Some differences:
– Tao uses more words. This makes sense: he’s busy explaining this stuff to himself as well as to his readers. To a statistician, these ideas are so basic that it’s hard for us to really elaborate. (Also, I had a word limit.)
– Tao emphasizes that a confidence interval is not a probability interval. In my experience, confidence intervals are always treated as probability intervals anyway, so I don’t spend time with the distinction.
– I emphasize that a poll is a snapshot, not a forecast.
– Tao says that the number of polled voters is fixed in advance. I don’t think this is exactly true, what with nonresponse.
– Tao fills his blog entry with Wikipedia links. Wikipedia is ok but I’m not so thrilled with it; I’m happy with people looking things up in it if they want but I won’t encourage it.
But we’re basically saying the same thing. I like how I put it, but I’m sure a lot of people prefer Tao’s style. Luckily there’s room on the web for both!
Not thrilled about wikipedia? Welcome to the 21st century! It's the best reference we have and if you don't like an entry, or you think there is a mistake, just fix it and take credit for it.
My 2 cents: science is too slow to move on and embrace these new media (go blogs!) … This is a great opportunity for science to build an Encyclopedia Galactica (http://en.wikipedia.org/wiki/Encyclopedia_Galactica)
In my experience, confidence intervals are always treated as probability intervals anyway, so I don't spend time with the distinction.
Do you just mean that people make the mistake no matter how many times you point it out to them, so there's no point in pointing it out? Or do you mean that in some sense this isn't really a mistake?
And Anonymous – on some topics Wikipedia tends to be great (I've noticed that this is often the case for research-level math – at least, the entries tend to be good for people who are already at the research math level, which I guess doesn't mean that they're good), on others a few subtle corrections are sufficient to make it a good article. But on some topics (and I would guess that topics connected to Bayesianism would fall into this category, because of the radically interdisciplinary group of people interested in the topic, and the controversy sometimes surrounding it) to make the article good would require a very serious rewrite, which takes a lot of time. It's probably easier to write a decent encyclopedia article from scratch than to try to turn lots of information from a mediocre-to-bad article into a decent article that contains all that information.
Anonymous:
Wikipedia is fine, and everybody knows about it. Readers can type in "confidence interval" if they want; I just don't see the benefit in putting in the Wikipedia link into my blog entry. For one thing, I find it distracting–I think it makes the article less readable.
Kenny:
The mistake is thinking that what is called the "confidence interval" is actually relevant to real problems. It is the probability interval that is relevant. In many cases the conf interval is almost the same as the prob interval, in which case there's no mistake; in other cases, they're different, in which case the mistake, in my opinion, would be to use the prob interval.
In either case, I don't see the point in going into a long explanation about how to interpret the wrong thing.