Here’s my take on ‘Linda’. Don’t know if anyone else has made the same or nearly the same point, but anyway I’ll try to be brief:
Let E be the background information about Linda, and imagine two scenarios:
We know E and someone comes up to us and tells us statement A, that Linda is a bank teller.
We know E and someone comes up to us and tells us statement B, that Linda is a bank teller who is active in the feminist movement.
Now obviously P(A | E) is greater than or equal to P(B | E). However, I think it’s quite reasonable for P(A | E + “someone told us A”) to be less than P(B | E + “someone told us B”), because if someone merely tells us A, we don’t have any particularly good reason to believe them, but if someone tells us B then it seems likely that they know this particular Linda, that they’re thinking of the right person, and that they know she’s a bank teller.
However, the ‘frequentist version’ of the Linda experiment cannot possibly be (mis?)-interpreted in this way, because we’re fixing the statements A and B and considering a whole bunch of people who are obviously unrelated to the processes by which the statements were formed.
(Perhaps there’s an analogous point to be made about your second example: Someone being tested at all is likely to be someone for whom there are independent reasons why they might have the disease (perhaps they exhibited some of the symptoms, got worried and went to see their doctor.)
But surely the experiment must have specified that the person being tested for the disease was picked at random from the population?)
However, I think it’s quite reasonable for P(A | E + “someone told us A”) to be less than P(B | E + “someone told us B”), because if someone merely tells us A, we don’t have any particularly good reason to believe them, but if someone tells us B then it seems likely that they know this particular Linda, that they’re thinking of the right person, and that they know she’s a bank teller.
I just skimmed through the 1983 Tversky & Kahneman paper, and the same thing occurred to me. Given the pragmatics of human natural language communication, I would say that T&K (and the people who have been subsequently citing them) are making too much of these cases. I’m not at all surprised that the rate of “fallacious” answers plummets when the question is asked in a way that suggests that it should be understood in an unnaturally literal way, free of pragmatics—and I’d expect that even the remaining fallacious answers are mostly due to casual misunderstandings of the question, again caused by pragmatics (i.e. people casually misinterpreting “A” as “A & non-B” when it’s contrasted with “A & B”).
The other examples of the conjunction fallacy cited by T&K also don’t sound very impressive to me when examined more closely. The US-USSR diplomatic break question sounds interesting until you realize that the probabilities actually assigned were so tiny that they can’t be reasonably interpreted as anything but saying that the event is within the realm of the possible, but extremely unlikely. The increase due to conjunction fallacy seems to me well within the noise—I mean, what rational sense does it make to even talk about the numerical values of probabilities such as 0.0047 and 0.0014 with regards to a question like this one? The same holds for the other questions cited in the same section.
It strikes me that academics have a blind spot to one of the major weaknesses of this research because to get to their position they have had to adapt to exam questions effectively during their formative years.
One of the tricks to success in exams is to learn how to read the questions in a very particular way where you ignore all your background knowledge and focus to an unusual degree on the precise wording and context of the question. This is a terrible (and un-Bayesian) practice in most real world scenarios but is necessary to jump through the academic hoops required to get through school and university.
Most people who haven’t trained themselves to deal with these type of questions will apply common sense and make ‘unwarranted’ assumptions. A wise strategy in the real world but not in exam style questions.
This [people casually misinterpreting “A” as “A & non-B” when it’s contrasted with “A & B”] is covered in Tversky and Khaneman, 1983.
You mean the medical question? I’m not at all impressed with that one. This question and the subsequent one about its interpretation were worded in a way that takes a considerable mental effort to parse correctly, and is extremely unnatural for non-mathematicians, even highly educated ones like doctors. What I would guess happened was that the respondents skimmed the question without coming anywhere near the level of understanding that T&K assume in their interpretation of the results.
Again, when thinking about these experiments, one must imagine realistic people in a realistic setting, who are extremely unlikely to be willing to grapple with semantic subtleties that go beyond what they’ll gather with effortless casual skimming of the questions and offhand answers given without much thought.
This question and the subsequent one about its interpretation were worded in a way that takes a considerable mental effort to parse correctly, and is extremely unnatural for non-mathematicians, even highly educated ones like doctors.
A 55-year-old woman had pulmonary embolism documented angiographically 10 days after a cholecstectomy. Please rank order the following in terms of the probability that they will be among the conditions experienced by the patient (use 1 for the most likely and 6 for the least likely). Naturally, the patient could experience more than one of these conditions.
Dyspnea and hemiparesis
Calf pain
Pleuritic chest pain
Syncope and tachycardia
Hemiparesis
Hemoptysis
In what way is this question difficult for a doctor to parse? Give the subjects a little credit here.
How do you know that subjects did not interpret “Linda is a bank teller” to mean “Linda is a bank teller and is not active in the feminist movement”? For one thing, dear readers, I offer the observation that most bank tellers, even the ones who participated in anti-nuclear demonstrations in college, are probably not active in the feminist movement. So, even so, Teller should rank above Teller & Feminist.
Do you mean this one (from Conjunction Controversy): [...]
Yes, that’s the one.
In what way is this question difficult for a doctor to parse? Give the subjects a little credit here.
In my experience, outside of some very exceptional situations like e.g. in-depth discussions of complex technical subjects, the overwhelming majority of people, including highly educated people, simply don’t operate under the assumption that language should be understood in a logically precise and strict way. The standard human modus operandi is to skim the text in a very superficial manner and interpret it according to intuitive hunches and casual associations, guided by some strong preconceptions about what the writer is likely to be trying to say—and it’s up to you to structure your text so that it will be correctly understood with such an approach, or at least to give a clear and prominent warning that it should be read painstakingly in an unnaturally careful and literal way.
Some people are in the habit of always reading in a precise and literal way (stereotypically, mathematicians tend to be like that). I am also like that, and I’m sure many people here are too. But this is simply not the way the great majority of people function—including many people whose work includes complex and precise formal reasoning, but who don’t carry over this mode of thinking into the rest of their lives. In particular, from what I’ve seen, doctors typically don’t practice rigid formal reasoning much, and it’s definitely not reasonable to expect them to make such an effort in a situation where they lack any concrete incentive to do so.
I thought about this post for awhile—partially because I’ve just been too busy for LW the past few days—and I’m still pretty skeptical. I general, I think you’re right—people don’t closely read much of anything, or interpret much literally. I’ve seen enough economic experiments to know that subject rarely have even a basic grasp of the rules coming into the experiment, and only when the experiment begins and they start trying things out do they understand the environment we put them in.
However, in the heuristics and biases experiments what subjects are reading is only a couple of sentences long. In my exerpience, people tend to only skim when what they’re reading is long or complicated. So I find it fairly hard to believe that most people aren’t reading something like the Linda problem close enough to understand it—especially undergraduates at high end universities and trained doctors.
In my experience, people tend to only skim when what they’re reading is long or complicated. So I find it fairly hard to believe that most people aren’t reading something like the Linda problem close enough to understand it—especially undergraduates at high end universities and trained doctors.
My thinking about this topic is strongly influenced by my experiences from situations where I was in charge of organizing something, but without any formal authority over the people involved, with things based on an honor system and voluntary enthusiasm. In such situations, when I send off an email with instructions, I often find it a non-trivial problem to word things in a such a way that I’ll have peace of mind that it will be properly understood by all recipients.
In my experience, even very smart people with a technical or scientific background who normally display great intelligence and precision of thought in the course of their work will often skim and misunderstand questions and instructions worded in a precise but unnatural way, unless they have an incentive to make the effort to read the message with extra care and accuracy (e.g. if it’s coming from someone whose authority they fear). Maybe some bad experiences from the past have made me excessively cautious in this regard, but if I caught myself writing an email worded the same way as the doctors’ question by T&K and directed at people who won’t be inclined to treat it with special care—no matter how smart, except perhaps if they’re mathematicians—I would definitely rewrite it before sending.
Why would you think that subjects are working from a different state of information for the two possibilities in the Linda question? Here’s the question again as the subjects read it:
Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations.
What is the probability that Linda is:
(a) a bank teller
(b) a bank teller and active in the feminist movement
After reading the question, the probability of (a) and (b) is evaluated—with the same state of information: the background knowledge (E in your terms), that someone told us (a) (A), and that someone told us (b) (A). So formally, the two probabilties are:
P(A | E, “someone told us both A and B”)
P(B | E, “someone told us both A and B”)
So the conjunction rule still holds. Now it’s certainly possible that subjects are interpreting the question in the way you suggest (with different states of information for A and B), but it’s also possible that they’re interpreting it in any number of incorrect ways. They could think it’s a thinly veiled question about how they feel about feminism, for example. So why do you think the possible interpretation you raise is plausible enough to be worrisome?
note: this comment was scrapped and rewritten immediately after it was posted
Why would someone tell us “Linda is a bank teller and Linda is a bank teller and active in the feminist movement.”? That would be indeed a strange sentence.
ETA: Maybe the parent comment can be formulated more clearly in the following way (using frequentist language): People parse the discussed question not as what fraction of people from category E belong also into category A?, but rather what fraction of people telling us that a person (who certainly belongs to E) belongs also to A speak truth?, or even better, what fraction of individual statements of the described type is true?
Although A may be proper subset of B, statements telling A about any particular Linda aren’t proper subset of statements telling B about her. Quite contrary, they are disjoint. (That is, people tend to count frequencies of statements of given precise formulation, i.e. don’t count each occurence of B as a simultaneous occurence of A, even if B can be reanalysed as A and C. Of course, I am relying on my intuition in that and can be guilty of mind projection here.)
It is entirely possible to imagine that among real world statements about former environmental activists, the exact sentence “she is a bank teller” is less often true than the exact sentence “she is a bank teller and an active feminist”. I am quite inclined to believe that more detailed information is more often true than less detailed one, since the former is more likely to be given by informed people, and this mechanism may have contributed to evolution of heuristics which produce the experimentally detected conjunction fallacy.
Here’s my take on ‘Linda’. Don’t know if anyone else has made the same or nearly the same point, but anyway I’ll try to be brief:
Let E be the background information about Linda, and imagine two scenarios:
We know E and someone comes up to us and tells us statement A, that Linda is a bank teller.
We know E and someone comes up to us and tells us statement B, that Linda is a bank teller who is active in the feminist movement.
Now obviously P(A | E) is greater than or equal to P(B | E). However, I think it’s quite reasonable for P(A | E + “someone told us A”) to be less than P(B | E + “someone told us B”), because if someone merely tells us A, we don’t have any particularly good reason to believe them, but if someone tells us B then it seems likely that they know this particular Linda, that they’re thinking of the right person, and that they know she’s a bank teller.
However, the ‘frequentist version’ of the Linda experiment cannot possibly be (mis?)-interpreted in this way, because we’re fixing the statements A and B and considering a whole bunch of people who are obviously unrelated to the processes by which the statements were formed.
(Perhaps there’s an analogous point to be made about your second example: Someone being tested at all is likely to be someone for whom there are independent reasons why they might have the disease (perhaps they exhibited some of the symptoms, got worried and went to see their doctor.)
But surely the experiment must have specified that the person being tested for the disease was picked at random from the population?)
AlephNeil:
I just skimmed through the 1983 Tversky & Kahneman paper, and the same thing occurred to me. Given the pragmatics of human natural language communication, I would say that T&K (and the people who have been subsequently citing them) are making too much of these cases. I’m not at all surprised that the rate of “fallacious” answers plummets when the question is asked in a way that suggests that it should be understood in an unnaturally literal way, free of pragmatics—and I’d expect that even the remaining fallacious answers are mostly due to casual misunderstandings of the question, again caused by pragmatics (i.e. people casually misinterpreting “A” as “A & non-B” when it’s contrasted with “A & B”).
The other examples of the conjunction fallacy cited by T&K also don’t sound very impressive to me when examined more closely. The US-USSR diplomatic break question sounds interesting until you realize that the probabilities actually assigned were so tiny that they can’t be reasonably interpreted as anything but saying that the event is within the realm of the possible, but extremely unlikely. The increase due to conjunction fallacy seems to me well within the noise—I mean, what rational sense does it make to even talk about the numerical values of probabilities such as 0.0047 and 0.0014 with regards to a question like this one? The same holds for the other questions cited in the same section.
It strikes me that academics have a blind spot to one of the major weaknesses of this research because to get to their position they have had to adapt to exam questions effectively during their formative years.
One of the tricks to success in exams is to learn how to read the questions in a very particular way where you ignore all your background knowledge and focus to an unusual degree on the precise wording and context of the question. This is a terrible (and un-Bayesian) practice in most real world scenarios but is necessary to jump through the academic hoops required to get through school and university.
Most people who haven’t trained themselves to deal with these type of questions will apply common sense and make ‘unwarranted’ assumptions. A wise strategy in the real world but not in exam style questions.
see this comment.
also:
This is covered in Tversky and Khaneman, 1983. Also in conjunction controversy
Matt_Simpson:
You mean the medical question? I’m not at all impressed with that one. This question and the subsequent one about its interpretation were worded in a way that takes a considerable mental effort to parse correctly, and is extremely unnatural for non-mathematicians, even highly educated ones like doctors. What I would guess happened was that the respondents skimmed the question without coming anywhere near the level of understanding that T&K assume in their interpretation of the results.
Again, when thinking about these experiments, one must imagine realistic people in a realistic setting, who are extremely unlikely to be willing to grapple with semantic subtleties that go beyond what they’ll gather with effortless casual skimming of the questions and offhand answers given without much thought.
Do you mean this one (from Conjunction Controversy):
In what way is this question difficult for a doctor to parse? Give the subjects a little credit here.
Also note this about the Linda problem (also from Conjunction Controversy):
Matt_Simpson:
Yes, that’s the one.
In my experience, outside of some very exceptional situations like e.g. in-depth discussions of complex technical subjects, the overwhelming majority of people, including highly educated people, simply don’t operate under the assumption that language should be understood in a logically precise and strict way. The standard human modus operandi is to skim the text in a very superficial manner and interpret it according to intuitive hunches and casual associations, guided by some strong preconceptions about what the writer is likely to be trying to say—and it’s up to you to structure your text so that it will be correctly understood with such an approach, or at least to give a clear and prominent warning that it should be read painstakingly in an unnaturally careful and literal way.
Some people are in the habit of always reading in a precise and literal way (stereotypically, mathematicians tend to be like that). I am also like that, and I’m sure many people here are too. But this is simply not the way the great majority of people function—including many people whose work includes complex and precise formal reasoning, but who don’t carry over this mode of thinking into the rest of their lives. In particular, from what I’ve seen, doctors typically don’t practice rigid formal reasoning much, and it’s definitely not reasonable to expect them to make such an effort in a situation where they lack any concrete incentive to do so.
I thought about this post for awhile—partially because I’ve just been too busy for LW the past few days—and I’m still pretty skeptical. I general, I think you’re right—people don’t closely read much of anything, or interpret much literally. I’ve seen enough economic experiments to know that subject rarely have even a basic grasp of the rules coming into the experiment, and only when the experiment begins and they start trying things out do they understand the environment we put them in.
However, in the heuristics and biases experiments what subjects are reading is only a couple of sentences long. In my exerpience, people tend to only skim when what they’re reading is long or complicated. So I find it fairly hard to believe that most people aren’t reading something like the Linda problem close enough to understand it—especially undergraduates at high end universities and trained doctors.
OTOH, I’m open to any evidence you have
Matt_Simpson:
My thinking about this topic is strongly influenced by my experiences from situations where I was in charge of organizing something, but without any formal authority over the people involved, with things based on an honor system and voluntary enthusiasm. In such situations, when I send off an email with instructions, I often find it a non-trivial problem to word things in a such a way that I’ll have peace of mind that it will be properly understood by all recipients.
In my experience, even very smart people with a technical or scientific background who normally display great intelligence and precision of thought in the course of their work will often skim and misunderstand questions and instructions worded in a precise but unnatural way, unless they have an incentive to make the effort to read the message with extra care and accuracy (e.g. if it’s coming from someone whose authority they fear). Maybe some bad experiences from the past have made me excessively cautious in this regard, but if I caught myself writing an email worded the same way as the doctors’ question by T&K and directed at people who won’t be inclined to treat it with special care—no matter how smart, except perhaps if they’re mathematicians—I would definitely rewrite it before sending.
Why would you think that subjects are working from a different state of information for the two possibilities in the Linda question? Here’s the question again as the subjects read it:
After reading the question, the probability of (a) and (b) is evaluated—with the same state of information: the background knowledge (E in your terms), that someone told us (a) (A), and that someone told us (b) (A). So formally, the two probabilties are:
P(A | E, “someone told us both A and B”)
P(B | E, “someone told us both A and B”)
So the conjunction rule still holds. Now it’s certainly possible that subjects are interpreting the question in the way you suggest (with different states of information for A and B), but it’s also possible that they’re interpreting it in any number of incorrect ways. They could think it’s a thinly veiled question about how they feel about feminism, for example. So why do you think the possible interpretation you raise is plausible enough to be worrisome?
note: this comment was scrapped and rewritten immediately after it was posted
Why would someone tell us “Linda is a bank teller and Linda is a bank teller and active in the feminist movement.”? That would be indeed a strange sentence.
ETA: Maybe the parent comment can be formulated more clearly in the following way (using frequentist language): People parse the discussed question not as what fraction of people from category E belong also into category A?, but rather what fraction of people telling us that a person (who certainly belongs to E) belongs also to A speak truth?, or even better, what fraction of individual statements of the described type is true?
Although A may be proper subset of B, statements telling A about any particular Linda aren’t proper subset of statements telling B about her. Quite contrary, they are disjoint. (That is, people tend to count frequencies of statements of given precise formulation, i.e. don’t count each occurence of B as a simultaneous occurence of A, even if B can be reanalysed as A and C. Of course, I am relying on my intuition in that and can be guilty of mind projection here.)
It is entirely possible to imagine that among real world statements about former environmental activists, the exact sentence “she is a bank teller” is less often true than the exact sentence “she is a bank teller and an active feminist”. I am quite inclined to believe that more detailed information is more often true than less detailed one, since the former is more likely to be given by informed people, and this mechanism may have contributed to evolution of heuristics which produce the experimentally detected conjunction fallacy.