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:
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 already though the capital was in Melbourne, for some reason - is no higher in the world where it's true than the one where it's false.
So the population opinion has to be truth-tracking, not in the sense that the majority opinion is correct, but in the sense that more people believe X is true, relatively, in a world where X is true versus a world where X is false.
Systematic bias in population proportion:
A second failure mode could happen when people are systematically biased in their estimate of the general opinion. Suppose, for instance, that the following headline went viral:
"Miss Australia mocked for claims she got a doctorate in the nation's capital, Canberra."
And suppose that those who believed the capital was in Sydney thought "stupid beauty contest winner, she thought the capital was in Canberra!". And suppose those know knew the true capital thought "stupid beauty contest winner, she claimed to have a doctorate!". So the actual proportion in the belief doesn't change much at all.
But then suppose everyone reasons "now, I'm smart, so I won't update on this headline, but some other people, who are idiots, will start to think the capital is in Canberra." Then they will update their estimate of the population proportion. And Canberra may no longer be surprisingly popular, just expectedly popular.
Purely subjective opinions
How would this method work on a purely subjective opinion, such as:
Is Picasso superior to Van Gogh?
Well, there are two ways of looking at this. The first is to claim this is a purely subjective opinion, and as such people's beliefs are not truth tracking, and so the answers don't give any information. Indeed, if everyone accepts that the question is purely subjective, then there is no such thing as private (or public) information that is relevant to this question at all. Even if there were a prior on this question, no-one can update on any information.
But now suppose that there is a judgement that is widely shared, that, I don't know, blue paintings are objectively superior to paintings that use less blue. Then suddenly answers to that question become informative again! Except now, the question that is really being answered is:
Does Picasso use more blue than Van Gogh?
Or, more generally:
According to widely shared aesthetic criteria, is Picasso superior to Van Gogh?
The same applies to moral questions like "is killing wrong?". In practice, that is likely to reduce to:
According to widely shared moral criteria, is killing wrong?