---
tag: 0FREPFKNS7
kind: concept
name: Maximum likelihood estimation
aliases: [mle, maximum-likelihood-estimator, maximum-likelihood]
msc: [62F10]
uses: [05FSTXM92W, 0TTNRP5PFQ, 0XJVF81QF6, 0QNRBHWV35]
sources:
  - url: https://en.wikipedia.org/wiki/Maximum_likelihood_estimation
    locator: Principles
  - url: https://en.wikipedia.org/wiki/Maximum_likelihood_estimation
    locator: Properties
same_as:
  wikidata: Q1045555
  wikipedia: Maximum likelihood estimation
---

## Summary

Maximum likelihood estimation is a method to build [[estimators|0TTNRP5PFQ]] of unknown parameters.
For observed data $x$, the maximum likelihood estimate is a value $\hat{\theta}$ that maximizes the [[likelihood function|05FSTXM92W]] $\theta \mapsto L(\theta \mid x)$ over the parameter set $\Theta$.
In other words, it is the parameter value under which the observed data are most probable.

## Remarks

The maximizer of $L$ is also the maximizer of the log-likelihood $\ell = \ln L$. For a smooth $\ell$ and an interior maximum, $\hat{\theta}$ solves the likelihood equations $\partial \ell / \partial \theta_j = 0$.
Example: for $k$ successes in $n$ Bernoulli($p$) trials, $\ell(p) = k \ln p + (n - k) \ln(1 - p)$, so $\hat{p} = k/n$.
Example: for an i.i.d. $\mathcal{N}(\mu, \sigma^2)$ sample, $\hat{\mu} = \bar{x}$ and $\hat{\sigma}^2 = \frac{1}{n}\sum_i (x_i - \bar{x})^2$. This estimator of $\sigma^2$ is biased.
Under regularity conditions, the estimator is [[consistent|0XJVF81QF6]] and asymptotically normal. Also, for a function $g$, the maximum likelihood estimate of $g(\theta)$ is $g(\hat{\theta})$.

## Explanations

### L1 human

Maximum likelihood chooses the parameter that makes the observed data most probable.
If a coin shows $7$ heads in $10$ tosses, the maximum likelihood estimate of the probability of heads is $7/10$.