Confusion of the inverse

Confusion of the inverse (also fallacy of the transposed conditional) is the statistical equivalent of the propositional fallacy of affirming the consequent. Essentially it is confusing the difference between the probability of a set of data given a hypothesis, and the probability of a hypothesis given a set of data.

The fallacy is a formal fallacy.

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., $$P(A=1)=\tfrac16$$ 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. $$P(A=\text{rain}|B=\text{overcast})$$ then represents the probability that it will rain given that the sky is overcast.

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. If our hypothesis is that it is raining we can be almost certain that we will observe clouds in the sky, however if we observe clouds in the sky we cannot say that it is almost certainly raining. So the probability of our data (observing clouds) is nearly 100 percent given our hypothesis (that it is raining), but our hypothesis (that it is raining) is not a 100 percent given only our data (observing clouds).

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.

In order to relate the probability of our data given our hypothesis, to the probability of the hypothesis itself we require additional information. We need to know the probability of our hypothesis compared to alternative hypotheses before the data was collected. We can then use our observations to update the probability of our main hypothesis. Bayesian statistics and Bayes' equation is the main method of doing this, however, most statistical analysis is usually frequentist and so any attempt to relate the calculated probabilities to the original hypothesis is a fallacy of the transposed conditional.

Examples

 * The proportion of Muslims who are terrorists (i.e., one's probability of being a terrorist given that he is a Muslim) is roughly equal to the proportion of terrorists who are Muslims (i.e., one's probability of being a Muslim given that he is a terrorist).
 * 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%).
 * Probability that a die roll's result is less than seven given that the die is six-sided is equal to the probability that a die is six-sided given that a single roll resulted in less than seven.
 * If most criminals are members of group X, most members of group X must be criminals.