This article is an attempt to summarize basic material, and thus probably won't have anything new for the hard core posting crowd. It'd be interesting to know whether you think there's anything essential I missed, though.
You've probably seen the word 'Bayesian' used a lot on this site, but may be a bit uncertain of what exactly we mean by that. You may have read the intuitive explanation, but that only seems to explain a certain math formula. There's a wiki entry about "Bayesian", but that doesn't help much. And the LW usage seems different from just the "Bayesian and frequentist statistics" thing, too. As far as I can tell, there's no article explicitly defining what's meant by Bayesianism. The core ideas are sprinkled across a large amount of posts, 'Bayesian' has its own tag, but there's not a single post that explicitly comes out to make the connections and say "this is Bayesianism". So let me try to offer my definition, which boils Bayesianism down to three core tenets.
We'll start with a brief example, illustrating Bayes' theorem. Suppose you are a doctor, and a patient comes to you, complaining about a headache. Further suppose that there are two reasons for why people get headaches: they might have a brain tumor, or they might have a cold. A brain tumor always causes a headache, but exceedingly few people have a brain tumor. In contrast, a headache is rarely a symptom for cold, but most people manage to catch a cold every single year. Given no other information, do you think it more likely that the headache is caused by a tumor, or by a cold?
If you thought a cold was more likely, well, that was the answer I was after. Even if a brain tumor caused a headache every time, and a cold caused a headache only one per cent of the time (say), having a cold is so much more common that it's going to cause a lot more headaches than brain tumors do. Bayes' theorem, basically, says that if cause A might be the reason for symptom X, then we have to take into account both the probability that A caused X (found, roughly, by multiplying the frequency of A with the chance that A causes X) and the probability that anything else caused X. (For a thorough mathematical treatment of Bayes' theorem, see Eliezer's Intuitive Explanation.)
There should be nothing surprising about that, of course. Suppose you're outside, and you see a person running. They might be running for the sake of exercise, or they might be running because they're in a hurry somewhere, or they might even be running because it's cold and they want to stay warm. To figure out which one is the case, you'll try to consider which of the explanations is true most often, and fits the circumstances best.
Core tenet 1: Any given observation has many different possible causes.
Acknowledging this, however, leads to a somewhat less intuitive realization. For any given observation, how you should interpret it always depends on previous information. Simply seeing that the person was running wasn't enough to tell you that they were in a hurry, or that they were getting some exercise. Or suppose you had to choose between two competing scientific theories about the motion of planets. A theory about the laws of physics governing the motion of planets, devised by Sir Isaac Newton, or a theory simply stating that the Flying Spaghetti Monster pushes the planets forwards with His Noodly Appendage. If these both theories made the same predictions, you'd have to depend on your prior knowledge - your prior, for short - to judge which one was more likely. And even if they didn't make the same predictions, you'd need some prior knowledge that told you which of the predictions were better, or that the predictions matter in the first place (as opposed to, say, theoretical elegance).
Or take the debate we had on 9/11 conspiracy theories. Some people thought that unexplained and otherwise suspicious things in the official account had to mean that it was a government conspiracy. Others considered their prior for "the government is ready to conduct massively risky operations that kill thousands of its own citizens as a publicity stunt", judged that to be overwhelmingly unlikely, and thought it far more probable that something else caused the suspicious things.
Again, this might seem obvious. But there are many well-known instances in which people forget to apply this information. Take supernatural phenomena: yes, if there were spirits or gods influencing our world, some of the things people experience would certainly be the kinds of things that supernatural beings cause. But then there are also countless of mundane explanations, from coincidences to mental disorders to an overactive imagination, that could cause them to perceived. Most of the time, postulating a supernatural explanation shouldn't even occur to you, because the mundane causes already have lots of evidence in their favor and supernatural causes have none.
Core tenet 2: How we interpret any event, and the new information we get from anything, depends on information we already had.
Sub-tenet 1: If you experience something that you think could only be caused by cause A, ask yourself "if this cause didn't exist, would I regardless expect to experience this with equal probability?" If the answer is "yes", then it probably wasn't cause A.
This realization, in turn, leads us to
Core tenet 3: We can use the concept of probability to measure our subjective belief in something. Furthermore, we can apply the mathematical laws regarding probability to choosing between different beliefs. If we want our beliefs to be correct, we must do so.
The fact that anything can be caused by an infinite amount of things explains why Bayesians are so strict about the theories they'll endorse. It isn't enough that a theory explains a phenomenon; if it can explain too many things, it isn't a good theory. Remember that if you'd expect to experience something even when your supposed cause was untrue, then that's no evidence for your cause. Likewise, if a theory can explain anything you see - if the theory allowed any possible event - then nothing you see can be evidence for the theory.
At its heart, Bayesianism isn't anything more complex than this: a mindset that takes three core tenets fully into account. Add a sprinkle of idealism: a perfect Bayesian is someone who processes all information perfectly, and always arrives at the best conclusions that can be drawn from the data. When we talk about Bayesianism, that's the ideal we aim for.
Fully internalized, that mindset does tend to color your thought in its own, peculiar way. Once you realize that all the beliefs you have today are based - in a mechanistic, lawful fashion - on the beliefs you had yesterday, which were based on the beliefs you had last year, which were based on the beliefs you had as a child, which were based on the assumptions about the world that were embedded in your brain while you were growing in your mother's womb... it does make you question your beliefs more. Wonder about whether all of those previous beliefs really corresponded maximally to reality.
And that's basically what this site is for: to help us become good Bayesians.
The problem you have is the one shared by everyone from devotees of parapsychology to people who believe Meredith Kercher was killed in an orgy initiated by Amanda Knox: your prior on your theory is simply way too high.
Simply put, the events of 9/11 are so overwhelmingly more likely a priori to have been the exclusive work of a few terrorists than the product of a conspiracy involving the U.S. government, that the puzzling details you cite, even in their totality, fail to make a dent in a rational observer's credence of (more or less) the official story.
You might try asking yourself: if the official story were in fact correct, wouldn't you nevertheless expect that there would be strange facts that appear difficult to explain, and that these facts would be seized upon by conspiracy theorists, who, for some reason or another, were eager to believe the government may have been involved? And that they would be able to come up with arguments that sound convincing?
I want to stress that it is not the fact that the terrorists-only theory is officially sanctioned that makes it the (overwhelming) default explanation; as the Kercher case illustrates, sometimes the official story is an implausible conspiracy theory! Rather, it is our background knowledge of how reality operates -- which must be informed, among other things, by an acquaintance with human cognitive biases.
"Not silencing skeptical inquiry" is a great-sounding applause light, but we have to choose our battles, for reasons more mathematical than social: there are simply too many conceivable explanations for any given phenomenon, for it it be worthwhile to consider more than a very small proportion of them. Our choice of which to consider in the first place is thus going to be mainly determined by our prior probabilities -- in other words, our model of the world. Under the models of most folks here, 9/11 conspiracy theories simply aren't going to get any time of day.
If it's different for you, I'd be curious to know what kind of ideas with substantial numbers of adherents you would feel safe in dismissing without bothering to research. (If there aren't any, then I think you severely overestimate the tendency of people's beliefs to be entangled with reality.)