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  "body": "\n## Definition\n\nLet $\\Theta$ be a set of parameters. For each $\\theta \\in \\Theta$, let $f(\\cdot \\mid \\theta)$ be a [[probability mass function|0WF4EJP91W]] or a [[density|04YG690N5V]] on $\\mathbb{R}^n$. It gives the law of the data $X = (X_1, \\dots, X_n)$ when $\\theta$ is the true parameter.\nFor observed data $x \\in \\mathbb{R}^n$, the likelihood function is the function $L \\colon \\Theta \\to [0, \\infty)$ with\n$$L(\\theta \\mid x) = f(x \\mid \\theta).$$\nThe log-likelihood is $\\ell(\\theta \\mid x) = \\ln L(\\theta \\mid x)$.\n\n## Remarks\n\nFor an [[i.i.d.|0AJN7WHBJZ]] sample with mass function or density $f_1(\\cdot \\mid \\theta)$ for each observation, $L(\\theta \\mid x) = \\prod_{i=1}^{n} f_1(x_i \\mid \\theta)$ and $\\ell(\\theta \\mid x) = \\sum_{i=1}^{n} \\ln f_1(x_i \\mid \\theta)$.\nThe likelihood is not a probability distribution on $\\Theta$. In general, its integral over $\\theta$ is not $1$.\nExample: for $k$ successes in $n$ Bernoulli($p$) trials, $L(p \\mid x) = p^k (1 - p)^{n - k}$ for $p \\in [0, 1]$.\nOnly ratios $L(\\theta_1 \\mid x) / L(\\theta_2 \\mid x)$ matter for inference. These ratios compare how well two parameter values explain the data.\n\n## Explanations\n\n### L1 human\n\nThe likelihood answers the question: \"If the parameter were $\\theta$, how probable would my data be?\"\nParameter values with a high likelihood explain the observed data well.\n",
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