What is the objective of maximum likelihood estimation?

Question

I was reading in "Using Maximum Likelihood Estimation" from "Econometrics for Dummies", and here's what the author had to say: "The objective of maximum likelihood (ML) estimation is to choose values for the estimated parameters (betas) that would maximize the probability of observing the Y values in the sample with the given X values. This probability is summarized in what is called the likelihood function. " (Source: http://www.dummies.com/how-to/content/using-maximum-likelihood-ml-estimation.html)

He seems to be saying that we want to find the parameters that make Y most likely to occur given an X value, but I thought the objective might be to find the parameters that most likely reveal the true P(Y|X).

Answer 1

The objective is to estimate the parameters or, more precisely, to get a method for their estimation (since the same form of likelihood can be applied to different data sets).

There are different ways to choose parameter estimators - maximum likelihood is just one of them, which uses as the criteria for choosing the estimator that the probability of getting the observed result is maximal. Maximum likelihood estimators have many convenient mathematical properties.

$P(Y|X, \theta)$ is a function relating the predictor variables $X$ and the output variables $Y$, parametrized by parameters $\theta$. Its functional form is chosen a priori and limits how close it can be to the "true" distribution (if such a "true" distribution exists at all): e.g., normal/Gaussian function can well approximate many distributions (gamma distribution, lognormal, etc.) but it will never reveal that the underlying distribution is not normal.

Answer 2

Oh, I think I understand a bit better now. The objective is to find parameters that maximize the likelihood that our observations will be similar if we take another, similar, sample. For example, let's say that we draw 5 marbles from a bag and 3 of them are black. We then place them back in the bag. Our objective, then, is to figure out the fraction of black marbles in the bag that would make us most likely to see 3 black marbles the next time we draw 5.