Open Phil gives $500,000 to Tetlock's research.

Great post. Another issue is why B doesn't believe Y in spite of believing X and in spite of A believing that X implies Y. Some mechanisms:

a) B rejects that X implies Y, for reasons that are good or bad, or somewhere in between. (Last case: reasonable disagreement.)

b) B hasn't even considered whether X implies Y. (Is not logically omniscient.)

c) Y only follows from X given some additional premises Z, which B either rejects (for reasons that are good or bad or somehwere in between) or hasn't entertained. (What Tyrrell McAllister wrote.)

d) B is confused over the meaning of X, and hence is confused over what X implies. (The dialect case.)

Your problem is called a clustering problem. First of all, you need to answer how you measure your error (information loss, as you call it). Typical error norms used are l1 (sum of individual errors), l2 (sum of squares of errors, penalizes larger errors more) and l-infinity (maximum error).

Once you select a norm, there always exists a partition that minimizes your error, and to find it there are a bunch of heuristic algorithms, e.g. k-means clustering. Luckily, since your data is one-dimensional and you have very few categories, you can just brute force it (for 4 categories you need to correctly place 3 boundaries, and naively trying all possible positions takes only n^3 runtime)

Hope this helps.

I have a maths question. Suppose that we are scoring n individuals on their performance in an area where there is significant uncertainty. We are categorizing them into a low number of categories, say 4. Effectively we're thereby saying that for the purposes of our scoring, everyone with the same score performs equally well. Suppose that we say that this means that all individuals with that score get assigned the mean actual performance of the individuals with that that score. For instance, if there were three people who got the highest score, and their perfomance equals 8, 12 and 13 units, the assigned performance is 11 units.

Now suppose that we want our scoring system to minimise information loss, so that the assigned performance is on average as close as possible to the actual performance. The question is: how do we achieve this? Specifically, how large a proportion of all individuals should fall into each category, and how does that depend on the performance distribution?

It would seem that if performance is linearly increasing as we go from low to high performers, then all categories should have the same number of individuals, whereas if the increase is exponential, then the higher categories should have a smaller number of individuals. Is there a theorem that proves this, and which exacty specifies how large the categories should be for a given shape of the curve? Thanks.

"Spreading quicker" may not be the best question to ask. The question I'm more interested in is, What is the relationship between speed of communication, and the curve that describes innovation over time?

A good model for this is the degree of genetic isolation in a genetic algorithm. Compare two settings for a GA. One allows mating between any two organisms in the population. Another has many subpopulations, and allows genetic exchange between subpopulations less frequently.

Plot the fitness of the most-fit organism in each population by generation. The first GA, which has fast genetic communication, will initially outstrip the second, but it will plateau at a lower level of fitness, and all the organisms in the population will be identical, and evolution will stop. This is called premature convergence.

The second GA, with restricted genetic communication, will catch up and pass the fitness of the first GA, usually continuing on to a much higher optimum, because it maintains homogenous subpopulations (which allows adaptation) but a diverse global population (which prevents premature convergence).

Think about the development of pop music. As communication technology improved, pop stars like Elvis could be heard, seen, and their records marketed and moved across the entire country more efficiently than marketing local musicians, and replaced live performers with recorded music. On one hand, you could live in Peoria and listen to the most-popular musicians in the country. On the other, by 1990, American pop music had nearly stopped evolving. Rebecca Black could become popular across the nation in a single week, but the amount of innovation or quality she produced was negligible.

Basically, rapid communication gives people too much choice. They choose things comfortably similar to what they know. Isolation is needed to allow new things to gain an audience before they're stomped out by the dominant things.

You need to state your preferences as a function of the long-term trajectory of the entropy of ideas, rather than as any instantaneous quantity.

What about non-elite groups? (...) they are likely to be heavily influenced by the cognitive elite, especially in the longer run.

I think they are likely to be influenced by whom they consider high-status. If you succeed to make a conspiracy theory website seem like an authoritative source (it must seem as professional as the mainstream media, except that it brings "news you will not hear elsewhere"), that's all you need.

Why would anyone make a professionally looking conspiracy theory website? Aren't "professionality" and "crackpot thinking" kinda opposed in real life? Yeah, the genuine crackpots usually also have low regards for mainstream design or marketing. But a professionally looking conspiracy theory website is a great vehicle for political propaganda. So a foreign government may spend a lot of money and professional work to make a high-status conspiracy website, where the conspiracies are filtered, and only those that are neutral or convenient for the owner are published.

This is not a hypothetical scenario. Russians already have such website with online radio in my country. I see advertisements for it almost everywhere. The content is more or less: "Things your government wants to keep secret from you: Vaccination causes autism. West is the source of all evil. Russia is a paradise. Being a member of European Union is bad for you, because they will kidnap and abuse your children!" And of course, the website is "independent and alternative".

I think I will want to see further the further debates in the US presidential race in this format. At the moment I don't see a clear link where I can express that preferance and get an email when the next debates get released with annotations.

In order to be fact-checked, a statement has to be truth-apt in the first place. That is, it has to be the sort of statement that is capable of being true or false.

A lot of political arguments aren't truth-apt; they amount to cheering ("Georgism, boo! Synarchism, yay!") as opposed to historical claims ("Countries that adopt goat control have seen their arson rate double") or even theoretical claims ("The erotic calculation problem predicts that college-educated adults will move out of states that ban vibrators").