Base rate fallacy

# Base rate fallacy

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Encyclopedia
The base rate fallacy, also called base rate neglect or base rate bias, is an error that occurs when the conditional probability
Conditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...

of some hypothesis H given some evidence E is assessed without taking into account the "base rate
Base rate
In probability and statistics, base rate generally refers to the class probabilities unconditioned on featural evidence, frequently also known as prior probabilities...

" or "prior probability
Prior probability
In Bayesian statistical inference, a prior probability distribution, often called simply the prior, of an uncertain quantity p is the probability distribution that would express one's uncertainty about p before the "data"...

" of H and the total probability of evidence E.

## Example

In a city of 1 million inhabitants there are 100 known terrorists and 999,900 non-terrorists. The base rate
Base rate
In probability and statistics, base rate generally refers to the class probabilities unconditioned on featural evidence, frequently also known as prior probabilities...

probability of one random inhabitant of the city being a terrorist is thus 0.0001 and the base rate
Base rate
In probability and statistics, base rate generally refers to the class probabilities unconditioned on featural evidence, frequently also known as prior probabilities...

probability of a random inhabitant being a non-terrorist is 0.9999. In an attempt to catch the terrorists, the city installs a surveillance camera with automatic facial recognition software
Facial recognition system
A facial recognition system is a computer application for automatically identifying or verifying a person from a digital image or a video frame from a video source...

. The software has two failure rates of 1%:
1. if the camera sees a terrorist, it will ring a bell 99% of the time, and mistakenly fail to ring it 1% of the time (in other words, the false-negative rate is 1%).
2. if the camera sees a non-terrorist, it will not ring the bell 99% of the time, but it will mistakenly ring it 1% of the time (the false-positive rate is 1%).

So, the failure rate of the camera is always 1%.

Suppose somebody triggers the alarm. What is the chance they are a terrorist?

Someone making the 'base rate fallacy' would incorrectly claim that there is a 99% chance that they are a terrorist, because 'the' failure rate of the camera is always 1%. Although it seems to make sense, it is actually bad reasoning. The calculation below will show that the chances they are a terrorist are actually near 1%, not near 99%.

The fallacy arises from confusing two different failure rates. The 'number of non-terrorists per 100 bells' and the 'number of non-bells per 100 terrorists' are unrelated quantities, and there is no reason one should equal the other. They don't even have to be roughly equal.

To show that they do not have to be equal, consider a camera that, when it sees a terrorist, rings a bell 20% of the time and fails to do so 80% of the time, while when it sees a nonterrorist, it works perfectly and never rings the bell. If this second camera rings, the chance that it failed by ringing at a non-terrorist is 0%. However if it sees a terrorist, the chance that it fails to ring is 80%. So, here 'non-terrorists per bell' is 0% but 'non-bells per terrorist' is 80%.

Now let's go back to our original camera, the one with 'bells per non-terrorist' of 1% and 'non-bells per terrorist' of 1%, and let's compute the 'non-terrorists per bell' rate.

Imagine that the city's entire population of one million people pass in front of the camera. About 99 of the 100 terrorists will trigger the alarmâ€”-and so will about 9,999 of the 999,900 non-terrorists. Therefore, about 10,098 people will trigger the alarm, among which about 99 will be terrorists. So the probability that a person triggering the alarm is actually a terrorist is only about 99 in 10,098, which is less than 1%, and very very far below our initial guess of 99%.

The base rate fallacy is only fallacious in this example because there are more non-terrorists than terrorists. If the city had about as many terrorists as non-terrorists, and the false-positive rate and the false-negative rate were nearly equal, then the probability of misidentification would be about the same as the false-positive rate of the device. These special conditions hold sometimes: as for instance, about half the women undergoing a pregnancy test are actually pregnant, and some pregnancy tests give about the same rates of false positives and of false negatives. In this case, the rate of false positives per positive test will be nearly equal to the rate of false positives per nonpregnant woman. This is why it is very easy to fall into this fallacy: it gives the correct answer in many common situations.

In many real-world situations, though, particularly problems like detecting criminals in a largely law-abiding population, the small proportion of targets in the large population makes the base rate fallacy very applicable. Even a very low false-positive rate will result in so many false alarms as to make such a system useless in practice.

## Mathematical formalism

In the above example, where P(A|B) means the probability of A given B, the base rate fallacy is the incorrect assumption that:

However, the correct expression uses Bayes' theorem
Bayes' theorem
In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....

to take into account the probabilities of both A and B, and is written as:

Thus, in the example the probability is overestimated by more than 100 times, due to the failure to take into account the fact that there are about 10000 times more nonterrorists than terrorists (a.k.a. failure to take into account the 'prior probability' of being a terrorist).

## Findings in psychology

In some experiments, students were asked to estimate the grade point averages (GPAs) of hypothetical students. When given relevant statistics about GPA distribution, students tended to ignore them if given descriptive information about the particular student, even if the new descriptive information was obviously of little or no relevance to school performance. This finding has been used to argue that interviews are an unnecessary part of the college admissions
University admission or college admissions is the process through which students enter tertiary education at universities and colleges. Systems vary widely from country to country, and sometimes from institution to institution....

process because interviewers are unable to pick successful candidates better than basic statistics.

Psychologist
Psychologist
Psychologist is a professional or academic title used by individuals who are either:* Clinical professionals who work with patients in a variety of therapeutic contexts .* Scientists conducting psychological research or teaching psychology in a college...

s Daniel Kahneman
Daniel Kahneman
Daniel Kahneman is an Israeli-American psychologist and Nobel laureate. He is notable for his work on the psychology of judgment and decision-making, behavioral economics and hedonic psychology....

and Amos Tversky
Amos Tversky
Amos Nathan Tversky, was a cognitive and mathematical psychologist, a pioneer of cognitive science, a longtime collaborator of Daniel Kahneman, and a key figure in the discovery of systematic human cognitive bias and handling of risk. Much of his early work concerned the foundations of measurement...

attempted to explain this finding in terms of the representativeness heuristic
Representativeness heuristic
The representativeness heuristic is a psychological term describing a phenomenon wherein people judge the probability or frequency of a hypothesis by considering how much the hypothesis resembles available data as opposed to using a Bayesian calculation. While often very useful in everyday life, it...

. Richard Nisbett has argued that some attributional bias
In psychology, an attributional bias is a cognitive bias that affects the way we determine who or what was responsible for an event or action...

es like the fundamental attribution error
In social psychology, the fundamental attribution error describes the tendency to over-value dispositional or personality-based explanations for the observed behaviors of others while under-valuing situational explanations for those behaviors...

are instances of the base rate fallacy: people underutilize "consensus information" (the "base rate") about how others behaved in similar situations and instead prefer simpler dispositional attribution
Dispositional attribution is the explanation of individual behavior as a result caused by internal characteristics that reside within the individual, as opposed to outside influences that stem from the environment or culture in which that individual is found...

s.

• Bayesian probability
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...

• Data dredging
Data dredging
Data dredging is the inappropriate use of data mining to uncover misleading relationships in data. Data-snooping bias is a form of statistical bias that arises from this misuse of statistics...