Integrating out an extra parameter in Maximum Liklihood estimation In estimation theory I have seen maximum likelihood being used assuming additive Gaussian AWGN where the signal is a function of multiple parameters (like frequency, time delay, phase, bit).
Sometimes you aren't interested in finding the value of a subset of the parameters (like the bit) and so I have seen them being integrated out by taking the expected value of the MLE over the specific parameter. 
I'm wondering how do I interpret this? What is actually going on? 
 A: This is a common problem in maximum likelihood estimation.  It arises whenever you have a likelihood function $L_\mathbf{x}(\theta, \psi)$ where $\theta$ is the parameter of interest and $\psi$ is a nuisance parameter.  In this case, one method of finding the MLE for the parameter of interest is to first find the conditional maximising value:
$$\hat{\psi}(\theta) = \underset{\psi \in \Psi}{\text{arg max }} L_\mathbf{x}(\theta, \psi).$$
By substitution, you then form the profile likelihood function:
$$\bar{L}_\mathbf{x}(\theta) = L_\mathbf{x}(\theta, \hat{\psi}(\theta)) = \underset{\psi \in \Psi}{\text{max }} L_\mathbf{x}(\theta, \psi).$$
You then obtain the MLE for $\theta$ by maximising this latter function:
$$\hat{\theta} = \underset{\theta \in \Theta}{\text{arg max }} \bar{L}_\mathbf{x}(\theta).$$
In terms of interpretation, what is happening is that you are maximising a multivariate function by first forming a smaller function over the "ridge" of values that maximise with respect to the nuisance parameter.  You first find the maximising value of the nuisance parameter $\psi$ for each fixed $\theta$ and then you form the profile likelihood as a new function that goes over the "ridge" of these maximised values for each $\theta$.  By maximising that latter function with respect to $\theta$ you obtain the maximising point of the original multivariate function.
As an analogy, suppose you have a mountain range over a square area (when looked at from above) and you want to find the highest point.  First you look at the mountain range from West-to-East and you see a horizon of maximising values where the ridges of the mountain are at their highest point (going North-to-South).  You trace the path of that ridge over the mountain range, and this gives you a squiggly line going from North-to-South.  Now you ignore the whole mountain range and just look at the ups and downs of the mountain over that single path, and you find the highest point along that path.  Stepping back and looking at the whole mountain range again, that point will be the highest point in the whole range.
