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I think I've seen some papers that don't integrate out random effects when computing the likelihood function and have also seen some papers where random effects are integrated out. Does this have something to do with whether a model is Bayesian or frequentist? Why is it integrated out?

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    $\begingroup$ Can you please give an example of what you mean and/or point to relevant papers. When fitting a mixed model often one uses a REML procedure where fixed effects $X$ are "factored out" but I am not sure this relates to your question. Please elaborate it further. $\endgroup$
    – usεr11852
    Mar 19, 2016 at 7:04
  • $\begingroup$ @usεr11852 there are many papers where the random effects are integrated out before going through the inference steps. The one that I'm looking at is this. And the one that doesn't integrate out the random effects is this. In the middle of this paper, there is a 'linear mixed model' section. What is the difference? $\endgroup$
    – Daeyoung
    Mar 19, 2016 at 8:12
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    $\begingroup$ Thank you for this additional information (you probably want to add this to the main body of your question, people routinely skip the comments). I see your point; I believe this relates closely to the particular application. These are essentially different estimation methods which try to encapsulate task specific information. I do not think it has to do anything with being "Bayesian" or "frequentist". We "integrate out" the random effects because we can only determine the joint density of $y$ and $\gamma$. We then integrate the density in respect with $\gamma$ to get the marginal. $\endgroup$
    – usεr11852
    Mar 19, 2016 at 8:36

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We integrate out the random effects to obtain the marginal distribution of the data. You may do this to obtain the sampling distribution of the observation.

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