2a:null 2b:["$","title",null,{"children":"Inverse Gamma distribution | Mattias Villani | Observable"}] 31:[] 2e:{"id":"0c8449cf2c627a44","slug":"inverse-gamma-distribution","title":"Inverse Gamma distribution","thumbnail":"8b5d774f7dc071108d739553a171f35f22c8abb26b45b8c1421fc838a1d84d7f","access_level":"public","create_time":"$D2025-02-16T19:12:06.259Z","update_time":"$D2025-02-16T19:38:51.251Z","view_time":null,"first_public_time":"$D2025-02-16T19:26:20.004Z","fork":{"id":"0bc2ad0532475682","slug":"gamma-distribution","title":"Gamma distribution","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":968,"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":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Inverse Gamma distribution","cells":[{"id":38,"value":"# Inverse Gamma distribution","pinned":false,"hidden":false,"mode":"md"},{"id":662,"value":"md`

${tex`X\\sim \\operatorname{Inv-Gamma}(\\alpha,\\beta)`}, for ${tex`X \\in (0,\\infty)`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":496,"value":"md`Probability density function \n${tex.block`f(x)=\\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} x^{-\\alpha - 1} e^{-\\frac{\\beta}{x} }`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([0.01, 10], {value: 3, step: 0.01, label: tex`\\text{shape }\\alpha`}),\n Inputs.range([0.01, 300], {value: 1, step: 0.01, label: tex`\\text{scale }\\theta`}),\n Inputs.range([0, 10], {value: 1, step: 0.01, label: \"quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\operatorname{Inv-Gamma}(${params[0]},${params[1]}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs"},{"id":555,"value":"tex`\n\\begin{aligned}\n&E( X) =\\frac{\\beta}{\\alpha-1} = ${moments.mean.toPrecision(2)}\\text { for }\\alpha > 1 \\\\[1em]\n&Var( X) = \\frac{\\beta^2}{(\\alpha-1)^2(\\alpha-2)} = ${moments.var.toPrecision(2)} \\text { for }\\alpha > 2 \\\\[1em]\n&P( X\\leq ${params[2]}) =${cdf.toPrecision(4)}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: {\n legend: true\n },\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n label: \"f(x)\",\n //axis: false,\n //domain: [0, d3.max(pdfvals.map(d => d.pdf))] \n },\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.line(pdfvals, {x: \"x\", y: \"pdf\", stroke : \"green\", strokeWidth: 2}),\n Plot.areaY(pdfvals, {filter: d => d.x <= params[2], x: \"x\", y: \"pdf\", fill: \"green\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nJust some code and details that you probably want to ignore.","pinned":false,"hidden":false,"mode":"md"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":958,"value":"extremes = {\n let lims = [0.001, 20]\n if (jstat.invgamma.cdf(lims[1], alpha, beta) < 0.975){\n lims = [0.001, jstat.invgamma.inv(0.975, alpha, beta )]\n }\n return lims\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"pdfvals = {\n const x = d3.range(extremes[0], extremes[1], (extremes[1]-extremes[0])/1000);\n const pdf = x.map(x => ({x: x, pdf: jstat.invgamma.pdf(x, alpha, beta)}))\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"cdf = jstat.invgamma.cdf(params[2], alpha, beta)","pinned":false,"hidden":false,"mode":"ojs"},{"id":808,"value":"alpha = params[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":811,"value":"beta = params[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":767,"value":"moments = {\n var densmean;\n var densvar;\n if (alpha > 1){\n densmean = beta/(alpha-1);\n }else{\n densmean = Infinity \n }\n if (alpha > 2){\n densvar = beta**2/((alpha-1)**2*(alpha-2))\n }else{\n densvar = Infinity \n }\n return {mean: densmean, var: densvar}\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}} 2f:"$2e:body"