komponisto comments on What is Bayesianism? - Less Wrong
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(Some folks have expressed disapproval of this conversation continuing in this thread; ironically, though, it's becoming more and more an explicit lesson in Bayesianism -- as this comment in particular will demonstrate. Nevertheless, after this comment, I am willing to move it elsewhere, if people insist.)
You're in Bayes-land here, not a debating society. Beliefs are what we're interested in. There's no distinction between an argument that a certain point of view should be taken seriously and an argument that the point of view in question has a significant probability of being true. If you want to make a case for the former, you'll necessarily have to make a case for the latter.
Here's how you do a Bayesian analysis: you start with a prior probability P(H). Then you consider how much more likely the evidence is to occur if your hypothesis is true (P(E|H)) than it is in general (P(E)) -- that is, you calculate P(E|H)/P(E). Multiplying this "strength of evidence" ratio P(E|H)/P(E) by the prior probability P(H) gives you your posterior (updated) probability P(H|E).
Alternatively, you could think in terms of odds: starting with the prior odds P(H)/P(~H), and considering how much more likely the evidence is to occur if your hypothesis is true (P(E|H)) than if it is false (P(E|~H)); the ratio P(E|H)/P(E|~H) is called the "likelihood ratio" of the evidence. Multiplying the prior odds by the likelihood ratio gives you the posterior odds P(H|E)/P(~H|E).
One of the two questions you need to answer is: by what factor do you think the evidence raises the probability/odds of your hypothesis being true? Are we talking twice as likely? Ten times? A hundred times?
If you know that, plus your current estimate of how likely your hypothesis is, division will tell you what your prior was -- which is the other question you need to answer.
If there's enough information for you to have a belief, then there's enough information to calculate the odds. Because, if you're a Bayesian, that's what these numbers represent in the first place: your degree of belief.
"Your failure to dismiss..." is simply an English-language locution that means "The fact that you did not dismiss..."