# Why is the Likelihood function NOT a case of the inverse fallacy?

This may be a trivial question, but as a research psychologist I do not have a robust statistics background to answer it.

It appears to me that the likelihood function--$L(\theta | \text{data}) = P(\text{data} | \theta)$--is committing the inverse fallacy which is exactly what using Bayes' theorem avoids. I'm sure that the logic behind the likelihood is sound, but I can't see why this is NOT a case of incorrectly equating two different conditional probabilities (i.e., the inverse fallacy).

Confusion of the inverse, also called the conditional probability fallacy or the inverse fallacy, is a logical fallacy whereupon a conditional probability is equivocated with its inverse

i.e. this is talking about making the mistake of thinking P(A|B) is the same as P(B|A).

The likelihood is, however, not regarded as a conditional probability in that sense at all. In other words, $L(\theta|\text{data})$ is clearly understood NOT to be $P(\theta|\text{data})$.

(Indeed, as a function of $\theta$, generally it doesn't even integrate to 1! It can't be a probability distribution in that sense.)

When discussing likelihood in probability terms, people always talk about $P(\text{data}|\theta)$ ... that is, the thing it's defined in terms of.

Given the likelihood $L(\theta|\text{data})$ is not taken to be a conditional probability $P(\theta|\text{data})$, in what sense is this the 'inverse fallacy'?

• Thanks. I was making the assumption that L(theta | data) was equivalent to P(theta | data).
– ATJ
Apr 26, 2013 at 0:26

It's not saying $P(\theta|\text{data})$ = $P (\text{data}|\theta)$. It's defining the likelihood function.

• Right, $L(\theta | data ) \ne P( \theta | data )$, the former is just notation for a likelihood. Apr 25, 2013 at 18:45