29:null 2a:["$","title",null,{"children":"Bayesian inference for Exponential iid data | Mattias Villani | Observable"}] 2f:T426,densdata = { const lowprob = 0.005 const highprob = 1 - lowprob const xlimlow = d3.min([ jstat.gamma.inv(lowprob, d.alpha, 1/d.beta), jstat.gamma.inv(lowprob, d.n, 1/(d.n*d.xbar)), jstat.gamma.inv(lowprob, d.alpha + d.n, 1/(d.beta + d.n*d.xbar)) ]) const xlimhigh = d3.max([ jstat.gamma.inv(highprob, d.alpha, 1/d.beta), jstat.gamma.inv(highprob, d.n, 1/(d.n*d.xbar)), jstat.gamma.inv(highprob, d.alpha + d.n, 1/(d.beta + d.n*d.xbar)) ]) const thetas = d3.range(xlimlow, xlimhigh, (xlimhigh-xlimlow)/500 ); const priorpdf = thetas.map(theta => ( { theta: theta, pdf: jstat.gamma.pdf(theta, d.alpha, 1/d.beta), type: "prior" } )); const likepdf = thetas.map(theta => ( { theta: theta, pdf: jstat.gamma.pdf(theta, d.n, 1/(d.n*d.xbar)), type: "likelihood" } )); const postpdf = thetas.map(theta => ( { theta: theta, pdf: jstat.gamma.pdf(theta, d.alpha + d.n, 1/(d.beta + d.n*d.xbar)), type: "posterior" } )); return priorpdf.concat(likepdf).concat(postpdf) }30:[] 2d:{"id":"dde99ac1c5a54388","slug":"bayesian-inference-for-exponential-iid-data","title":"Bayesian inference for Exponential iid data","thumbnail":"e5b232d8e28b479f5bc2aa7781d6003a9a504f7df1b989bfb692362aac423f50","access_level":"public","create_time":"$D2023-08-30T16:50:08.499Z","update_time":"$D2026-08-07T13:00:41.278Z","view_time":null,"first_public_time":"$D2023-08-30T16:50:08.499Z","has_unpublished_edits":null,"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":"education_2026"},"version":1568,"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"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":2,"collection":{"id":"636adfa34cc7a030","slug":"sda3","title":"SDA3","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Bayesian inference for Exponential iid data","cells":[{"id":38,"value":"# Bayesian inference for Exponential iid data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":700,"value":"model = md`### Model\n\nAssume data from the iid Exponential model \n\n> ${tex`X_1,\\ldots,X_n \\mid \\lambda \\overset{\\mathrm{iid}}{\\sim} \\operatorname{Expon}(\\lambda)`}
\n\nwhere the [Exponential distribution](https://observablehq.com/@mattiasvillani/exponential-distribution) is parametrized so that ${tex`\\mathbb{E}(X)=\\frac{1}{\\lambda}.`}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1345,"value":"prior = md`\n### Conjugate prior\n\nThe conjugate prior for ${tex`\\lambda`} in the iid Exponential model is the [Gamma distribution](https://observablehq.com/@mattiasvillani/gamma-distribution)\n\n> ${tex`\\lambda \\sim \\operatorname{Gamma}(\\alpha, \\beta)`}
\n\nwhere we assume that the Gamma distribution is given in the so called *rate parameterization* so that ${tex`\\mathbb{E}(X)=\\frac{\\alpha}{\\beta}`}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":719,"value":"titletext1 = md`### Posterior\nThe 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 exponential likelihood is

\n\n> ${tex`p(x_1,\\ldots,x_n \\vert \\lambda) = \\prod_{i=1}^n \\lambda e^{-\\lambda x_i} = \\lambda^n e^{-\\lambda\\sum_{i=1}^n x_i} = \\lambda^n e^{-\\lambda n \\bar x}`}

\n\nand ${tex`p(\\lambda)`} is 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
\n\nThe posterior is therefore
\n\n> ${tex`p(\\lambda \\vert x_1,\\ldots,x_n ) \\propto \\lambda^n e^{-\\lambda n \\bar x}\\lambda^{\\alpha-1}e^{-\\lambda \\beta} = \\lambda^{\\alpha + n -1} e^{-\\lambda (\\beta + n \\bar x)}`}
\n\nwhich is proportional to the ${tex`\\operatorname{Gamma}(\\alpha+n, \\beta + n\\bar x )`} density. \n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"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{Expon}(\\lambda)`}.
Prior:	${tex`\\lambda \\sim \\operatorname{Gamma}(\\alpha,\\beta)`}
Posterior:	${tex`\\lambda \\mid x_1,\\ldots,x_n \\sim \\operatorname{Gamma}(\\alpha+n, \\beta + n\\bar x )`}

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"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 ${tex`\\beta`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1265,"value":"**Data**:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":118,"value":"viewof datasettings = Inputs.form([\n Inputs.range([1, 100], {value: 5, step: 1, label: tex`n`}),\n Inputs.range([Number.EPSILON, 10], {value: 1, step: 0.1, label: tex`\\bar x`}),\n ])","pinned":false,"hidden":false,"mode":"ojs","author":"$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([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","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"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)/(n*xbar + 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+n)/(nxbar + beta)*2)).toPrecision(3)}`}

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":305,"value":"plt_pdf = Plot.plot({\n color: {\n legend: true,\n domain: [\"prior\",\"likelihood\",\"posterior\"],\n range: [\"#D6B656\", \"#6C8EBF\", \"#B85450\"]\n },\n x: {\n label: \"theta\",\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","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$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":"$undefined"},{"id":221,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$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":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1447,"value":"alpha = priorsettings[0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1450,"value":"beta = priorsettings[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1465,"value":"n = datasettings[0]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1467,"value":"xbar = datasettings[1]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":777,"value":"d = \n{\n const n = datasettings[0]\n const xbar = datasettings[1]\n const alpha = priorsettings[0]\n const beta = priorsettings[1]\n return {n:n, xbar: xbar, alpha: alpha, beta: beta}\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":621,"value":"maxpdf = d3.max(densdata.map(d => d.pdf))","pinned":false,"hidden":false,"mode":"ojs","author":"$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":1526,"value":"colors = [\"#6C8EBF\", \"#D6B656\", \"#B85450\", \"#006d77\", \n \"#bf9d6c\", \"#55a12a\", \"#a6bbd8\",\"#e6d399\", \"#d49896\", \"#66a7ad\", \"#bf8d6c\", \"#407d24\"]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F0F0F0\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W30","stdlib":"1","titling":"auto"}} 2e:"$2d:body"