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What do people call the maximum of the integrated likelihood function (i.e. marginal likelihood function)?

This is, suppose that $x_i\stackrel{iid}{\sim} f(\vert\theta)$, $\theta=(\alpha,\beta)$, and $\beta \sim \pi$ is a nuisance parameter. The integrated likelihood of $\alpha$ is

$$L(\alpha) = \int L(\theta)\pi(\beta)d\beta,$$

where $L(\theta)$ is the full likelihood. So, what do people call $\hat{\alpha}= \text{argmax} \,L(\alpha)$?

I have not found references about this.

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  • $\begingroup$ Welcome to our site! I've edited your answer to mention marginal likelihood function so that someone in future who searches on that term rather than "integrated likelihood" can still find this question; feel free to revert if you disagree with this claim. $\endgroup$ – Silverfish Apr 23 '16 at 8:50
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The problem you are analysing is referred to in the literature as "maximum marginal likelihood" estimation, and is usually solved using the EM-algorithm. It arises in cases where you have a specified distribution for the nuisance parameter so that you can integrate it out. This is quite rare since it generally involves a mixture of methodologies where you are using classical methods and want the MLE, but you have a (partial) parameter distribution like in a Bayesian analysis.

There is quite an extensive academic literature on this topic, most of which concentrates on the computational aspects of the problem, but some of which also discusses the context in which the problem arises. Some key papers that will get you started in this field are listed below.

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