{
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  "body": "\n## Summary\n\nMaximum likelihood estimation is a method to build [[estimators|0TTNRP5PFQ]] of unknown parameters.\nFor 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 $\\Theta$.\nIn other words, it is the parameter value under which the observed data are most probable.\n\n## Remarks\n\nThe 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 $\\partial \\ell / \\partial \\theta_j = 0$.\nExample: for $k$ successes in $n$ Bernoulli($p$) trials, $\\ell(p) = k \\ln p + (n - k) \\ln(1 - p)$, so $\\hat{p} = k/n$.\nExample: 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.\nUnder regularity conditions, the estimator is [[consistent|0XJVF81QF6]] and asymptotically normal. Also, for a function $g$, the maximum likelihood estimate of $g(\\theta)$ is $g(\\hat{\\theta})$.\n\n## Explanations\n\n### L1 human\n\nMaximum likelihood chooses the parameter that makes the observed data most probable.\nIf a coin shows $7$ heads in $10$ tosses, the maximum likelihood estimate of the probability of heads is $7/10$.\n",
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