Confusion of the inverse

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Confusion of the inverse is the statistical equivalent of the propositional fallacy of affirming the consequent.

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[edit] Background

In statistics there are random variables representing events with uncertain outcomes, such as die tosses. The random variable is a function mapping each possible outcome of the event to a different number; in the die toss case, this would just be the number that was rolled.

A marginal probability (just probability for short) is then assigned to each number in the range of the random variable (e.g., A). If the range is finite or countable, this will be written P(A = x) for some possible outcome x (e.g., P(A = 1) = 1/6 in the case of the die toss).

A conditional probability involves two random variables (e.g., A and B) and gives the probability of a certain outcome for A given a certain outcome for B. It is written P(A = x|B = y).

Suppose, for example, that A represents whether or not it will rain, and B represents the state of the clouds in the sky. P(A = rain|B = overcast) then represents the probability that it will rain given that the sky is overcast.

[edit] The fallacy

The fallacy of confusion of the inverse is to assume that P(A|B) = P(B|A) – that is, the probability of A being true given B (i.e., a hypothesis is true given the evidence) is the same as the probability of B being true given A (the probability of the evidence given a hypothesis). These are not the same thing, for example: The probability that it is cloudy outside given that it is raining does not equal the probability that it is raining given it is cloudy outside. Evidently, there are many times when it can be cloudy without rain, but rain in a completely cloudless sky is considerably rare.

The distinction between these and how to convert between them is used in Thomas Bayes's eponymous 1763 theorem;

P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}

P(A|B) = P(B|A) are equal only in the limited cases where the prior probabilities are the same P(A) = P(B). If this was the case, then you can't draw a new conclusion about your hypothesis from your evidence.

[edit] Examples

  • The proportion of Muslims who are terrorists (i.e., one's probability of being a Muslim given that he is a terrorist) is roughly equal to the proportion of terrorists who are Muslims (i.e., one's probability of being a terrorist given that he is a Muslim).[1]
  • The probability that a species exists given that it evolved (100%) is exactly equal to the probability that a species evolved given that it exists (< 100%).[2]

[edit] Footnotes

  1. Ann Coulter implied this one when she said, "Maybe not all Muslims are terrorists, but all terrorists are Muslims."
  2. This is cited by Answers in Genesis as an example of a fallacious argument against creationism; probably one of their straw men.
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