Maximum Likelihood Estimation › Advantages

Maximum likelihood is a general construction rather than a single formula. Once we can write down a probability model for the data, we can form a likelihood, take its logarithm, and optimize. The same recipe produces the sample proportion for Bernoulli data, the sample mean and variance for Gaussian data, logistic regression coefficients for binary outcomes, and many survival and time-series models. This consistency is valuable because you do not need a new estimation principle for every model family.

The method also gives us more than a point estimate. The first derivative of the log-likelihood, called the score, tells us where the likelihood is flat. The second derivative, or curvature, tells us how sharply the likelihood peaks. That curvature produces approximate standard errors and confidence intervals. In regular models, the maximum likelihood estimator is asymptotically normal and asymptotically efficient. Under the regular asymptotic theory, it attains the inverse-information variance bound within the class of regular estimators. You should read that as a large-sample statement, not a promise that every finite-sample estimate is unbiased or precise.

Finally, maximum likelihood connects statistical modeling to machine learning losses. Gaussian likelihood leads to squared error, Bernoulli likelihood leads to log loss, and Poisson likelihood leads to a deviance loss. This makes likelihood a bridge between probability models, optimization objectives, and predictive evaluation. When we see a loss function in a machine learning model, it often has a likelihood interpretation behind it.