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)

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.

Open Thread: June 2010

by Morendil 1 min read1st Jun 2010663 comments

5


To whom it may concern:

This thread is for the discussion of Less Wrong topics that have not appeared in recent posts. If a discussion gets unwieldy, celebrate by turning it into a top-level post.

(After the critical success of part II, and the strong box office sales of part III in spite of mixed reviews, will part IV finally see the June Open Thread jump the shark?)