Is the likelihood function $L(\theta|X)$ equal to or proportional to $p(X|\theta)$? Sometimes I find that the likelihood function is written as $L(\theta|X)=p(X|\theta)$ while other times $L(\theta|X)\propto p(X|\theta)$ where it is mentioned that $L(\theta|X)= K\times p(X|\theta)$ for some positive contant $K$.
which is correct?
If $L(\theta|X)\propto p(X|\theta)$, then what is the constant?

In the context of MLE, I would write that $L(\theta\lvert X)=p(X\lvert\theta)$. But I think that there could be other contexts/interpretations of $L$ where the other one is true. Maybe if you link the concept of MLE to bayesian statistics ? Then $K$ would be $\frac{P(\theta)}{P(X)}$.
EDIT:
Your source states that $K$ is arbitrary. So I think that they are just defining a likelihood measure in order to compare different hypotheses. Do you have other sources where the proportionality factor is defined as "arbitrary"?
