2a:null 2b:["$","title",null,{"children":"Prior predictive - iid Poisson model with Gamma prior | Mattias Villani | Observable"}] 31:[] 2e:{"id":"00ece4fe98411275","slug":"prior_pred_poismodel","title":"Prior predictive - iid Poisson model with Gamma prior","thumbnail":"bcabdac5ee5403de2fe655f11e3f1e93f1a90e022003a0400014469127eafd01","access_level":"public","create_time":"$D2025-02-09T14:21:21.116Z","update_time":"$D2025-10-16T14:36:23.679Z","view_time":null,"first_public_time":"$D2025-02-09T16:01:04.279Z","fork":{"id":"d39a8714e5c419e7","slug":"negative-binomial-distribution","title":"Negative binomial 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":995,"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":"Prior predictive - iid Poisson model with Gamma prior","cells":[{"id":38,"value":"# Prior predictive - iid Poisson model with Gamma prior","pinned":false,"hidden":false,"mode":"md"},{"id":979,"value":"It is often useful to set the prior hyperparameters, for example ${tex`\\alpha`} and ${tex`\\beta`} in a ${tex`\\mathrm{Gamma}(\\alpha,\\beta)`} prior, from what you believe about the *data*.
\nThe **prior predictive distribution**\n${tex.block`p(x)=\\int p(x\\vert\\theta)p(\\theta) \\mathrm{d}\\theta`}\ndescribe the distribution of the data from a model, ${tex`p(x\\vert\\theta)`}, averaged over the parameters from the prior, ${tex`p(\\theta)`}. This makes it possible to test different values for the prior hyperparameters until the user agrees with the implied data distribution.\n\nThe interactive plot below plot the prior predictive distribution from an iid Poisson model with a ${tex`\\mathrm{Gamma}(\\alpha,\\beta)`} prior. Experiment with the prior hyperparameters ${tex`\\alpha`} and ${tex`\\beta`} and see how the prior predictive distribution changes.","pinned":false,"hidden":false,"mode":"md"},{"id":864,"value":"**Model**: \n${tex.block`X_1,\\ldots,X_n \\vert \\theta \\overset{\\mathrm{iid}}{\\sim} \\mathrm{Poisson}(\\theta)`}\n**Prior**: \n${tex.block`\\theta \\sim \\mathrm{Gamma}(\\alpha,\\beta)`}\n**Prior predictive**:\n${tex.block`X \\sim \\mathrm{Negbin}\\bigg(\\alpha,\\frac{\\beta}{\\beta + 1}\\bigg)`}\n","pinned":false,"hidden":false,"mode":"md"},{"id":496,"value":"viewof params = Inputs.form([\n Inputs.range([0.01, 5], {value: 2, step: 0.01, label: tex`\\alpha:`}),\n Inputs.range([Number.EPSILON, 5], {value: 1/2, step: 0.01, label: tex`\\beta:`}),\n Inputs.range([Number.EPSILON, 25], {value: 15, step: 1, label: \"Quantile:\"})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":461,"value":"textmoments = {\n var text;\n \n text = tex`\n\\text{\\textbf{Prior predictive properties}:}\\\\[1em]\n\\begin{aligned}\n&\\mathrm{E}(X) = \\frac{\\alpha}{\\beta} = ${mean.toPrecision(3)} \\\\[1em]\n&\\mathrm{StDev}(X) = \\frac{\\alpha(1+\\beta)}{\\beta^2} = ${variance.toPrecision(3)} \\\\[1em]\n&P( X\\leq ${params[2]}) =${negbincdf.toPrecision(4)}\n\\end{aligned}\n`\n return text\n}\n\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":952,"value":"bothplots = html`\n \n \n \n

\n ${plt_pred_pdf}\n

\n ${plt_prior_pdf}\n

\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 do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":305,"value":"plt_pred_pdf = Plot.plot({\n color: { \n legend: false\n },\n title: \"Prior predictive distribution\",\n x: {\n label: \"x\",\n axis: true\n },\n y: {\n label: \"p(x)\",\n axis: true\n },\n tooltip: {\n fill: colors[1],\n stroke: colors[1],\n opacity: 1,\n },\n marks: [\n Plot.ruleY([0]),\n Plot.barY(negbinpdf,{x: \"x\", y: \"pdf\", fill: colors[1], strokeWidth: 0, opacity: .5,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`}),\n Plot.barY(negbinpdf, {filter: d => d.x <= params[2], x: \"x\", y: \"pdf\", fill: colors[0], opacity: 1,\n title: (d) => `P(X=${d.x}) = ${(d.pdf).toPrecision(4)}`})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":930,"value":"plt_prior_pdf = Plot.plot({\n color: {\n legend: true\n },\n title: \"Prior distribution\",\n x: {\n label: \"θ\",\n axis: true\n },\n y: {\n label: \"p(θ)\",\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(priorpdf, {x: \"x\", y: \"pdf\", stroke : \"steelblue\", strokeWidth: 3}),\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":764,"value":"import {Plot} from \"@mkfreeman/plot-tooltip\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":960,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":801,"value":"alpha = params[0]","pinned":true,"hidden":false,"mode":"ojs"},{"id":846,"value":"beta = params[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":914,"value":"r = alpha","pinned":true,"hidden":false,"mode":"ojs"},{"id":917,"value":"p = beta/(1+beta)","pinned":true,"hidden":false,"mode":"ojs"},{"id":805,"value":"mean = r*(1-p)/p","pinned":true,"hidden":false,"mode":"ojs"},{"id":807,"value":"variance = r*(1-p)/(p**2)","pinned":true,"hidden":false,"mode":"ojs"},{"id":17,"value":"negbinpdf = {\n const x = d3.range(0, mean + 5*Math.sqrt(variance), 1);\n const data = x.map(x => ({x: x, pdf: jstat.negbin.pdf(x, r, p)}));\n return data\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"negbincdf = jstat.negbin.cdf(params[2], r, p);","pinned":false,"hidden":false,"mode":"ojs"},{"id":934,"value":"priorpdf = {\n const x = d3.range(0, 4, 0.01);\n return x.map(x => ({x: x, pdf: jstat.gamma.pdf(x, alpha, beta)}));\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"