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passive_fist comments on Am I Understanding Bayes Right? - Less Wrong Discussion
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Probability is never used as a degree of belief. Likelihood is used as a degree of belief. See this thread: http://stats.stackexchange.com/questions/2641/what-is-the-difference-between-likelihood-and-probability
Once you understand likelihood it's easy to see why it represents degree of belief.
In light of the downvotes, I just wanted to explain that probability is frequently used to refer to a degree of belief by LessWrong folks. You're absolutely right that statistical literature will always use "probability" to denote the true frequency of an outcome in the world, but the community finds it a convenient shorthand to allow "probability" to mean a degree of belief.
I haven't seen this shorthand explained anywhere here.
This would be the explanation http://lesswrong.com/lw/oj/probability_is_in_the_mind/ It really should be talked about more explicitly elsewhere though.
I must have missed that thread, thanks. Though I can't see why I'm wrong. It has nothing to do with frequentism vs. bayesianism (I'm a bayesian). It's simply that likelihood is relative to a model, whereas probability is not relative to anything (or, alternatively, is relative to everything), as they're saying in that thread. Through this interpretation it's easy to see why likelihood represents a degree of belief.
Likelihood is the probability of the data given the model, not the probability of the model given the data. A likelihood function gives you a number between 0 and 1 for every model, but that number does not mean anything like "how certain is it that this model is true".
Probability (for a Bayesian) is relative to a prior. There is always a prior: P(A|B) is the fundamental concept, not P(A). See, for example, Jaynes, chapter 1, pp.112ff., which is the point where he begins to construct a calculus for reasoning about "plausibilities", and eventually, in chapter 2, derives their measurement by numbers in the range 0-1.
This is true, and I can see why it could create some conflict in interpreting this question. Thanks.
It's a quirk of the community, not an actual mistake on your part. LessWrong defines probability as Y, the statistics community defines probability as X. I would recommend lobbying the larger community to a use of the words consistent with the statistical definitions but shrug...
Okay, that clears up it up a lot.
Probability is always used as degree of belief by Bayesians (and we all seem to be Bayesians here). For example. Frequentists take probabilities to be long-run relative frequencies. There are other schools.