2b:null 2c:["$","title",null,{"children":"Poisson-Gamma distribution | Mattias Villani | Observable"}] 31:[] 2f:{"id":"cda1c447cb4ade03","slug":"poisson-gamma-distribution","title":"Poisson-Gamma distribution","thumbnail":"70eb6c5d3a2efa0233289e4a364676df6bc37127424bc62c63299812708821f6","access_level":"public","create_time":"$D2024-04-28T08:40:09.253Z","update_time":"$D2024-04-28T10:34:16.225Z","view_time":null,"first_public_time":"$D2024-04-28T09:21:04.616Z","fork":{"id":"93ddcc9cbf763f75","slug":"poisson-distribution","title":"Poisson 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":887,"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":"Poisson-Gamma distribution","cells":[{"id":38,"value":"# Poisson-Gamma distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":780,"value":"disttext = md`

${tex`X\\sim \\operatorname{PoissonGamma}(\\alpha,\\beta,\\nu)`}, for ${tex`X \\in \\{0,1,2,\\ldots \\}`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":496,"value":"A random variable following the Poisson-Gamma distribution has probability mass function\n\n${tex.block`P(X=x)= \\frac{\\beta^\\alpha}{\\Gamma(\\alpha)}\\frac{\\Gamma(\\alpha+x)}{x!} \\frac{\\nu^x}{(\\beta+\\nu)^{\\alpha+x}}`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":118,"value":"viewof params = Inputs.form([\n Inputs.range([0.1, 20], {value: 2, step: 0.1, label: tex`\\alpha:`}),\n Inputs.range([0.1, 20], {value: 2, step: 0.1, label: tex`\\beta:`}),\n Inputs.range([0.1, 20], {value: 2, step: 0.1, label: tex`\\nu:`}),\n Inputs.range([0, 20], {value: 1, step: 1, label: \"quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":461,"value":"dist_quantile = tex`\\text{If } X \\sim \\operatorname{Poisson-Gamma}(${params[0]},${params[1]},${params[2]}) \\text{ then } `","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":555,"value":"moments = tex`\n\\begin{aligned}\n&\\mathrm{E}( X) = \\nu\\frac{\\alpha}{\\beta} = ${mean.toPrecision(3)} \\\\[0.5em]\n&\\mathrm{Var}( X) = \\nu\\frac{\\alpha}{\\beta}\\Big(1+\\frac{\\nu}{\\beta}\\Big) = ${variance.toPrecision(3)} \\\\[0.5em]\n&\\mathrm{P}( X\\leq ${quantile}) =${poisgamcdf.toPrecision(4)}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: { \n legend: false\n },\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n label: \"f(x)\",\n axis: true\n },\n tooltip: {\n fill: \"#005073\",\n stroke: \"#005073\",\n opacity: 1,\n },\n marks: [\n Plot.ruleY([0]),\n Plot.barY(pdfdata,{x: \"x\", y: \"pdf\", fill: \"#71c7ec\", strokeWidth: 0, opacity: 1,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`}),\n Plot.barY(pdfdata, {filter: d => d.x <= quantile, x: \"x\", y: \"pdf\", fill: \"#189ad3\", opacity: 1,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":877,"value":"#### Characterization\nThe Poisson-Gamma distribution can be derived by the following continuous mixture of Poissons ([Bernardo. J.M. and Smith, A.F.M. (1994), p. 119](https://onlinelibrary.wiley.com/doi/book/10.1002/9780470316870))\n\n${tex.block`\\mathrm{PoisGamma}(x \\vert \\alpha, \\beta, \\nu) = \\int \\mathrm{Pois}(x\\vert \\nu\\lambda)\\mathrm{Gamma}(\\lambda \\vert \\alpha, \\beta) d\\lambda`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nJust some code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":764,"value":"import {Plot} from \"@mkfreeman/plot-tooltip\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":822,"value":"alpha = params[0]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":824,"value":"beta = params[1]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":826,"value":"nu = params[2]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":846,"value":"quantile = params[3]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":859,"value":"mean = nu*alpha/beta","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":857,"value":"variance = (nu*alpha/beta)*(1+nu/beta)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":854,"value":"pdfdata = {\n const x = d3.range(0, Math.floor(mean + 5*Math.sqrt(variance) ) )\n const pdf = x.map((x) => ({ x: x, pdf: poissongammapdf(x, alpha, beta, nu) }));\n return pdf;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":19,"value":"poisgamcdf = poissongammacdf(quantile, alpha, beta, nu)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":833,"value":"#### Function","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":838,"value":"function poissongammapdf(x, alpha, beta, nu){\n return ((beta**alpha)/jstat.gammafn(alpha))*(jstat.gammafn(alpha+x)/jstat.factorial(x))*((nu**x)/((beta+nu)**(alpha+x)))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":835,"value":"function poissongammacdf(quantile, alpha, beta, nu){\n return d3.sum(pdfdata.filter(d => d.x <= quantile).map(d => d.pdf))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$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"}],"files":"$W31","stdlib":"1","titling":"auto"}} 30:"$2f:body"