Would you tell me what Bayesian methodology is used for?
I can tell you what I use it for, but that would be a biased sample. Commensurately, telling you
what to use it for is easy, without a background in statistics, as long as you don't ask
how to use it.
The heart of the method is Bayes' theorem, which is one of several ways of looking at a conditional probability -- the probability of one outcome given the known outcome of a related event. If there are six glasses on the bar, one filled with vodka and the rest with water, what are the odds of picking the one with the vodka after I already picked one with water?
Statistics is the study of numbers that for whatever reason cannot be determined with certainty. There are different ways of describing such indeterminate numbers. Classical statistics uses one set of methods. Bayesian methods use a different set. Specifically, Bayesian methods let you make various guesses at what you think the number might be, without knowing anything at first about the number. You don't have to justify the guess to the
nth degree, but you do have to figure out what the effect would be upon your guess if data came in about the number. That's where Bayes' theorem helps.
Much here has been made that your initial guesses about a number (the "prior probability") can be purely subjective beliefs. That is true. But as with any methodology, if you put garbage into it, you get garbage out of it. The most effective use of Bayesian inference comes when you put a great deal of thought into choosing your priors.
One of the most effective use of Bayesian methods is searching, and one of the success stories in the literature is finding flight recorders from airliners lost at sea. It lets us combine our prior belief about how easy or hard it would be to find "black boxes" at every point along the ocean floor with the results from actual search operations. We can base that belief on something scientific such as depth and murkiness of the water as long as it's external to the search at hand. We can say that if the water is deep, we will stand a greater chance of missing the recorder, if it's there. And we work this out numerically at first, without worrying about such things as where the plane actually went down or whether anyone has already searched that spot.
If you go up-thread a few days, you'll find another example of Bayesian inference used in computer network traffic analysis.
There is a controversy among statisticians over which method works best. I view it largely as a tempest in a teapot because both methods have valid applications and the only mistake you can make is to use one method when you should be using the other. That said, among fringe theorists Bayesian inference is attractive because they can posture it simplistically as a "rigorous" treatment of subjective beliefs. Since much of what constitutes the fringe is subjective belief, it is often erroneously claimed that Bayesian analysis can confirm the truth of a subjective belief without the use of evidence or data.
