2c:null 2d:["$","title",null,{"children":"Bayesian inference for Gaussian iid data | Mattias Villani | Observable"}] 32:[] 30:{"id":"b6ebdc269f09fbbe","slug":"bayesian-inference-for-gaussian-iid-data","title":"Bayesian inference for Gaussian iid data","thumbnail":"a06f0a0fd280497d32adf6c92f0810d2364719482102530dbed7fb737b9d7143","access_level":"public","create_time":"$D2024-04-25T18:41:05.988Z","update_time":"$D2024-04-26T07:33:56.909Z","view_time":null,"first_public_time":"$D2024-04-25T20:45:55.833Z","fork":{"id":"58afe28db9cfc1ee","slug":"bayes-iid-gaussian-known-var","title":"Bayesian inference for Gaussian iid data with known variance","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":1599,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"},{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":0,"collection":{"id":"9fdb7c9ba7be6ba8","slug":"bayesian-learning","title":"Bayesian Learning","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Bayesian inference for Gaussian iid data","cells":[{"id":38,"value":"# Bayesian inference for Gaussian iid data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":700,"value":"model = md` \nConsider data from the iid Gaussian model with both parameters unknown\n\n${tex.block`X_1,\\ldots,X_n \\mid \\theta, \\sigma^2 \\sim \\operatorname{N}(\\theta,\\sigma^2)`}\n` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1518,"value":"prior = md` \nwith the conjugate prior\n${tex.block`\\theta \\vert \\sigma^2 \\sim N \\Big(\\mu_0, \\frac{\\sigma^2}{\\kappa_0}\\Big)`} \n ${tex.block`\\sigma^2 \\sim \\mathrm{Inv-}\\chi^2 (\\nu_0, \\sigma_0^2)`}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1569,"value":"The posterior distribution is\n${tex.block`\\theta \\vert \\sigma^2, \\boldsymbol{x} \\sim N \\Big(\\mu_n, \\frac{\\sigma^2}{\\kappa_n}\\Big)`} \n${tex.block`\\sigma^2 \\vert \\boldsymbol{x} \\sim \\mathrm{Inv-}\\chi^2 (\\nu_n, \\sigma_n^2)`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1575,"value":"where \n\n${tex.block`\\mu_n = w\\bar x + (1-w)\\mu_0 = ${d.mun.toPrecision(3)}`} \n${tex.block`w = \\frac{n}{\\kappa_0 + n} = ${d.w.toPrecision(3)}`}\n${tex.block`\\kappa_n = \\kappa_0 + n = ${d.kappan.toPrecision(3)}`} \n${tex.block`\\nu_n = \\nu_0 + n = ${d.nun.toPrecision(3)} = ${d.nun.toPrecision(3)}`} \n${tex.block`\\sigma_n^2 = \\frac{\\nu_0\\sigma_0^2 + (n-1)s^2 + \\frac{\\kappa_0 n}{\\kappa_0 + n}(\\bar x - \\mu_0)^2}{\\nu_n} = ${(d.sigman**2).toPrecision(3)}`} \n\nThe marginal posterior for ${tex`\\theta`} is \n${tex.block`\\theta \\vert \\boldsymbol{x} \\sim t_{\\nu_n}\\Big(\\mu_n, \\frac{\\sigma_n^2}{\\kappa_n}\\Big)`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":719,"value":"titletext1 = md`### Prior-to-Posterior updating`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1265,"value":"**Data**:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":118,"value":"viewof datasettings = Inputs.form([\n Inputs.range([1, 100], {value: 5, step: 1, label: tex`n`}),\n Inputs.range([0, 20], {value: 16, step: 0.1, label: tex`\\text{sample mean, }\\bar x`}),\n Inputs.range([0.1, 10], {value: 5, step: 0.1, label: tex`\\text{sample stdev, }s`})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1268,"value":"**Prior**:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1258,"value":"viewof priorsettings = Inputs.form([\n Inputs.range([0, 100], {value: 20, step: 1, label: tex`\\mu_0`}),\n Inputs.range([0.1, 100], {value: 1, step: 0.1, label: tex`\\kappa_0`}),\n Inputs.range([0.1, 100], {value: 6, step: 0.1, label: tex`\\nu_0`}),\n Inputs.range([0.1, 100], {value: 5, step: 0.1, label: tex`\\sigma_0`})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1550,"value":"\n \n \n \n

\n ${plt_pdf_theta}\n

\n \n

\n ${plt_pdf_sigma2}\n

\n \n","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":109,"value":"
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\n\n---\n#### 🙈 Appendix 🙈\n\nHere is just some data, imports and function defintions that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1559,"value":"#### Plots","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1502,"value":"plt_pdf_sigma2 = Plot.plot({\n color: {\n legend: true,\n domain: [\"prior\",\"likelihood\",\"posterior\"],\n range: [\"#D6B656\", \"#6C8EBF\", \"#B85450\"]\n },\n title: \"Marginal prior and posterior for σ²\",\n x: {\n label: \"σ²\",\n axis: true\n },\n y: {\n axis: false,\n domain: [0,1.02*maxpdfsigma2]\n },\n caption: \"The likelihood is normalized.\",\n marks: [\n Plot.ruleY([0]),\n Plot.line(densdatasigma2, {x: \"sigma2\", y: \"pdf\", stroke: \"type\", strokeWidth: 2.5}),\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":305,"value":"plt_pdf_theta = Plot.plot({\n color: {\n legend: true,\n domain: [\"prior\",\"likelihood\",\"posterior\"],\n range: [\"#D6B656\", \"#6C8EBF\", \"#B85450\"]\n },\n title: \"Marginal prior and posterior for θ\",\n x: {\n label: \"θ\",\n axis: true\n },\n y: {\n axis: false,\n domain: [0,1.02*maxpdf]\n },\n caption: \"The likelihood is normalized.\",\n marks: [\n Plot.ruleY([0]),\n Plot.line(densdata, {x: \"theta\", y: \"pdf\", stroke: \"type\", strokeWidth: 2.5}),\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1195,"value":"function toc(selector = \"h2,h3\", heading = \"Table of Contents\") {\n return Generators.observe(notify => {\n let headings = [];\n\n function observed() {\n const h = Array.from(document.querySelectorAll(selector));\n if (h.length !== headings.length || h.some((h, i) => headings[i] !== h)) {\n notify(html`${heading}

${DOM.text(h.textContent)}`,\n {onclick: e => (e.preventDefault(), h.scrollIntoView())}\n );\n })}`);\n }\n }\n\n const observer = new MutationObserver(observed);\n observer.observe(document.body, {childList: true, subtree: true});\n observed();\n return () => observer.disconnect();\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1426,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1431,"value":"function tpdf(theta, mu, sigma, df){\n return jstat.studentt.pdf( (theta-mu)/sigma, df )/sigma\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1434,"value":"function tcdf(theta, mu, sigma, df){\n return jstat.studentt.cdf((theta-mu)/sigma, df)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1484,"value":"function scaledinvchi2pdf(x, nu, tau2){\n // ScaledInverseChiSq(ν,τ²) = InverseGamma(ν/2,ν*τ²/2)\n return jstat.invgamma.pdf(x, nu/2, nu*tau2/2)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1486,"value":"function scaledinvchi2cdf(x, nu, tau2){\n return jstat.invgamma.cdf(x, nu/2, nu*tau2/2)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1460,"value":"n = datasettings[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":940,"value":"xbar = datasettings[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1463,"value":"s = datasettings[2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":777,"value":"d = \n{\n const mu0 = priorsettings[0]\n const kappa0 = priorsettings[1]\n const nu0 = priorsettings[2]\n const sigma0 = priorsettings[3]\n const s2 = s**2\n const kappan = kappa0 + n\n const w = n/kappan\n const mun = w*xbar + (1-w)*mu0\n const nun = nu0 + n\n const sigma2n = (nu0*sigma0**2 + (n-1)*s2 + ((kappa0*n)/(kappa0 + n))*(xbar - mu0)**2)/nun\n return {n:n, xbar: xbar, s2: s2, mu0: mu0, kappa0: kappa0, nu0:nu0, sigma0: sigma0, \n kappan: kappan, w: w, mun: mun, nun: nun, sigman: Math.sqrt(sigma2n)}\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"densdata = {\n const thetas = d3.range(0, 40, 0.05);\n const priorpdf = thetas.map(theta => (\n {\n theta: theta, \n pdf: tpdf(theta, d.mu0, d.sigma0/Math.sqrt(d.kappa0), d.nu0),\n type: \"prior\"\n }\n ));\n const likepdf = thetas.map(theta => (\n {\n theta: theta, \n pdf: tpdf(theta, d.xbar, s/Math.sqrt(n), n),\n type: \"likelihood\"\n }\n ));\n const postpdf = thetas.map(theta => (\n {\n theta: theta, \n pdf: tpdf(theta, d.mun, d.sigman/Math.sqrt(d.kappan), d.nun),\n type: \"posterior\"\n }\n ));\n return priorpdf.concat(likepdf).concat(postpdf)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":621,"value":"maxpdf = Math.max(tpdf(d.mu0, d.mu0, d.sigma0/Math.sqrt(d.kappa0), d.nu0), tpdf(d.xbar, d.xbar, s/Math.sqrt(n), n), tpdf(d.mun, d.mun, d.sigman/Math.sqrt(d.kappan), d.nun))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1491,"value":"densdatasigma2 = {\n const sigma2s = d3.range(0, 150, 0.1);\n const priorpdf = sigma2s.map(sigma2 => (\n {\n sigma2: sigma2, \n pdf: scaledinvchi2pdf(sigma2, d.nu0, d.sigma0**2),\n type: \"prior\"\n }\n ));\n const likepdf = sigma2s.map(sigma2 => (\n {\n sigma2: sigma2, \n pdf: scaledinvchi2pdf(sigma2, n, s**2),\n type: \"likelihood\"\n }\n ));\n const postpdf = sigma2s.map(sigma2 => (\n {\n sigma2: sigma2, \n pdf: scaledinvchi2pdf(sigma2, d.nun, d.sigman**2),\n type: \"posterior\"\n }\n ));\n return priorpdf.concat(likepdf).concat(postpdf)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1499,"value":"maxpdfsigma2 = Math.max(scaledinvchi2pdf((d.nu0*d.sigma0**2)/(d.nu0+2), d.nu0, d.sigma0**2), \n scaledinvchi2pdf((n*s**2)/(n+2), n, s**2), \n scaledinvchi2pdf((d.nun*d.sigman**2)/(d.nun+2), d.nun, d.sigman**2)\n )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 31:"$30:body"