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