Instinctive Frequentists, the Outside View, and de-Biasing

by Stuart_Armstrong2 min read20th Sep 201323 comments

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Heuristics & BiasesProbability & Statistics
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In "How to Make Cognitive Illusions Disappear: Beyond Heuristics and Biases", Gerd Gigerenzer attempts to show that the whole "Heuristics and Biases" approach to analysing human reasoning is fundamentally flawed and incorrect.

In that he fails. His case depends on using the frequentist argument that probabilities cannot be assigned to single events or situations of subjective uncertainty, thus removing the possibility that people could be "wrong" in the scenarios where the biases were tested. (It is interesting to note that he ends up constructing "Probabilistic Mental Models", which are frequentist ways of assigning subjective probabilities - just as long as you don't call them that!).

But that dodge isn't sufficient. Take the famous example of the conjunction fallacy, where most people are tricked to assigning a higher probability to "Linda is a bank teller AND is active in the feminist movement" than to "Linda is a bank teller". This error persists even when people take bets on the different outcomes. By betting more (or anything) on the first option, people are giving up free money. This is a failure of human reasoning, whatever one thinks about the morality of assigning probability to single events.

However, though the article fails to prove its case, it presents a lot of powerful results that may change how we think about biases. It presents weak evidence that people may be instinctive frequentist statisticians, and much stronger evidence that many biases can go away when the problems are presented in frequentist ways.

Now, it's known that people are more comfortable with frequencies that with probabilities. The examples in the paper extend that intuition. For instance, when people are asked:

There are 100 persons who fit the description above (i.e., Linda's). How many of them are:
(a) bank tellers
(b) bank tellers and active in the feminist movement.

Then the conjunction fallacy essentially disappears (22% of people make the error, rather than 85%). That is a huge difference.

Similarly, overconfidence. When people were 50 general knowledge questions and asked to rate their confidence for their answer on each question, they were systematically, massively overconfident. But when they were asked afterwards "How many of these 50 questions do you think you got right?", they were... underconfident. But only very slightly: they were essentially correct in their self-assessments. This can be seen as a use of the outside view - a use that is, in this case, entirely justified. People know their overall accuracy much better than they know their specific accuracy.

A more intriguing example makes the base-rate fallacy disappear. Presenting the problem in a frequentist way makes the fallacy vanish when computing false positives for tests on rare diseases - that's compatible with the general theme. But it really got interesting when people actively participated in the randomisation process. In the standard problem, students were given thumbnail description of individuals, and asked to guess whether they were more likely to be engineers or lawyers. Half the time the students were told the descriptions were drawn at random from 30 lawyers and 70 engineers; the other half, the proportions were reversed. It turns out that students assigned similar guesses to lawyer and engineer in both setups, showing they were neglecting to use the 30/70 or 70/30 base-rate information.

Gigerenzer modified the setups by telling the students the 30/70 or 70/30 proportions and then having the students themselves drew each description (blindly) out of an urn before assessing it. In that case, base-rate neglect disappears.

Now, I don't find that revelation quite as superlatively exciting as Gigerenzer does. Having the students draw the description out of the urn is pretty close to whacking them on the head with the base-rate: it really focuses their attention on this aspect, and once it's risen to their attention, they're much more likely to make use of it. It's still very interesting, though, and suggests some practical ways of overcoming the base-rate problem that stop short of saying "hey, don't forget the base-rate".

There is a large literature out there critiquing the heuristics and biases tradition. Even if they fail to prove their point, they're certainly useful for qualifying the biases and heuristics results, and, more interestingly, for suggesting practical ways of combating their effects.

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That's quite amazing, really. If, by rephrasing a question, one can remove a bias, then it makes sense to learn to detect poorly phrased questions and to ask better questions of oneself and others. This seems like a cheaper alternative than fighting one's nature with Bayesian debiasing. Or maybe the first step toward it.

I'm pretty sure this is one of the main areas Prof David Spiegelhalter is trying to cover with experiments like this one. He advises the British government on presenting medical statistics, and his work is worth a read if you want to know about how to phrase statistical questions so people get them more right.

The strategy to avoid a bias depends on the type of bias. Asking a different question is not (always) enough or the best idea.

The outside view helps in the cited cases of overconfidence,

For evalutation of false negative rates (the often used example of the mammographies e.g. in http://www.yudkowsky.net/rational/bayes) the method that works best is to use an example population and a graphical representation (a grid with the 4 sub populations marked). This is the same debiasing as used in the conjunction fallacy case above. Same goes for the base rate neglect.

For anchoring think an enemy provided the 'misleading' achor-information and instead ask what a friend might say about it.. Confirmation can be countered by imagining arguments an enemy might use (against which to defend) and then to use these as (counter)-examples. Using an imagined enemy is a simple and specific form of the outside view.

Framing can be avoided by trying to come up with differnt frames and averaging.

Availability bias and e.g. illusory correlation cannot really be coutnered because negative examples are just unavailable by nature. I'd recommend honesty and modesty in this case: If you see/remember some salient detail you can imaging the vast number of forgotten non-salient counter-examples and say: I will infer nothing until proven otherwise.

I started to rephrase "how often might I be wrong?" into "how many of my acquintances (who are alike me) might be wrong on this question?". And "which range am I confident this value must be in?" becomes "which range of values might my friends guess this to be?".

This seems impossible to test, but I wonder if the earlier versions of the questions were intuitively framed in ways that people were likely get wrong, as a way of displaying status and/or trying to get interesting results.

stop short of saying "hey, don't forget the base-rate".

Why is this a goal? Sure, if you're trying to design an experiment to measure the fallacy, you likely don't want to say "Hey, don't forget about the thing I'm trying to determine whether you'll forget or not". But outside of this specific context, why not hit people over the head with the base rate? We want them to get probabilities right, presumably!

You can say "hey, don't forget the base-rate" if people are doing a specific problem where you've identified that they need to remember the base-rate. But people do all sorts of problems all the time, in all sorts of circumstances; it might be possible to use approaches to make people naturally pay attention to the base-rate, whenever it comes up, without having to figure out when they're making the mistake and reminding them of it.

I wonder whether using a space metaphor might help. Gigerenzer seems to apply this: drawing from an urn, imainging a population. If you have an equals distribution (base rate 50%) you might use left/right, but this doesn't gain much as 50/50 is already unbiased. How about imagining the more likely candidates in front of you and the other behind you - that will likely put them out of your awareness and thus reducing their saliency. The question is what percentage that represents. The same approach could be used to imagine positive candidates near and negative ones far away.

Downvoted for misrepresenting frequentism, but otherwise a great post. Frequentism does not prohibit assigning probabilities to single events, and approximation of probabilities using counts is not exclusive to frequentist models. The impact of presenting the questions to people using numbers vs percentages has little to do with frequentism, and much more to do with cognitive ease.

What's more probable:

A: "Linda is a bank teller AND is active in the feminist movement"

B: "Linda is a bank teller".

When the question is formulated this way, people assume, that it is in fact asked:

What's more probable:

A: "Linda is a bank teller AND is active in the feminist movement"

B: "Linda is a bank teller AND she gave up her previous views".

If giving up her views isn't very probable for Linda, then A is more likely.

The format of a question sometimes conveys some additional information, beware!

Nope. People thought that this might be what's going on, so they tested it. People definitely weren't misinterpreting the question; the conjunction fallacy really is a thing.

What is more likely?

A: People gave wrong answer AND weren't misinterpreting the question

B: People gave wrong answer

Wrong framing*. The question is between

A: People gave wrong answer AND weren't misinterpreting the question

B: People gave wrong answer AND were misinterpreting the question

*Unless that was a joke, in which case disregard

No.

If you ask one person to assign probability to "Linda is a bank teller and active in the feminist movement" and another person to assign probability to "Linda is a bank teller" (not numerically, but by comparing this hypothesis with other verbal descriptions), the first person will still on average give higher estimates than the second person.

I think that conveying more information, such as with the statement "Linda is a bank teller and is active in the feminist movement" subtly suggests a greater familiarity with or knowledge of the subject (in this case Linda) and so seems more authoritative. I believe that is what is happening here. If you included even more information it would create that impression even more strongly. For example "Linda is a bank teller and is active in the feminist movement and she is dating Fred and lives near the train station and her phone number is 555-3213" sounds like its her best friend talking and who knows more about Linda? Her best friend or someone who only knows that she is a bank teller? I think the extra information pulls on an intuition that someone who knows a lot about something is very familiar with it and likely to be correct.

Not sure if I am rationalizing now, but this explanation seems compatible with the official one; or more precisely, it is a subset. As in:

Q: Why do people think that more complex explanations have higher probability, when mathematically it is the other way round?

A: Because detailed stories feel more credible.

Q: Okay, but why do detailed stories feel more credible?

A: Because detailed stories are often told by people who have more information about the subject.

I think this is part of what is going on.

Of course the mathematical probability of a given statement decreases by adding an additional condition. But the average person uses common sense and basic social skills -- not formal probability -- in the Linda example.

If someone approaches you and spouts unsolicited information about some lady you don't know and then asks you which of several statements about her is most probably true, then the average person assumes they are being asked to make their best guess at choosing the statement that contains the most truth.

There probably (ha) is some large % of the population that is ignorant of even basic formal mathematic probability (I'm one of them) and will choose the wrong statement however the question is phrased...

But the average Joe isn't doing the math anyway when they are asked about Linda. In polite society, if we are asked what we think about a matter of trivial consequence, we give it our best shot using the all the data we have.

If someone tells us a bunch of stuff about Linda that indicates she might have the desire and capability to be active in a social movement, we might guess the conjunction statement to be the one containing more truth. Or we might grab a calculator and ask the person to restate the question (more slowly this time, please) so we can arrive at the answer that would score highly on a math test.

If you tend to do the latter, no one will talk to you after a while. Thus, using formal probability in situations like this is not very rational.

Did you read the post Viliam_Bur and I both linked to?

Yes. Did you read my post?

I acknowledged "some large % of the population that is ignorant of even basic formal mathematic probability (I'm one of them) and will choose the wrong statement however the question is phrased..."

In EY's post, among lots of other stuff, he says research shows the incidence of the conjunction fallacy is reduced by taking measures to ensure those asked about Linda realize they are being asked for the formal mathematical probability and there is some incentive to get the correct answer.

My point was that in real life, especially in matter of trivial significance, people make quick decisions using the context and clues they are presented with. Of course this leads to mistakes, and I therefore do not deny or dismiss the conjunction fallacy exists... but treating every social situation like a math problem is a sure fire way to make sure you are not taken seriously outside of the math classroom.

In real life, the Linda example might go something like this:

Random Person: "Hey you! Linda is all these awesome things. Do you think she is a bank teller? Or do you think she is a bank teller who is also doing some awesome things that other awesome people do? Which one?

Me: "Um. Who are you? And who is Linda?"

RP: "Just answer."

Me: "Um. Okay. I don't know, and I'm not sure why you are asking me. But from what you said, it sounds like Linda is probably a bank teller who is doing some awesome things."

RP: "Nope!! You are wrong. Conjunction fallacy. It is more probable that Linda is just a bank teller. Ha!"

Me: "Wow. I guess you really got me there. I've no idea why you asked, no reason to believe anything you said about Linda was true and no incentive to provide anything other than my inkling about what seemed to be the most true scenario given all that. Of course, if you would have presented this as a question about the formal probabilities of each statement, assured me all the data was accurate and this wasn't some kind of trick question, and then provided even the slightest incentive to answer correctly beyond just wanting to end this horribly awkward interaction, then I posit my response would have been significantly more likely to be correct. Though I suck at math, so you still might have...erm...won. If your point is that the conjunction fallacy it real, you are preaching to the converted, and you're doing it in a pretty off-putting way. Have a nice day."

RP: {yelling as I walk away} "Conjunction fallacy. You can't deny it! Eliezer wrote a massive post, so..."

The conjunction fallacy is a great reason to take time to closely examine the probabilities of important life decisions. It exists. It is limited when the question is framed so that it is clearly a math problem.

Given Eliezer's massive post of references for why the standard interpretation of the conjunction fallacy is correct, the burden of proof is on you if you want to argue with it. Go and read the research!

Which is more probable?

A. Some guy wrote a massive post defending the standard interpretation of the conjunction fallacy.

B. Some guy wrote a massive post defending the standard interpretation of the conjunction fallacy AND his massive post is evidence for said interpretation.

Why do people keep on replying like this to my comments? It doesn't make any sense. If I'm arguing for B and you're not arguing for A then it doesn't matter if P(A)>P(B). It can still be the case that P(B) is high!

EDIT: Especially if A is a know historical fact which we have easy access to. If P(A)=1 then that's no restriction on B at all!

Indeed, wording is a problem. The first time I read about this fallacy (elsewhere on this website), it was between "A: Paul is a laywer AND plays the sax" and "B: Paul plays the sax." I understood B as meaning that Paul's job was as a musician, which I judged less likely than being a lawyer with a music hobby.