Say you have a likelihood function $L(X|\theta_1,\theta_2)$ and you marginalise out $\theta_1$ and then take the log of the marginalised likelihood function. Would this be the same as taking the log and then marginalising?
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I think what you're asking is under which circumstances logarithm and integration are exchangeable. There's a related question here. – Durden Jan 07 '24 at 16:41