# Is this a correct interpretation of maximum likelihood estimation?

Here is an excerpt from Pattern Recognition and Machine Learning by Christopher Bishop:

This seems to be not quite right—"the probability of the data set", when the data set is drawn from a Gaussian (or any continuous) distribution, is zero. Right?

The book goes on to explain how maximum likelihood estimation finds the parameters that maximize "the probability of the data". But that probability is always zero, so what are we actually maximizing?

It is clear that what the author means to say is that we are maximizing the joint probability density of the observations—that's actually what is expressed by $p(\mathbf{x}|\mu, \sigma^2)$ above. But that's a little unsatisfying to me—it would be nice to have a statement in terms of probability, not probability density.

I am thinking that maybe the correct way to put it is something like this: the maximum likelihood solution finds parameters such that there exists a neighborhood $M$ centered at $\mathbf{x}$ where for any neighborhood $N\subset M$ centered at $\mathbf{x}$, $P(\mathbf{x}\in N)$ is maximized.

But I just made that up. Is it true? Is it a good way to think about what maximum likelihood is actually finding?