What does "Bayesian" mean?

I've been using the term "Bayesian" for a while now on this blog, usually linking to the Wikipedia article each time to help newcomers to the concept understand exactly what I mean. But I still occasionally get questions about what I mean when I use the word, mostly because the wiki page focuses just a little too much on the mathematics involved, and is not really readable by non-math people yet. Since quite a few of the current draft articles I am writing require a fair understanding of the term, I've decided to write a short, readable essay on what I mean when I use the term "Bayesian". Hopefully this will clear up any confusion about the purpose and importance of the term.

The Problem Bayes Theorem Solves

For most people, science is just something that is taken for granted. Whatever those scientists are doing seems to bring us new inventions and technology that improves our daily lives, so we leave well enough alone about the specifics of how what they do works. We know that for some reason, the knowledge acquisition methods employed by scientists seems to bring results, so we praise science as a good method of knowledge of science acquisition.

Compare this to another field of knowledge acquisition, such as the one psychics or astrologers use. They employ a method that does not seem to produce results, so we generally call such fields unscientific. By this, we seem to be implying that science's methods work while these alternatives do not. This is, of course, completely correct. The predictions of psychics and astrologers are never believable to a rational human being, even when they happen to occasionally correct. This is because the method by which they purport to gain new knowledge is completely unbelievable.

But why is it unbelievable? What is it about science's methods that makes us tend to put trust in it when they are correct, when we simultaneously fail to put trust in pseudoscience's methods even if they also turn out to be correct? Why do we say one is correct by design and one is only correct by accident?

The Bayesian Solution

[...the rest of this entry was left unwritten. I am publishing anyway because I have no goal of ever finishing this old essay.]