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passive_fist comments on Am I Understanding Bayes Right? - Less Wrong Discussion
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Comments (20)
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.