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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?

user3911153
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  • Formally, the former is the likelihood function. The latter would not define a single function, but rather an equivalence class of functions. A special member of that equivalence class, the one in which all multiplicative constants w.r.t. $\theta$ are omitted, is called the likelihood kernel. – statmerkur Jul 22 '22 at 23:54

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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"?

FP0
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