Why maximum likelihood and not expected likelihood? Why is it so common to obtain maximum likelihood estimates of parameters, but you virtually never hear about expected likelihood parameter estimates (i.e., based on the expected value rather than the mode of a likelihood function)? Is this primarily for historical reasons, or for more substantive technical or theoretical reasons?
Would there be significant advantages and/or disadvantages to using expected likelihood estimates rather than maximum likelihood estimates?
Are there some areas in which expected likelihood estimates are routinely used?
 A: There is an interesting paper proposing to maximize not the observed likelihood, but the expected likelihood Expected Maximum Log Likelihood Estimation. In many examples this gives the same results as MLE, but in some examples where it is different, it as arguably better, or at least different in an interesting way. 
Note that this is a pure frequentist idea, so is different from what is discussed in the other answers, where it is assumed that expectation is of the parameter itself, so some (quasi-)bayesian idea.
One example: Take the usual multiple linear regression model, with normal errors. Then the log-likelihood function is (up to a constant): 
$$ \log L(\beta) = -\frac{n}{2}\log \sigma^2  - \frac1{2\sigma^2} (Y-X\beta)^T (Y-X\beta)
$$ which can be written (with $\hat{\beta}=(X^TX)^{-1} X^T Y$, the usual least-squares estimator of $\beta$)
$$
\left[ -\frac{n}{2\sigma^2}+\frac1{2\sigma^4}(Y-X\hat{\beta})^T(Y-X\hat{\beta})\right]+\frac1{2\sigma^4}(\hat{\beta}-\beta)^T X^T X(\hat{\beta}-\beta)
$$
The second term here is $\frac12 (\frac{\partial \log L}{\partial \beta})^T (X^T X)^{-1} \frac{\partial \log L}{\partial \beta})$ with expectation $\frac{p}{2\sigma^2}$, so the estimating equation for $\sigma^2$ becomes
$$
-\frac{n}{2\sigma^2}+\frac1{2\sigma^4}(Y-X\hat{\beta})^T (Y-X\hat{\beta}) + \frac{p}{2\sigma^4}
$$ where  $p$ is the number of columns in $X$. The solution is the usual bias-corrected estimator, with denominator $n-p$, and not $n$, as for the MLE. 
A: This approach exists and it is called Minimum Contrast Estimator. The example of related paper (and see other references from inside) 
https://arxiv.org/abs/0901.0655
A: The method proposed (after normalizing the likelihood to be a density) is equivalent to estimating the parameters using a flat prior for all the parameters in the model and using the mean of the posterior distribution as your estimator.  There are cases where using a flat prior can get you into trouble because you don't end up with a proper posterior distribution so I don't know how you would rectify that situation here.
Staying in a frequentist context, though, the method doesn't make much sense since the likelihood doesn't constitute a probability density in most contexts and there is nothing random left so taking an expectation doesn't make much sense.  Now we can just formalize this as an operation we apply to the likelihood after the fact to obtain an estimate but I'm not sure what the frequentist properties of this estimator would look like (in the cases where the estimate actually exists).
Advantages: 


*

*This can provide an estimate in some cases where the MLE doesn't actually exist.  

*If you're not stubborn it can move you into a Bayesian setting (and that would probably be the natural way to do inference with this type of estimate).  Ok so depending on your views this may not be an advantage - but it is to me.


Disadvantages: 


*

*This isn't guaranteed to exist either.   

*If we don't have a convex
parameter space the estimate may not be a valid value for the
parameter.

*The process isn't invariant to reparameterization.  Since the process is equivalent to putting a flat prior on your parameters it makes a difference what those parameters are (are we talking about using $\sigma$ as the parameter or are we using $\sigma^2$)

A: One reason is that maximum likelihood estimation is easier: you set the derivative of the likelihood w.r.t. the parameters to zero and solve for the parameters.  Taking an expectation means integrating the likelihood times each parameter.
Another reason is that with exponential families, maximum likelihood estimation corresponds to taking an expectation.  For example, the maximum likelihood normal distribution fitting data points $\{x_i\}$  has mean $\mu=E(x)$ and second moment $\chi=E(x^2)$.
In some cases, the maximum likelihood parameter is the same as the expected likelihood parameter.  For example, the expected likelihood mean of the normal distribution above is the same as the maximum likelihood because the prior on the mean is normal, and the mode and mean of a normal distribution coincide.  Of course that won't be true for the other parameter (however you parametrize it).
I think the most important reason is probably why do you want an expectation of the parameters?   Usually, you are learning a model and the parameter values is all you want.  If you're going to return a single value, isn't the maximum likelihood the best you can return?
