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Sorry about the title not being that clear, but I wouldn't know how to summarize it. My question is a simple one, but it's been bugging me for a while.

In bayesian probability, quite often, we consider the probability density function $f : x \mapsto f(x;\theta)$ of a variable $x$, given a parameter $\theta$.

Now, in some parameter estimation methods (maximum likelihood estimation, for instance):

we take the expression of this function, and treat it as a function of $\theta$ given $x$ instead:

$L : \theta \mapsto L(\theta;x)$ (where $L$ is commonly denoted as the likelihood function).

Now, my question is the following:

Does it make sense to write, on the same line?

$f(x;\theta) = (\text{mathematical expression}) = L(\theta;x)$

Because the left and right hand side are not treated as functions of the same variable, so is there any justification to "rigorously get away with it"? Or is it just a shortcut we use to not have to write the expression twice- whilst keeping in mind it's not a very "legal" move?

Thanks for your answers :)

Azur
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Yes equality makes perfect sense, because you are writing equality of real numbers. Both $f(x;\theta)$ and $L(\theta;x)$ are real numbers, and thus can be equal (and they are, by definition, in your case).

It would be different if you asked whether the functions $f$ and $L$ are equal: they are not equal because the variable are switched.

Captain Lama
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