Wednesday, September 9, 2015

The Heuristic About Representativeness Heuristic

Some people think that the problem with representativeness heuristic is a base rate neglect. I hold that this is incorrect, and the problem is deeper than that, and simple use of a base rate isn't going to fix it. This makes the idea about "look at the base rate!" a heuristic as well.

The thing is, there is a fundamental difference between "How strongly E resembles H" and "How strongly H implies E". The latter question is about P(E|H), and this number could be used in Bayesian reasoning, if you add P(E|!H) and P(H)[1]. The former question — the question humans actually answer when asked to judge about whether something is likely — sometimes just could not be saved at all.

Several examples to get point across:

1) Conspiracy theorists / ufologists: naively, their existence strongly points to a world where UFOs exist, but really, their existence is very weak evidence of UFOs (human psychology suggests that ufologists could exist in a perfectly alienless world), and even could be an evidence against them, because if Secret World Government was real, we expect it to be very good at hiding, and therefore any voices who got close to the truth will be quickly silenced.

So, the answer to "how strongly E resembles H?" is very different from "how much is P(E|H)?". No amount of accounting for base rate is going to fix this.

2) Suppose that some analysis comes too good in a favor of some hypothesis

Maybe some paper argues that leaded gasoline accounts for 90% variation in violent crime (credit for this example goes to /u/yodatsracist on reddit). Or some ridiculously simple school intervention is claimed to have a gigantic effect size.

Let's take leaded gasoline, for example. On the surface, this data strongly "resembles" a world where leaded gasoline is indeed causing a violence, since 90% suggest that effect is very large and is very unlikely to be a fluke. On the other hand, this effect is too large, and 10% of "other factors" (including but not limited to: abortion rate, economic situation, police budget, alcohol consumption, imprisonment rate) is too small of percentage.

The decline we expect in a world of harmful leaded gasoline is more like 10% than 90%, so this model is too good to be true; instead of being very strong evidence in favor, this analysis could be either irrelevant (just a random botched analysis with faked data, nothing to see here) or offer some evidence against (for reasons related to the conservation of expected evidence, for example).

So, how it should be done? Remember that P(E|H) would be written as P(H -> E), were the notation a bit saner. P(E|H) is a "to which degree H implies E?", so the correct method for answering this query involves imagining world-where-H-is-true and asking yourself about "how often does E occur here?" instead of answering the question "which world comes to my mind after seeing E?".

[1] And often just using base rate is good enough, but this is another, even less correct heuristic. See: Prosecutor's Fallacy.

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