Maximum likelihood estimation

Maximum likelihood estimation finds the parameter values that make the observed data most probable under the assumed model. It produces consistent, asymptotically efficient estimators and underpins methods such as logistic regression.

ByHoang TruongUpdated July 3, 2026

See it move

Loading infographic...

The estimator theta-hat is the value of theta that maximises the log-likelihood, the sum of log f(xi; theta) across all observations. Maximising the log-likelihood is equivalent to maximising the likelihood itself but far easier to compute. As sample size grows, the estimator becomes consistent and asymptotically efficient, and it underlies logistic regression, probit models and survival analysis.

Where it fits

SubjectData Analysis & StatisticsAdvancedTopicEstimation & Sampling DistributionsAdvanced

The formula

LaTeX

\ell(\theta) = \sum_{i=1}^{n} \log f(x_i;\, \theta)

Variables

$ℓ (θ)$: log-likelihood function (dimensionless)
$θ$: parameter vector to be estimated (varies by model)
$f (x_{i}; θ)$: probability density (or mass) of observation xᵢ given θ (dimensionless)
$n$: sample size (dimensionless)

Log-likelihood is used instead of the likelihood product for numerical tractability; maximising both is equivalent.

LaTeX

\hat{\theta} = \arg\max_{\theta}\, \ell(\theta)

Variables

$\hat{θ}$: maximum likelihood estimate of θ (varies by model)
$ℓ (θ)$: log-likelihood function (dimensionless)
$θ$: parameter vector (varies by model)

For normal data, MLE recovers the sample mean and (biased) variance. For logistic regression, numerical optimisation is required.

Check yourself

PracticeCORE

Which of the following best describes what maximum likelihood estimation (MLE) does when fitting a statistical model?

Select an answer to check your understanding.