2a:T422,densdata = { const lowprob = 0.005 const highprob = 1 - lowprob const xlimlow = d3.min([ jstat.gamma.inv(lowprob, d.alpha, 1/d.beta), jstat.gamma.inv(lowprob, d.n*d.xbar, 1/n), jstat.gamma.inv(lowprob, d.alpha + d.n*d.xbar, 1/(d.beta + d.n)) ]) const xlimhigh = d3.max([ jstat.gamma.inv(highprob, d.alpha, 1/d.beta), jstat.gamma.inv(highprob, d.n*d.xbar, 1/(d.n)), jstat.gamma.inv(highprob, d.alpha + d.n*d.xbar, 1/(d.beta + d.n)) ]) const thetas = d3.range(xlimlow, xlimhigh, (xlimhigh-xlimlow)/500 ); const priorpdf = thetas.map(theta => ( { theta: theta, pdf: jstat.gamma.pdf(theta, d.alpha, 1/d.beta), type: "prior" } )); const likepdf = thetas.map(theta => ( { theta: theta, pdf: jstat.gamma.pdf(theta, d.n*d.xbar, 1/(d.n)), type: "likelihood" } )); const postpdf = thetas.map(theta => ( { theta: theta, pdf: jstat.gamma.pdf(theta, d.alpha + d.n*d.xbar, 1/(d.beta + d.n)), type: "posterior" } )); return priorpdf.concat(likepdf).concat(postpdf) }2b:[] 28:{"title":"Bayesian inference for iid Poisson counts","cells":[{"id":38,"value":"# Bayesian inference for iid Poisson counts","pinned":false,"hidden":false,"mode":"md"},{"id":700,"value":"model = md`### Model\n\nAssume data from the iid Poisson model \n\n${tex`X_1,\\ldots,X_n \\mid \\lambda \\overset{\\mathrm{iid}}{\\sim} \\operatorname{Pois}(\\lambda)`}

\nwhere the [Poisson distribution](https://observablehq.com/@mattiasvillani/poisson-distribution) is such\nthat ${tex`\\mathbb{E}(X)=\\mathbb{V}(X)=\\lambda.`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1345,"value":"prior = md`\n### Conjugate prior\n\nThe conjugate prior for ${tex`\\lambda`} in the iid Poisson model is the [Gamma distribution](https://observablehq.com/@mattiasvillani/gamma-distribution)\n\n${tex`\\lambda \\sim \\operatorname{Gamma}(\\alpha, \\beta)`}

\nwhere the Gamma distribution is given in the so called *rate parameterization* so that ${tex`\\mathbb{E}(X)=\\frac{\\alpha}{\\beta}`}.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":719,"value":"\n\ntitletext1 = md`### Posterior\n\nBy Bayes' theorem, the posterior distribution is proportional to the likelihood times the prior\n\n${tex`p(\\lambda \\vert x_1,\\ldots,x_n ) \\propto p(x_1,\\ldots,x_n \\vert \\lambda)p(\\lambda)`}
\n\nwhere the Poisson likelihood is
\n\n${tex`p(x_1,\\ldots,x_n \\vert \\lambda) = \\prod_{i=1}^n \\frac{\\lambda^{x_i} e^{-\\lambda}}{x_i!} \\propto \\lambda^{\\sum_{i=1}^n x_i} e^{-\\lambda n} = \\lambda^{n \\bar x} e^{-\\lambda n}`}
\n\nwith the ${tex`\\mathrm{Gamma}(\\alpha,\\beta)`} prior density
\n\n${tex`p(\\lambda)= \\frac{\\beta^\\alpha}{\\Gamma(\\alpha)}\\lambda^{\\alpha-1}e^{-\\lambda \\beta}`}
\n\nThe posterior is therefore
\n\n${tex`p(\\lambda \\vert x_1,\\ldots,x_n ) \\propto \\lambda^{n \\bar x} e^{-\\lambda n}\\lambda^{\\alpha-1}e^{-\\lambda \\beta} = \\lambda^{\\alpha + n \\bar x - 1} e^{-\\lambda( \\beta + n)}`}
\n\nwhich is proportional the ${tex`\\mathrm{Gamma}(\\alpha+n\\bar x,\\beta +n)`} density.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1036,"value":"resultstext = html`

\n\n \n \n \n \n\n \n \n \n \n\n \n \n \n \n

Model:	${tex`X_1,\\ldots,X_n \\mid \\lambda \\sim \\operatorname{Pois}(\\lambda)`}.
Prior:	${tex`\\lambda \\sim \\operatorname{Gamma}(\\alpha,\\beta)`}
Posterior:	${tex`\\lambda \\mid x_1,\\ldots,x_n \\sim \\operatorname{Gamma}(\\alpha+n \\bar x, \\beta + n)`}

\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1493,"value":"The interactive widget below lets you experiment with different data, summarized by ${tex`\\bar x`}, and with different values for the prior hyperparameters ${tex`\\alpha`} and ${tex`\\beta`}.","pinned":false,"hidden":false,"mode":"md"},{"id":1265,"value":"datatext = md`**Data**:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"viewof datasettings = Inputs.form([\n Inputs.range([1, 200], {value: 5, step: 1, label: tex`n`}),\n Inputs.range([Number.EPSILON, 10], {value: 2, step: 0.1, label: tex`\\bar x`}),\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":1268,"value":"**Prior**:","pinned":false,"hidden":false,"mode":"md"},{"id":1258,"value":"viewof priorsettings = Inputs.form([\n Inputs.range([Number.EPSILON, 5], {value: 1.5, step: 0.1, label: tex`\\alpha`}),\n Inputs.range([Number.EPSILON, 5], {value: 1, step: 0.1, label: tex`\\beta`})\n ])","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"plt_pdf = Plot.plot({\n \n color: {\n legend: true,\n domain: [\"prior\",\"likelihood\",\"posterior\"],\n range: [\"#D6B656\", \"#6C8EBF\", \"#B85450\"]\n },\n x: {\n label: \"λ\",\n axis: true\n },\n y: {\n axis: false,\n domain: [0,1.02*maxpdf]\n },\n caption: \"Prior-to-Posterior mapping. The likelihood is normalized.\",\n marks: [\n Plot.ruleY([0]),\n Plot.line(densdata, {x: \"theta\", y: \"pdf\", stroke: \"type\", strokeWidth: 2.5}),\n //Plot.areaY(densdata, {filter: d => d.theta <= post_quantile && d.type == \"posterior\", x: \"theta\", y: \"pdf\", \n // fill: \"orange\", opacity: 0.2})\n ]\n })","pinned":false,"hidden":false,"mode":"ojs"},{"id":614,"value":"postresultstext = html`

\n\n \n \n \n \n \n \n \n \n \n \n \n \n

Prior mean:	${tex`\\mathbb{E}( \\lambda) = ${(alpha/beta).toPrecision(3)}`}	Posterior mean:	${tex`\\mathbb{E}( \\lambda \\vert \\boldsymbol{x}) = ${((alpha+n*xbar)/(n + beta)).toPrecision(3)}`}
Prior st.dev:	${tex`\\mathbb{S}( \\lambda) = ${Math.sqrt((alpha/beta**2)).toPrecision(3)}`}	Posterior st.dev:	${tex`\\mathbb{S}( \\lambda \\vert \\boldsymbol{x}) = ${Math.sqrt(((alpha+nxbar)/(n + beta)*2)).toPrecision(3)}`}

\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\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"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":26,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"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"},{"id":1575,"value":"import {roundbox, md_colour} from \"@mattiasvillani/imports\"","pinned":true,"hidden":false,"mode":"ojs"},{"id":1587,"value":"roundbox","pinned":true,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":1447,"value":"alpha = priorsettings[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1450,"value":"beta = priorsettings[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1465,"value":"n = datasettings[0]","pinned":true,"hidden":false,"mode":"ojs"},{"id":1467,"value":"xbar = datasettings[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":777,"value":"d = \n{\n const n = datasettings[0]\n const xbar = datasettings[1]\n const alpha = priorsettings[0]\n const beta = priorsettings[1]\n return {n:n, xbar: xbar, alpha: alpha, beta: beta}\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"$2a","pinned":false,"hidden":false,"mode":"ojs"},{"id":621,"value":"maxpdf = d3.max(densdata.map(d => d.pdf))","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W2b","stdlib":"1","titling":"auto"}