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.)
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.)