I've been trying to get my head around why a likelihood isn't a probability density function. My understanding says that for an event $X$ and a model parameter $m$:


$P(X\mid m)$ is a probability density function


$P(m\mid X)$ is not


It feels like it should be, and I can't find a clear explanation of why it's not. Does it also mean that a Likelihood can take a value greater than 1?



Why isn't likelihood a probability density function?
If $X$ is data and $m$ are the parameters, then the likelihood function $l(m) = p(X | m)$. I.e. it's $p(X | m)$, considered as a function of $m$. 


Both $p(X|m)$ and $p(m|X)$ are pdfs: $p(X|m)$ is a density on $X$ and $p(m|X)$ is a density on $m$. But the likelihood is $p(X|m)$, not as a function of $X$ (it would indeed be a density as a function of X), but as a function of m. So it's not a pdf; in particular, it's not necessarily true that $$\sum\_m p(X|m) = 1.$$


Edit: just to clarify, $p(m|X)$ isn't the likelihood. $p(X|m)$ is.


