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komponisto comments on What is Bayesianism? - Less Wrong

81 Post author: Kaj_Sotala 26 February 2010 07:43AM

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Comment author: komponisto 01 March 2010 02:06:30AM *  6 points [-]

(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.)

Again, I think this question is a diversion from what I have been arguing; its truth or falseness does not substantially affect the truth or falseness of my actual claims (as opposed to beliefs mentioned in passing)

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.

That said, I made a start at a Bayesian analysis, but ran out of mental swap-space. If someone wants to suggest what I need to do next, I might be able to do it.

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.

Is there enough information there to calculate some odds, or are there still bits missing?

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.

I'm not sure what you're suggesting by your use of the word "failure" here

"Your failure to dismiss..." is simply an English-language locution that means "The fact that you did not dismiss..."