“an explanation for MLE” I used to get by on, “it’s the parameters that make the data most likely”, like it says on the name. I think that’s what you are after

“an explanation for MLE” I used to get by on, “it’s the parameters that make the data most likely”, like it says on the name. I think that’s what you are after. Then I took a stats class, and I know to say, “the MLE is minimum variance within the class of asymptotically unbiased estimators” … that is, “efficient” and “asymptotically consistent“ in the jargon. (Subject to caveats.) Then I took a Bayesian stats class and learned to say, “it’s minimum risk under a (improper) uniform prior.” I also recall there is a general result showing that any estimator which makes the score function zero has good properties with respect to average loss. So zero’ing the score by maximizing likelihood is a good strategy. (If someone could remind me of specifics, that would be great.) But perhaps Gauss had it right when he exploited the (known, but yet un-named) central limit theorem and used how easy it is to maximize the quadratic that sits atop its “e”. (https://arxiv.org/pdf/0804.2996 https://arxiv.org/pdf/0804.2996, page 3, top). It’s so easy we had to find a justification for using it?