7
$\begingroup$

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).

$\endgroup$
9
$\begingroup$

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'?

$\endgroup$
  • $\begingroup$ Thanks. I was making the assumption that L(theta | data) was equivalent to P(theta | data). $\endgroup$ – ATJ Apr 26 '13 at 0:26
7
$\begingroup$

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

$\endgroup$
  • 2
    $\begingroup$ Right, $L(\theta | data ) \ne P( \theta | data )$, the former is just notation for a likelihood. $\endgroup$ – Cam.Davidson.Pilon Apr 25 '13 at 18:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.