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?)
In 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; not an expert's opinion.)