## LESSWRONGLW

Do you (or anyone else reading this) know of any attempts to give a precise non-frequentist interpretation of the exact numerical values of Bayesian probabilities? What I mean is someone trying to give a precise meaning to the claim that the "degree of plausibility" of a hypothesis (or prediction or whatever) is, say, 0.98, which wouldn't boil down to the frequentist observation that relative to some reference class, it would be right 98/100 of the time, as in the above quoted example.

Or to put it in a way that might perhaps be clearer, suppose ... (read more)

Have you seen my What Are Probabilities, Anyway? post?

0Morendil10yIn the Bayesian interpretation, the numerical value of a probability is derived via considerations such as the principle of indifference - if I know nothing more about propositon A than I know about proposition B, then I hold both equally probable. (So, if all I know about a coin is that it is a biased coin, without knowing how it is biased, I still hold Heads or Tails equally probable outcomes of the next coin flip.) If we do know something more about A or B, then we can apply formulae such as the sum rule or product rule, or Bayes' rule which is derived from them, to obtain a "posterior probability" based on our initial estimation (or "prior probability"). (In the coin example, I would be able to take into account any number of coin flips as evidence, but I would first need to specify through such a prior probability what I take "a biased coin" to mean in terms of probability; whereas a frequentist approach relies only on flipping the coin enough times to reach a given degree of confidence.) (Note, this is my understanding based on having partially read through precisely one text - Jaynes' Probability Theory - on top of some Web browsing [http://en.wikipedia.org/wiki/Checking_whether_a_coin_is_fair]; not an expert's opinion.)
0JoshuaZ10yYes, you can do this precisely with measure theory, but some will argue that that is nice math but not a philosophically satisfying approach. Edit: A more concrete approach is to just think about it as what bets you should make about possible outcomes.