Relationship Between "Profile Likelihood" and "EM Algorithm"? I was reading Rao (2017) (Ch3) on profile likelihood.  An example is provided which shows how the parameters of a Weibull Distribution can be estimated using the "profile likelihood approach":

It seems here what they are doing :

*

*They are able to re-arrange the original likelihood so that its now only in terms of "alpha". This re-arranged likelihood is now called the "profile likelihood"


*Then, they take the derivative of these re-arranged likelihood, set it to zero, and obtain a formula for the optimal value of "thetha" in terms of "alpha"


*Then, they go back to the original likelihood and replace "thetha" with this "optimal formula", and obtain an formula for the value optimal value of "alpha" based on thetha
Supposedly, using this approach (i.e. Profile Likelihood) can save on computational time - but I had the following question:
This procedure in some ways reminds of the EM Algorithm (different aspects of the likelihood function are sequentially maximized) - but in the case of the EM Algorithm, the EM algorithm is not guaranteed to provide parameter estimates that are "globally optimum". Thus, in the case of Profile Likelihood - do we know if:

*

*The parameter estimates provided from maximizing the original likelihood will always be equal to the parameter estimates provided from maximizing the profile likelihoods?


*If 1) is not true - do we expect that the estimates provided from one of these approaches will necessarily "dominate" the estimates provided from the other approach?
Thanks!
Note: I think the EM algorithm is used in situations when re-arranging the original likelihood in terms of the parameters is not always possible - but I am not sure about this.

The profile likelihood approach is essentially just splitting a multivariate optimisation into two sequential parts.  Given a likelihood function $\mathcal{L}_\mathbf{x}: \mathcal{A} \times \Theta \rightarrow \mathbb{R}_+$ we can write the maximum as:
$$\max_{\alpha \in \mathcal{A} \\ \theta \in \Theta} \mathcal{L}_\mathbf{x}(\alpha, \theta)
= \max_{\alpha \in \mathcal{A}} \mathcal{L}_\mathbf{x}(\alpha, \hat{\theta}(\alpha)),$$
where we use the conditional maximiser:
$$\hat{\theta}(\alpha) = \underset{\theta \in \Theta}{\text{arg max}} \ \mathcal{L}_\mathbf{x}(\alpha, \theta)
\quad \quad \quad \quad \quad 
\text{for all } \alpha \in \mathcal{A}.$$
The function $\alpha \mapsto \mathcal{L}_\mathbf{x}(\alpha, \hat{\theta}(\alpha))$ in the former expression is the profile likelihood function, and it depends on a previous step of finding the conditional maximiser for the nuisance parameter $\theta$.  So long as the relevant maxima exist, the profile likelihood method will always properly define the MLE.
The profile likelihood approach differs from the EM algorithm, but they have some heuristic similarities.  In the profile likelihood method the goal is to split the optimisation based on two sets of parameter values whereby we conduct a conditional optimisation in one parameter and then conduct the marginal optimisation in the other.  In the EM algorithm we are instead concerned with problems where the goal is to split the optimisation based on two sets of sample vectors, one of which is an unobserved "latent variable".  This latter algorithm is used in cases where the marginal likelihood (integrating out the latent variable) is intractable, but the full likelihood is simpler.
