.