So then this initial probability estimate, 0.5, is not repeat not a “prior”.
1:1 odds seems like it would be a default null prior, especially because one round of Bayes’ Rule updates it immediately to whatever your first likelihood ratio is, kind of like the other mathematical identities. If your priors represent “all the information you already know”, then it seems like you (or someone) must have gotten there through a series of Bayesian inferences, but that series would have to start somewhere, right? If (in the real universe, not the ball & urn universe) priors aren’t determined by some chain of Bayesian inference, but instead by some degree of educated guesses / intuition / dead reckoning, wouldn’t that make the whole process subject to a “garbage in, garbage out” fallacy(?).
For a use case: A, low internal resolution rounded my posterior probability to 0 or 1, and now new evidence is not updating my estimations anymore, or B, I think some garbage crawled into my priors, but I’m not sure where. In either case, I want to take my observations, and rebuild my chain of inferences from the ground up, to figure out where I should be. So… where is the ground? If 1:1 odds is not the null prior, not the Bayesian Identity, then what is?
1:1 odds seems like it would be a default null prior, especially because one round of Bayes’ Rule updates it immediately to whatever your first likelihood ratio is, kind of like the other mathematical identities. If your priors represent “all the information you already know”, then it seems like you (or someone) must have gotten there through a series of Bayesian inferences, but that series would have to start somewhere, right? If (in the real universe, not the ball & urn universe) priors aren’t determined by some chain of Bayesian inference, but instead by some degree of educated guesses / intuition / dead reckoning, wouldn’t that make the whole process subject to a “garbage in, garbage out” fallacy(?).
For a use case: A, low internal resolution rounded my posterior probability to 0 or 1, and now new evidence is not updating my estimations anymore, or B, I think some garbage crawled into my priors, but I’m not sure where. In either case, I want to take my observations, and rebuild my chain of inferences from the ground up, to figure out where I should be. So… where is the ground? If 1:1 odds is not the null prior, not the Bayesian Identity, then what is?