The conditional probability of A given B is my shepherd, I shall not want. It makes me multiply through marginal probabilities, it leadeth me beside flat priors...<p>And the people sang "P of A given B is equal to the prior probability of A times P of B given A over the marginal probability of B"
This is a bit more technical, but still somewhat on topix. Andrew Gelman did a great "criticism" of Bayesian analysis on April Fools 2008:<p><a href="http://www.stat.columbia.edu/~cook/movabletype/archives/2008/04/problems_with_b.html" rel="nofollow">http://www.stat.columbia.edu/~cook/movabletype/archives/2008...</a><p>Followed later by his response to himself with some other statisticians:<p><a href="http://www.stat.columbia.edu/~cook/movabletype/archives/2008/07/responses_to_my.html" rel="nofollow">http://www.stat.columbia.edu/~cook/movabletype/archives/2008...</a>
The trouble with applying Bayesianism in science is that your conclusion becomes dependent on what priors you think are reasonable. If different people disagree about that, then it becomes a debate about beliefs, not science. (Unless the priors can actually be determined, of course.)<p>This seems to largely be the case in climate issues, where it's clear that some people have such strong priors against anthropogenic climate change that the evidence needed to convince them outstrip what science reasonably can provide.
Here's a simple, applied example I came up with while I was applying to med school: <a href="http://nielsolson.us/MedSchool/#odds" rel="nofollow">http://nielsolson.us/MedSchool/#odds</a><p>[edit]: there was subtraction error, now fixed.
I listened last year to a great set of lectures about the Philosophy of Science. Too much to go into here, save for the fact that, at the end of the day, it very well may end up that probabilities are all we have to go on for a wide swath of things.<p>I'm not yet ready to make the leap to total Bayesianism -- those priors can really bite you when you least expect it -- but there is some really good stuff here. Thanks Kaj.<p>Perhaps we'll see some kind of structured cross between Peirce and Bayes in the next few decades.
It's been a while since I've worked with conditional probabilities, but isn't he missing the case in which the headache is caused by both a cold and a tumor?