# Difference between “Base Rate Fallacy” and “Confusion of the Inverse Fallacy”

I am learning about fallacies and came accross the Base Rate Fallacy and the Confusion of the Inverse Fallacy.

I can't really distinguish the two. So my question is: are they the same? And if not, what is the difference? Both seem to emerge from the False Positive Paradox.

EDIT: Basically I am looking for a formal definition of the two (using formal logic) and then a rigid comparison.

In terms of non-statistical fallacies, the base-rate fallacy is analogous to generalization from an anecdote, and the confusion of the inverse is analogous to affirming the consequent.

In the base rate fallacy, you focus entirely upon the desired result about which you have the best information, and you ignore all other cases, although they are more numerous or more likely, and end up outweighing the target case by sheer bulk. If you stay home because you live in a terrible neighborhood, where you have significant odds of being attacked by a stranger on the street, in any given meeting and are only a hundredth as likely to be attacked by an individual you have invited into your home, in any given meeting, but you meet the people you invite into your home several hundreds of times as often because you don't go out, you have committed a base-rate fallacy. You have focused on the significant case, without looking at the context. You are not necessarily any safer at home in this case.

If you feel significantly safer avoiding men because a vast majority of murderers are men, but only a tiny number of men are murderers, and those men are unlikely to be among the men you are avoiding, you have committed the fallacy of confusion with the inverse. Your relative comfort is not proportionate to the change in risk, which is very, very small because you have reasoned from the wrong direction of the implication.

The two are indirectly linked because they both indicate how bad humans are at grasping what can happen unexpectedly to a Bayesian ratio.

The base rate fallacy assumes the odds of A (I get attacked) must be proportional to the odds of A|B (I get attacked because I go outside), ignoring to the odds of B (I go outside). But when B is not common, it is unwise to base your estimate of A on what happens to it when B is true.

The confusion with the inverse assumes that the odds of A|B (men murder) must be proportional to the odds of B|A (murders are men). But these two are just not really related in any important way.

• On both wiki pages they are explained in terms of Bayes. I get that point, but both wiki pages seem to make the same point. What is the difference of these two fallacies in terms of Bayes? – GambitSquared Aug 1 '17 at 15:45
• Put the response to the comment into the answer. – user9166 Aug 1 '17 at 19:26

The base rate fallacy and the confusion of the inverse fallacy are not the same. The base rate fallacy is related to base rate, so let’s first clear about base rate. Base rate is an unconditional (or prior) probability that relates to the feature of the whole class or set. These are examples of the base rate: the probability that a randomly chosen person is an Asian in California is 13% (https://en.wikipedia.org/wiki/Demography_of_California); the admission rate of Stanford is 5% (https://www.usnews.com/best-colleges/rankings/lowest-acceptance-rate). The base rate fallacy is committed when a person arrives at a conclusion without taking consideration of the base rate. So if someone says, “Only 5% of applicants make it into Stanford University, but my daughter is brilliant! So she will certainly be accepted by the school." However smart his daughter is, given the extremely low acceptance rate, the person’s optimism is unwarranted as it ignores the base rate.

Confusion of the Inverse Fallacy is related to misunderstanding conditional probabilities. The conditional probability of A given B is different from the conditional probability of B given A, mathematically.

For example, let A be the positive test result for cancer and B be having the cancer. The probability of A given B will be very high (e.g., P(A|B) = .9). On the other hand, the probability of B given A will be significantly lower since the test result can be due to the false positive (e.g., P(B|A)=.2). Similarly, the probability of a person coughing given that she has cold is high, but the probability that she has cold given that she coughs is much lower. Confusion of the Inverse Fallacy occurs when a person thinks that there is no difference between the probability of A given B and the probability of B given A, thus attributing the same probability on both cases.

These two fallacies are related to Bayes’ Theorem since the posterior probability is calculated by means of likelihood and prior probability.

• Isn't it true that the Base Rate of people having the cold is low; and therefore if someone coughs it is likely that something else caused the cough? What I mean to say is: both fallacies seem to come from the same error in reasoning, namely underestimating the base-rate and as a consequence overestimating the opposite of the base-rate (does that have a name?) – GambitSquared Aug 1 '17 at 22:03
• their difference can be best explained by the Bayesian inference ideal which states that we should update our belief based on Bayes' thm: posterior probability is proportional to prior probability and likelihood. the base rate fallacy ignores the prior probability, and the confusion fallacy mistakes posterior probability with likelihood. – Nanhee Byrnes PhD Aug 1 '17 at 23:46
• Since I see a clear difference between the two fallacies, I do not find any further discussion useful. Maybe you should post your own answer that explains why the two are the same. – Nanhee Byrnes PhD Aug 2 '17 at 15:29
• @ImreVégh The base rate fallacy does not assume P(B|A) ~ P(A|B). Its false answer is not about any conditional probability at all. it assumes that P(A) ~ P(A|B), even when B is small. It misestimates the probability of the focal outcome by misusing a conditional probability. – user9166 Aug 2 '17 at 23:43
• @ImreVégh Both fallacies ignore two terms. If you want to write Bayes theorem as P(B|A) x P(A) = P(A|B) x P(B), then they both omit P(B) but the confusion of the inverse also omits P(A) while the base rate fallacy omits P(B|A). – user9166 Aug 2 '17 at 23:52

Base Rate Fallacy:

This occurs when you estimate P(a|b) to be higher than it really is, because you didn’t take into account the low value (Base Rate) of P(a).

Example 1: Even if you are brilliant, you are not guaranteed to be admitted to Harvard: P(Admission|Brilliance) is low, because P(Admission) is low.

Example 2: The chance that you are an astronaut given that you live in the USA P(Astronaut|USA) is low, because the Base-Rate of Astronauts P(Astronaut) is very low.

Confusion of the Inverse Fallacy:

We often come to such false conclusions because we see that the inverse is true. Most people who are admitted to Harvard are brilliant and most astronauts live in the USA.

P(USA|Astronaut)>>P(~USA|Astronaut)

However, this does not imply that if you are brilliant you will be admitted to Harvard or if you live in the USA you are an Astronaut.

When this occurs you are falsely assuming that P(a|b)=P(b|a) while P(a|b) and P(b|a) are unrelated.

Are they really different?

When the base-rates are equal P(a)=P(b) then P(a|b)=P(b|a). So if there are an equal amount of Americans and Astronauts then the following is valid: if most Astronauts are American then most Americans are Astronaut.

Therefore, another way to explain the Confusion of the Inverse Fallacy is by seeing that you falsely conclude that the base-rates are equal, while in reality one is bigger than the other. In other words: in both fallacies you don't take the Base Rate into account. Hence the confusion of the OP.

My conclusion is therefore that the Confusion of the Inverse Fallacy is a variant of the Base Rate Fallacy, which is more general.

• Good explanation! I did not like the wiki page explanations which you mentioned since they make the fallacies unnecessarily complicated (and thus confusing). Happy to see that you came up with better explanations than wiki. – Nanhee Byrnes PhD Aug 4 '17 at 15:25