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  "body": "\n## Statement\n\nConsider a [[hypothesis test|087P03NEND]] of the simple null hypothesis $H_0 \\colon \\theta = \\theta_0$ against the simple alternative $H_1 \\colon \\theta = \\theta_1$. Let $L(\\theta \\mid x)$ be the [[likelihood|05FSTXM92W]] of the data $x$.\nFor $\\eta \u003e 0$, let the test reject $H_0$ when\n$$\\Lambda(x) = \\frac{L(\\theta_0 \\mid x)}{L(\\theta_1 \\mid x)} \\le \\eta,$$\nand let $\\alpha = P_{\\theta_0}(\\Lambda(X) \\le \\eta)$ be its size.\nThen this likelihood-ratio test is a most powerful test of size $\\alpha$. Each other test with size at most $\\alpha$ has at most the same power at $\\theta_1$.\n\n## Remarks\n\nConsider all tests with a probability of a [[type I error|0R41PVFA39]] at most $\\alpha$. Among them, the likelihood-ratio test has the smallest probability of a type II error.\nFor a discrete law, an exact size $\\alpha$ can need a randomized test on the boundary $\\Lambda(x) = \\eta$.\nExample: for an i.i.d. $\\mathcal{N}(\\mu, 1)$ sample and $\\mu_1 \u003e \\mu_0$, the test rejects $H_0$ when $\\bar{x}$ is large.\n\n## History\n\nJerzy Neyman and Egon Pearson introduced the lemma in a paper in 1933.\n\n## Explanations\n\n### L1 human\n\nTo choose between two exact hypotheses, the best test compares how well each hypothesis explains the data.\nIf the alternative explains the data much better than the null hypothesis, reject the null hypothesis. No other test with the same false-alarm rate finds more true effects.\n",
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