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  "body": "\n## Counterexample\n\nLet $X_1, \\dots, X_n$ be i.i.d. [[normal|07RXBVTVZB]] $\\mathcal{N}(\\theta, 1)$, so that $\\bar{X}_n \\sim \\mathcal{N}(\\theta, 1/n)$.\nTest $H_0 \\colon \\theta = 0$ against $H_1 \\colon \\theta \\ne 0$.\nSuppose the data give $\\sqrt{n}\\,\\bar{x}_n = z$ with $z = 1.96$, for every $n$.\nThe two-sided [[p-value|0EHYZFZ4PK]] is then $P(\\abs{Z} \\ge 1.96) \\approx 0.05$ for $Z \\sim \\mathcal{N}(0, 1)$, whatever $n$ is.\nSo a test of level $0.05$ is at the edge of rejecting $H_0$ for every $n$.\nFor the Bayesian comparison, give $\\theta$ the prior $\\mathcal{N}(0, \\tau^2)$ under $H_1$, with $\\tau \u003e 0$ fixed.\nUnder $H_0$ the density of $\\bar{X}_n$ is that of $\\mathcal{N}(0, 1/n)$, and under $H_1$ it is that of $\\mathcal{N}(0, \\tau^2 + 1/n)$.\nA direct computation gives the [[Bayes factor|NSBYE143KS]] of $H_0$ against $H_1$:\n$$B_{01} = \\sqrt{1 + n\\tau^2} \\, \\exp\\left(-\\frac{z^2}{2} \\cdot \\frac{n\\tau^2}{1 + n\\tau^2}\\right).$$\nAs $n \\to \\infty$ with $z$ fixed, $B_{01} \\to \\infty$.\nSo for any prior probability $P(H_0) \\in (0, 1)$, the posterior probability of $H_0$ tends to $1$, while the p-value stays at $0.05$.\n\n## Remarks\n\nThe p-value and the posterior probability answer different questions, and for large samples they can point in opposite directions.\nFor large $n$, a fixed $z$ means that $\\bar{x}_n$ is very close to $0$.\nSuch a value is much better explained by $\\theta = 0$ than by a $\\theta$ spread over a range of width about $\\tau$.\nThe conflict also depends on the prior: as $\\tau \\to \\infty$ with $n$ fixed, $B_{01} \\to \\infty$ as well, so a vague prior under $H_1$ favours $H_0$.\n\n## Explanations\n\n### L1 human\n\nWith a huge sample, a small but \"significant\" deviation from zero can appear.\nA classical test then says \"reject the null hypothesis at the $5\\%$ level\".\nA Bayesian comparison of \"the effect is exactly zero\" with \"the effect is some reasonable size\" can say the opposite: the data favour exactly zero.\nThe paradox shows that significance and evidence are not the same thing.\n",
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