# Name of maximum of integrated likelihood?

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 $\hat{\alpha}= \text{argmax} \,L(\alpha)$?