Sometimes I find that the likelihood function is written as $L(\theta|X)=p(X|\theta)$ while other times $L(\theta|X)\propto p(X|\theta)$.
which is correct?
If $L(\theta|X)\propto p(X|\theta)$, then what is the constant?
2026-04-05 09:32:53.1775381573
Is the likelihood function $L(\theta|X)$ equal to or proportional to $p(X|\theta)$?
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In one sense, it doesn't matter. The absolute value of the likelihood doesn't mean much, it's the likelihood ratio that is what is actually useful.
Typically, you normalize the likelihood function $L(\theta|x)$ by dividing by $L(\theta^*|x) = \max_{\theta} L(\theta|x)$ to get the standard likelihood ratio.
The likelihood ratio as developed above is quite useful. A very powerful/useful asymptotic result called Wilks Theorem defines the null distribution of the log-likelihood ratio. We use it to find the range $\theta$ that cannot be rejected in favor of the MLE.
The most general statement is $L(\theta|x) \propto p(\theta|x)$, since that encompasses but doesn't require, strict equality.