Does MLE always mean we know the underlying PDF of our data, and does EM mean we don't? I have some simple conceptual questions that I would like clarified regarding MLE (Maximum Likelihood Estimation), and what link it has, if any, to EM (Expectation Maximization). 
As I understand it, if someone says "We used the MLE", does that automatically mean that they have an explicit model of their data's PDF? It seems to me that the answer to this is yes. Put another way, if at any time someone says "MLE", it is fair to ask them what PDF they are assuming. Would this be correct?
Lastly, on EM, my understanding is that in EM, we do not actually know - or need to know, the underlying PDF of our data. This is my understanding. 
Thank you.
 A: MLE requires knowledge of at least the marginal distributions. When using MLE, we usually estimate the parameters of a joint distribution by making an iid assumption, then factoring the joint distribution as a product of the marginals, which we know. There are variations, but this is the idea in most cases. So MLE is a parametric method.
The EM algorithm is a method for maximizing the likelihood functions that come up as part of a MLE algorithm. It is often (usually?) used for numerical solutions.
Whenever we use MLE, we need at least the marginal distributions, and some assumption about how the joint is related to the marginals (independence, etc.). Therefore both methods rely on knowledge of distributions.
A: The MLE method can be applied in cases where someone knows the basic functional form of the pdf (e.g., it's Gaussian, or log-normal, or exponential, or whatever), but not the underlying parameters; e.g., they don't know the values of $\mu$ and $\sigma$ in the pdf: $$f(x|\mu, \sigma) = \frac{1}{\sqrt{2\pi\sigma^{2}}} \exp\left[\frac{-(x-\mu)^{2}}{2 \sigma^{2}}\right]$$  or whatever other type of pdf they are assuming.  The job of the MLE method is to choose the best (i.e., most plausible) values for the unknown parameters, given the particular data measurements $x_{1}, x_{2}, x_{3}, ...$ which were actually observed.  So to answer your first question, yes, you are always within your rights to ask someone what form of pdf they are assuming for their maximum likelihood estimate; indeed, the estimated parameter values that they tell you aren't even meaningful unless they first communicate that context.
The EM algorithm, as I've seen it applied in the past, is more of a sort of meta algorithm, where some of the metadata is missing, and you have to estimate that also. So, for example, perhaps I have a pdf which is a mixture of several Gaussians, e.g.: $$ f(x|A_{1},...,A_{N},\mu_{1},...,\mu_{N}, \sigma_{1},...\sigma_{N}) = \sum_{k=1}^{N} \frac{A_{k}}{\sqrt{2\pi\sigma_{k}^{2}}} \exp\left[\frac{-(x-\mu_{k})^{2}}{2 \sigma_{k}^{2}}\right] $$  Superficially, except for the addition of the amplitude parameter $A_{k}$, this looks a lot like the previous problem, but what if I told you that we also don't even know the value of $N$ (i.e, the number of modes in the Gaussian mixture) and we want to estimate that from the data measurements $x_{1}, x_{2}, x_{3}, ...$ too?
In this case, you have a problem, because each possible value of $N$ (this is the "meta" part that I was alluding to above) really generates a different model, in some sense.  If $N=1$, then you have a model with three parameters ($A_{1}$, $\mu_{1}$, $\sigma_{1}$) whereas if $N=2$, then you have a model with six parameters ($A_{1}$, $A_{2}$, $\mu_{1}$, $\mu_{2}$, $\sigma_{1}$, $\sigma_{2}$).  The best fit values that you obtain for ($A_{1}$, $\mu_{1}$, $\sigma_{1}$) in the $N=1$ model can't directly be compared to the best fit values that you obtain for those same parameters in the $N=2$ model, because they are different models with a different number of degrees of freedom.
The role of the EM algorithm is to provide a mechanism for making those types of comparisons (usually by imposing a "complexity penalty" that prefers smaller values of $N$) so that we can choose the best overall value for $N$.
So, to answer your original question, the EM algorithm requires a less precise specification of the form of the pdf; one might say that it considers a range of alternative options (e.g., the option where $N=1$, $N=2$, $N=3$, etc.) but it still requires you to specify something about the basic mathematical form of those options--you still have to specify a "family" of possible pdfs, in some sense, even though you are letting the algorithm decide for you which "member" of the family provides the best fit to the data.
