Remember Linda: we had a description of her and were asked to say whether she was more likely to be a bank employee or a Greenpeace supporter (I recently made a few changes to the post, treating the problem in odds form, which makes it clearer).
Imagine now that we are looking for Linda. We know she is in a hotel where two meetings are taking place: one, in the West wing conference room, is held by the local banking association; the other, in the East wing conference room, is held by the local Greenpeace support group. We have time to visit only one room. Where do we go?
If we knew nothing at all about Linda, it would be natural for us to enquire about the number of attendees in each room. If we were told, for example, that there are 70 people in the first room and 30 people in the second, the first room would be our obvious choice.
But once we have Linda’s description, all we care about, as we have seen, is that Linda looks more like a Greenpeace supporter than a bank employee. Hence we are drawn towards the second room, disregarding the fact that bank employees are much more common than Greenpeace supporters and neglecting to find out the number of attendees in the two meetings.
Let’s now ask: What would happen if we had Linda’s description as well as the number of attendees? Where would we go in this case? Would knowing the Base Rate influence our choice?
This is the Engineers vs. Lawyers problem in Kahneman and Tversky (1973) (see also Chapter 14 of Thinking, Fast and Slow). They ran an experiment where participants were given a description of someone who looked more like an engineer than a lawyer. They told half of the participants that the description was randomly selected from a sample of 30 engineers and 70 lawyers and the other half that it came from a sample of 70 engineers and 30 lawyers. Then they asked all participants to express the probability that the described person was an engineer. Results showed very clearly that Base Rate information was almost completely ignored. Given the description, the median probability estimate of participants who were given a 30/70 Base Rate were very similar to the median estimate of those who were given the opposite 70/30 Base Rate. The same was true if the description was closer to a lawyer than to an engineer and, crucially, if it was a neutral and uninformative description: in that case, the median estimate was 50/50 on both sides. The conclusion was that any evidence, irrespective of its accuracy, can be enough to cause a neglect of Base Rates. Whether people are aware of them or not does not matter: they almost completely ignore them anyway.
The experiment made two notable assumptions:
1) By giving x% as an estimate of the probability of engineer, respondents were tacitly implying that the probability of lawyer was 1-x%.
2) The probability that the person would fit the description, given that he was an engineer, was equal to one minus the probability that he would fit the description, given that he was a lawyer. In our notation, TPR=1-FPR: the description was interpreted as symmetric evidence.
As we know, with symmetric evidence, accuracy equals TPR and, according to Bayes’ Theorem, the relationship between accuracy and posterior probability is as follows:
Figure 1
With BR=50%, the relationship is a 45° line: PP=TPR. But if BR>50% the relationship is concave, and increasingly so as BR tends to 1; and if BR<50% it is convex, and increasingly so as BR tends to 0. K&T’s experiment showed that, by failing to appreciate such non-linearity, people fall prey to the Inverse Fallacy: they tend to assume PP=TPR, irrespective of the Base Rate. So, for example, with TPR=80% – the description is strongly suggestive of an engineer – the correct Bayesian posterior probability is 63% if BR=30%, and 90% if BR=70%; but actual estimates gravitate around 80%. Likewise, with TPR=20% – the description is strongly suggestive of a lawyer – the numbers are 10% and 37% respectively; but estimates gravitate around 20%. And, crucially, with TPR=50% – the description is totally uninformative as to whether the person is an engineer or a lawyer – Bayesian probabilities should equal the Base Rates: 30% and 70%, but actual estimates gravitate around 50%.
Such is the Inverse Fallacy: in looking for Linda, our behaviour would resemble K&T’s low priors group. As long as Linda looks more like a Greenpeace supporter, we would be drawn towards the East wing, even if we knew that there are only 30 people in the room, versus 70 in the other room: we are blinded by evidence. With a strongly accurate description, the Inverse Fallacy would not matter: if, for example, TPR=80%, both the low priors and the high priors Bayesian estimates would be above 50%, hence both groups would look for Linda in the East wing. But with TPR between 50% and 70% – Linda still looks more like a Greenpeace supporter, but not as strongly – a correct Bayesian estimate should lead the low priors group to the West wing, to look for Linda in the more numerous banking meeting. With symmetric evidence, the Bayesian cut-off point is TPR<1-BR. In the graph above, the grey area indicates the region where room choice should be determined by the Base Rate: TPR<70% for low priors; TPR>30% for high priors. The Inverse Fallacy collapses the grey region to a vertical line at 50%: room choice is entirely determined by perceived accuracy.
How strong is the Inverse Fallacy? The curve in the following figure is the locus of low priors-high priors Bayesian pairs for each level of accuracy: (10%,37%) for A=20%, (30%,70%) for A=50%, (63%,90%) for A=80%, and so on. Correct Bayesian estimates lie on the curve. Perfect prior indifference, on the other hand, coincides with the 45° line, where low and high priors estimates coincide.
Figure 2
Figure 1 in K&T’s paper showed five dots very close to the prior indifference line: very strong evidence for the Inverse Fallacy. More recently, Baratgin and Noveck (2000) ran a similar experiment, with Engineers and Lawyers replaced by Math and Literature teachers. They gave 40 participants a description of five teachers: Jacques looked like a math teacher; Anne looked like a literature teacher; Françoise also resembled a literature teacher, but less so than Anne; Paul had a neutral description – neither math nor literature; and Raphaël had a conflicting description – both math and literature. Half of the participants were told that the descriptions came from a low priors 30/70 sample and the other half that they came from a high priors 70/30 sample. All participants were asked the same question: What is the probability that the described person is a math teacher? Average answers (not median, as in K&T) are reported in the following table (extracted from B&N’s Table 1), and as yellow dots in the figure above.
Table 1
As in K&T, the yellow dots were close to the 45° indifference line – again a clear indication that, once they were given the five descriptions, respondents almost completely disregarded Base Rates. The dots were not exactly on the indifference line: there was some difference between the low priors and the high priors average estimates: 65% vs. 73% for Jacques, for example. But they were very distant from the Bayesian curve. Notice that we don’t know the point on the Bayesian curve corresponding to the perceived accuracy of Jacques’s description. We know that Jacques was perceived to be more like a math teacher – i.e. TPR>50% – but we don’t know by how much. The same is true for the other descriptions, except for Paul, where, provided that his description was effectively neutral, we can say that TPR=50%. Paul’s neutral description corresponded to Dick’s description in K&T. However, while the median estimates for Dick were spot on 50/50, the average estimates for Paul were 53% for the low priors group and 59% for the high priors group. This may perhaps be a sign that Paul’s description was not as neutral as it was intended to be (the two descriptions were very similar, except that Dick was portrayed as “a man of high ability” while Paul as “a man of a great intellectual capacity”). Also, the median (not reported in B&N) may be a more accurate estimate than the average. Be that as it may, the neutral description is the point of maximum distance between the Bayesian curve and the prior indifference line: 30% vs. 50% for the low priors group and 70% vs. 50% for the high priors group. At that point, probability estimates should coincide with Base Rates: PP=BR. So there should be no question as to who the described person is: a lawyer/a literature teacher/a bank employee for the low priors group; and an engineer/a math teacher/a Greenpeace supporter for the high priors group. But in K&T participants were perfectly undecided about whether Dick was an engineer or a lawyer, while in B&N even the low priors group was marginally in favour of Paul being a math teacher. To the extent that this is due to a description bias, we can re-centre Paul at 47% and 53%. In that case, taking 47% as the implied Base Rate, Figure 2 becomes:
Figure 3
Near prior indifference would explain most of B&N’s results.
But the authors pursued another line of inquiry: What happens – they asked – if we relax the first of the two assumptions implicit is K&T’s experiment? We’ll see.
