In Nature, there's been a recent publication arguing that the best way of gauging the truth of a question is to get people to report their views on the truth of the matter, and their estimate of the proportion of people who would agree with them.

Then, it's claimed, the surprisingly popular answer is likely to be the correct one.

In this post, I'll attempt to sketch a justification as to why this is the case, as far as I understand it.

First, an example of the system working well:

Capital City

Canberra is the capital of Australia, but many people think the actual capital is Sydney. Suppose only a minority knows that fact, and people are polled on the question:

Is Canberra the capital of Australia?

Then those who think that Sydney is the capital will think the question is trivially false, and will generally not see any reason why anyone would believe it true. They will answer "no" and put high proportion of people answering "no".

The minority who know the true capital of Australia will answer "yes". But most of them will likely know a lot of people who are mistaken, so they won't put a high proportion on people answering "yes". Even if they do, there are few of them, so the population estimate for the population estimate of "yes", will still be low.

Thus "yes", the correct answer, will be surprisingly popular.

A quick sanity check: if we asked instead "Is Alice Springs the capital of Australia?", then those who believe Sydney is the capital will still answer "no" and claim that most people would do the same. Those who believe the capital is in Canberra will answer similarly. And there will be no large cache of people believing in Alice Springs being the capital, so "yes" will not be surprisingly popular.

What is important here is that adding true information to the population, will tend to move the proportion of people believing in the truth, more than that moves people's estimate of that proportion.

No differential information:

Let's see how that setup could fail. First, it could fail in a trivial fashion: the Australian Parliament and the Queen secretly conspire to move the capital to Melbourne. As long as they aren't included in the sample, nobody knows about the change. In fact, nobody can distinguish a world in which that was vetoed from one where where it passed. So the proportion of people who know the truth - that being those few deluded souls who...

Aggregation