maximum likelihood estimation

Imagine you are handed a die and you roll it 60 times, getting a six 30 times. Someone asks: was this a fair die, or a loaded one? Common sense says a die rigged to favor sixes would explain this data far better than a fair die. Maximum likelihood estimation turns that common sense into a method: among all the candidate settings of the unknown parameters, pick the setting under which the data you actually observed was the most probable. You choose the explanation that makes your data least surprising.

Concretely, you write the likelihood — the probability (or density) of the observed data, viewed as a function of the unknown parameters — and then find the parameter values that maximize it. Because multiplying many small probabilities is awkward, in practice you maximize the log-likelihood (the sum of the logs), which has its peak at the same place. For a coin flipped n times with k heads, the likelihood is maximized at p = k / n, matching intuition exactly. For harder models a computer climbs the log-likelihood surface numerically. The maximizing values are the maximum likelihood estimates.

Maximum likelihood is the workhorse of modern actuarial estimation: it is how mortality rates, loss-severity parameters, and the coefficients of a generalized linear model are fitted. Its appeal is deep — under broad conditions MLEs are consistent, asymptotically efficient (no estimator does better for large samples), and come with a built-in formula for their standard errors via the curvature of the log-likelihood. The honest caveats: it can be biased in small samples, it trusts your chosen model fully (a wrong distribution gives confidently wrong answers), and the numerical search can stall on the wrong peak.