Heeding others' impressions often increases accuracy.  But "agreement"  and "majoritarianism" are not magic;  in a given circumstance, agreement is or isn't useful for *intelligible* reasons. 

You and four other contestants are randomly selected for a game show.  The five of you walk into a room.  Each of you is handed a thermometer drawn at random from a box; each of you, also, is tasked with guessing the temperature of a bucket of water.  You’ll each write your guess at the temperature on the card; each person who is holding a card that is within 1° of the correct temperature will win $1000.

The four others walk to the bucket, place their thermometers in the water, and wait while their thermometers equilibrate.  You follow suit.  You can all see all of the thermometers’ read-outs: they’re fairly similar, but a couple are a degree or two off from the rest.  You can also watch, as each of your fellow-contestants stares fixedly at his or her own thermometer and copies its reading (only) onto his or her card.

Should you:

1. Write down the reading on your own thermometer, because it’s yours;
2. Write down an average* thermometer reading, because probably the more accurate thermometer-readings will cluster;
3. Write down an average of the answers on others’ cards, because rationalists should try not to disagree;
4. Follow the procedure everyone else is following (and so stare only at your own thermometer) because rationalists should try not to disagree about procedures?

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.)