The idea
Given data and a parametric family, choose the parameter value under which the observed data is most probable.
In practice maximise the log-likelihood, since independence makes the likelihood a product and logs turn it into a sum that differentiates cleanly.
Worked example
For n independent normal observations, maximising the log-likelihood gives:
- Mean: the sample mean. Unbiased.
- Variance: the sum of squared deviations divided by n, not n-1.
That second result is the classic interview point. The MLE of variance is biased downward, which is exactly why the usual sample variance divides by n-1 - Bessel's correction restores unbiasedness at the cost of no longer being the MLE.
Being able to say both facts, and that they are in tension, is the complete answer.
Properties
Consistency - converges to the true value as n grows. Asymptotic efficiency - achieves the smallest possible variance asymptotically (the Cramer-Rao bound). Invariance - the MLE of a function of a parameter is that function of the MLE, which is a genuinely useful shortcut.
Not necessarily unbiased in finite samples, as above.
Where it fails
Model misspecification. MLE finds the best parameters within the family you assumed. If returns are heavy-tailed and you assume normality, you get a confidently wrong answer - and the confidence intervals will be wrong too.
Small samples. Asymptotic guarantees do not apply, and bias can be substantial.
Unbounded likelihoods. Mixture models can drive the likelihood to infinity by collapsing a component onto a single point, so the maximum is degenerate.
Relation to other methods
Least squares regression is maximum likelihood under normal errors - which is why OLS is optimal when that assumption holds and unmotivated when it does not.
MLE is also the Bayesian posterior mode under a flat prior, which is a useful bridge if a follow-up heads toward Bayesian inference.
Practise in statistics.