2b:null 2c:["$","title",null,{"children":"Bayesian linear regression | Mattias Villani | Observable"}] 32:T601,function bayeslinreg_sampling(y, X, μ0, Ω0, ν0, σ20, nSim){ // Compute posterior hyperparameters // OLS const n = X.size()[0] const p = X.size()[1] const Xty = math.multiply(math.transpose(X), y); const XX = math.multiply(math.transpose(X), X) const invXtX = math.inv(XX); const βhat = math.multiply(invXtX, Xty); // Bayes const νn = ν0 + n const Ωn = math.add(XX, Ω0) const invΩn = math.inv(Ωn) const μn = math.multiply(invΩn, math.add( math.multiply(XX,βhat), math.multiply(Ω0,μ0) ) ) const residuals = math.subtract(y, math.multiply(X,βhat) ) const RSS = math.multiply(math.transpose(residuals), residuals) const SSmuOLS = quadform(math.subtract(μn, βhat), XX) // (μn-βhat)'*XX*(μn-βhat) const SSmuPrior = quadform(math.subtract(μn, μ0), Ω0) // (μn-μ0)'*Ω0*(μn-μ0) const σ2n = (ν0*σ20 + RSS + SSmuOLS + SSmuPrior )/νn // Sampling from posterior let σ2sim = math.zeros(nSim) let βsim = math.zeros(nSim,p) let βindex = d3.range(p) var β = βhat var σ2 for (let i = 1; i<=nSim; i++){ // Simulate from p(σ²|y,X) σ2 = randscaledinvchi2(νn, σ2n) σ2sim.subset(math.index(i), σ2) // Simulate from p(β|σ²,y,X) let covar = math.multiply(σ2,invΩn) β = mvnrand(μn, covar._data) βsim.subset(math.index(i, βindex), β) } return {βsim: βsim, σ2sim: σ2sim, μn: μn, invΩn: invΩn, νn: νn, σ2n:σ2n} }33:[{"id":"e4edc9b12c5d4b65381d35bae8c9499efc26152a172975a159a01bc0486fecc1d9f899fcebae7da05d6b135511bc53ff7fb3f2b2602022a2c5ea4d88d11e0833","href":"https://static.observableusercontent.com/files/e4edc9b12c5d4b65381d35bae8c9499efc26152a172975a159a01bc0486fecc1d9f899fcebae7da05d6b135511bc53ff7fb3f2b2602022a2c5ea4d88d11e0833","name":"bikesday.csv","type":"text/csv","lastModified":1676066102860.85,"size":56836},{"id":"dd6cd934b3dcb87078a18d4a81b9944504ab3edd08902a5833df36ffad06260dbb8190bebab91d9fb9e22054120ec713852c562e8e7ec87331a1f84997861d83","href":"https://static.observableusercontent.com/files/dd6cd934b3dcb87078a18d4a81b9944504ab3edd08902a5833df36ffad06260dbb8190bebab91d9fb9e22054120ec713852c562e8e7ec87331a1f84997861d83","name":"bikesday_reduced.csv","type":"text/csv","lastModified":1676067014293.117,"size":38177}] 2f:{"id":"cdcd1e7a9ac08be1","slug":"bayesian-linear-regression-bike-share-data","title":"Bayesian linear regression","thumbnail":"22075ee1ecdd4dcf2bbac9356ad13234e7df1cad8f811fa7a514f2b803f31bae","access_level":"public","create_time":"$D2023-01-26T15:34:15.562Z","update_time":"$D2023-12-10T12:56:36.334Z","view_time":null,"first_public_time":"$D2023-01-26T15:34:15.562Z","fork":false,"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":1872,"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":true,"trashed":false,"subscribed":null,"forks":1,"collection":{"id":"9fdb7c9ba7be6ba8","slug":"bayesian-learning","title":"Bayesian Learning","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Bayesian linear regression","cells":[{"id":0,"value":"# Bayesian linear regression","pinned":false,"hidden":false,"mode":"md"},{"id":1866,"value":"toc()","pinned":false,"hidden":false,"mode":"ojs"},{"id":241,"value":"## Bike share data\n\nThe bike share dataset collected by Fanaee-T and Gama (2013) and made available in the [UCI repository](https://archive.ics.uci.edu/ml/\ndatasets/bike+sharing+dataset) records the number of daily rides (`nrides`) with the bike share company capital bikeshare. The dataset contains the number of daily bike rides on 731 days during the two years 2011 and 2012 and a number of variables that\nmay affect the demand for bikes, e.g. weather conditions, day of the week and holidays; The table below shows the dataset.","pinned":false,"hidden":false,"mode":"md"},{"id":157,"value":"bikes = FileAttachment(\"bikesday_reduced.csv\").csv({typed: true});","pinned":false,"hidden":false,"mode":"ojs"},{"id":254,"value":"Inputs.table(bikes)","pinned":false,"hidden":false,"mode":"ojs"},{"id":288,"value":"The time series `nrides` is plotted below.","pinned":false,"hidden":false,"mode":"md"},{"id":284,"value":"plottime = Plot.plot({\n marks: [\n Plot.ruleY([0]),\n Plot.lineY(bikes, {x: \"dteday\", y: \"nrides\", stroke: \"steelblue\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1771,"value":"Here are some scatter plot of `nrides` against some the covariates used in the model:","pinned":false,"hidden":false,"mode":"md"},{"id":1736,"value":"plt_data1 = html`\n \n \n \n

\n ${pltdata1}\n

\n \n

\n ${pltdata2}\n

\n\n \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1755,"value":"plt_data2 = html`\n \n \n \n

\n ${pltdata3}\n

\n \n

\n ${pltdata4}\n

\n\n \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":246,"value":"We will first ignore the time series nature of `nrides` here and model it by regression; toward the end of the notebook we will incorporate time series effects in the regression model. The variable `nrides` are count data, but we will nevertheless model it by a Gaussian linear regression since large counts are often approximately Gaussian.","pinned":false,"hidden":false,"mode":"md"},{"id":442,"value":"## Feature construction - one-hot for categorical\n\nLet's first define our response variable `nrides`:","pinned":false,"hidden":false,"mode":"md"},{"id":186,"value":"y = math.matrix(bikes.map(d => d.nrides))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1168,"value":"For the covariates, we need to do some feature construction to turn the categorical variables into dummy or one-hot variables. A variable with K categories is turned into K-1 dummy variables, to avoid perfect collinearity. We use the default here and take the first category as the reference.","pinned":false,"hidden":false,"mode":"md"},{"id":371,"value":"season_onehot = onehot(bikes.map(d => d.season))","pinned":false,"hidden":false,"mode":"ojs"},{"id":430,"value":"weather_onehot = onehot(bikes.map(d => d.weathersit))","pinned":false,"hidden":false,"mode":"ojs"},{"id":432,"value":"weekday_onehot = onehot(bikes.map(d => d.weekday))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1170,"value":"The dummy variables are finally concatenated to the continuous covariates, to produce or final matrix of features/transformed covariates, Z.","pinned":false,"hidden":false,"mode":"md"},{"id":177,"value":"X = math.matrix(bikes.map(d => [1, d.feeltemp, d.hum, d.windspeed, d.year, d.holiday]))","pinned":false,"hidden":false,"mode":"ojs"},{"id":380,"value":"Z = cbind(cbind(cbind(X,season_onehot),weather_onehot),weekday_onehot) // feature matrix","pinned":false,"hidden":false,"mode":"ojs"},{"id":1202,"value":"varnames = [\"intercept\", \"feeltemp\", \"hum\", \"wind\", \"year\", \"holiday\", \"spring\", \"summer\", \"fall\", \"mist\", \"rain/snow\", \"monday\",\"tuesday\",\"wednesday\",\"thursday\",\"friday\",\"saturday\"]","pinned":false,"hidden":false,"mode":"ojs"},{"id":440,"value":"## Least squares estimation","pinned":false,"hidden":false,"mode":"md"},{"id":1173,"value":"Let's start with computing least squares estimates/maximum likelihood estimator and its the standard error. We'll also put the posterior mean estimator and the posterior standard deviation in the same table.","pinned":false,"hidden":false,"mode":"md"},{"id":1820,"value":"table(ols_summary,{\n sortable: true,\n rank: true})","pinned":false,"hidden":false,"mode":"ojs"},{"id":802,"value":"## Bayesian inference","pinned":false,"hidden":false,"mode":"md"},{"id":1176,"value":"We will use the conjugate prior:","pinned":false,"hidden":false,"mode":"md"},{"id":1670,"value":"priormath = md`\n${tex`\n\\boldsymbol{\\beta}\\vert \\sigma^2 \\sim \\mathrm{N}(\\mu,\\sigma^2 \\Omega_0^{-1}) \\\\\n\\sigma^2 \\sim \\mathrm{Inv-}\\chi^2(\\nu_0,\\sigma_0^2)\n`}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1687,"value":"zellnermath = md`\nwith the following two choice for specifying the precision matrix ${tex`\\Omega_0`}\n\n- Zellner's *g*-prior: ${tex`\\Omega_0 = \\frac{\\kappa_0}{n} (\\boldsymbol{X}^\\top \\boldsymbol{X})^{-1}`}\n- Ridge prior: ${tex`\\Omega_0 = \\kappa_0 \\cdot I_p`},\n\nwhere ${tex`\\kappa_0`} plays the role of prior sample size, so that ${tex`\\kappa_0 = 1`} in the *g*-prior makes the information in the prior equivalent to one data observation. \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1321,"value":"viewof settings = Inputs.form([\n Inputs.radio([\"Zellner g-prior\", \"Identity\"], {label: tex`\\text{prior type:}`, value: \"Zellner g-prior\"}),\n Inputs.range([1, 100000], {value: 1, label: tex`\\text{prior sample size, }\\kappa_0`, step: 1, transform: Math.log}),\n Inputs.range([-5000, 5000], {value: 1000, label: tex`\\text{prior intercept, }\\mu_0`, step: 1, transform: Math.log}),\n Inputs.range([0, 100], {value: 5, label: tex`\\nu_0`, step: 1}),\n Inputs.range([1, 3000], {value: 1000, label: tex`\\sigma_0`, step: 1})\n])","pinned":false,"hidden":false,"mode":"ojs"},{"id":1446,"value":"plt_first_panel = html`\n \nMarginal posteriors for selected variables. Least squared estimates are indicated by points.\n \n \n

\n ${plt1}\n

\n \n

\n ${plt2}\n

\n \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1460,"value":"plt_second_panel = html`\n \n \n \n

\n ${plt3}\n

\n \n

\n ${plt4}\n

\n \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"
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\n\n---\n#### 🙈 Appendix 🙈\nHere follow code and details that you probably do not want to read.","pinned":false,"hidden":false,"mode":"md"},{"id":683,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md"},{"id":607,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":31,"value":"math = require(\"mathjs\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":687,"value":"import {randn, mvnpdf, mvnrand} from '@mattiasvillani/multivariate-normal-distribution'","pinned":false,"hidden":false,"mode":"ojs"},{"id":786,"value":"import {randscaledinvchi2, scaledinvchi2pdf} from '@mattiasvillani/scaled-inverse-chi-2-distribution'","pinned":false,"hidden":false,"mode":"ojs"},{"id":702,"value":"import {table} from '@binarmaya/table-from-array-of-data'","pinned":false,"hidden":false,"mode":"ojs"},{"id":705,"value":"import {defaultOptions} from \"@binarmaya/table-from-array-of-data\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":1863,"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":685,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":745,"value":"function quadform(x,A){\n // computes the quadratic form x'*A*x\n return math.multiply(math.multiply(math.transpose(x), A),x)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":390,"value":"function cbind(X,Z){\n const nrows = X.size()[0]\n const ncolsX = X.size()[1]\n const ncolsZ = Z.size()[1]\n let D = math.zeros(nrows, ncolsX + ncolsZ)\n D.subset(math.index(d3.range(nrows), d3.range(ncolsX) ), X)\n return D.subset(math.index(d3.range(nrows), d3.range(ncolsX, ncolsX + ncolsZ) ), Z)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":294,"value":"function onehot(x, ref_category = \"first\"){\n var uniqueItems = [...new Set(x)].sort();\n if (ref_category == \"first\"){\n ref_category = uniqueItems[0]\n }\n uniqueItems = uniqueItems.filter(x => x != ref_category) // remove reference category\n let X = math.zeros(x.length, uniqueItems.length-1)\n for (let i = 0; i<=(uniqueItems.length-1); i++){\n let x_i = x.map(d => d == uniqueItems[i])\n X.subset(math.index(d3.range(x_i.length),i), x_i) // X[:,i] = x_i\n }\n return X\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":102,"value":"function least_squares(y,X){\n const n = X.size()[0];\n const p = X.size()[1];\n const Xty = math.multiply(math.transpose(X), y);\n const invXtX = math.inv(math.multiply(math.transpose(X), X));\n const betahat = math.multiply(invXtX, Xty);\n const residuals = math.subtract(y,math.multiply(X, betahat));\n const s2 = math.multiply(math.transpose(residuals), residuals)/(n-p);\n const covbeta = math.multiply(s2, invXtX);\n return {betahat: betahat, covbeta: covbeta}\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1154,"value":"function zellner_prior(kappa0, priorintercept, X){\n const n = X.size()[0]\n const p = X.size()[1]\n const μ0 = math.zeros(p)\n μ0.subset(math.index(0), priorintercept);\n return {μ0: μ0, Ω0: math.multiply(kappa0/n, math.multiply(math.transpose(X), X ))}\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":445,"value":"function bayeslinreg(y, X, μ0, Ω0, ν0, σ20){\n\n // Compute posterior hyperparameters\n\n // OLS\n const n = X.size()[0]\n const p = X.size()[1]\n const Xty = math.multiply(math.transpose(X), y);\n const XX = math.multiply(math.transpose(X), X)\n const invXtX = math.inv(XX);\n const βhat = math.multiply(invXtX, Xty);\n\n // Bayes\n const νn = ν0 + n\n const Ωn = math.add(XX, Ω0)\n const invΩn = math.inv(Ωn)\n const μn = math.multiply(invΩn, math.add( math.multiply(XX,βhat), math.multiply(Ω0,μ0) ) )\n const residuals = math.subtract(y, math.multiply(X,βhat) )\n const RSS = math.multiply(math.transpose(residuals), residuals) \n const SSmuOLS = quadform(math.subtract(μn, βhat), XX) // (μn-βhat)'*XX*(μn-βhat) \n const SSmuPrior = quadform(math.subtract(μn, μ0), Ω0) // (μn-μ0)'*Ω0*(μn-μ0)\n const σ2n = (ν0*σ20 + RSS + SSmuOLS + SSmuPrior )/νn\n\n return {μn: μn, invΩn: invΩn, νn: νn, σ2n:σ2n}\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1520,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":1184,"value":"function studentt_pdf(x, mu, sigma, nu){\n return jstat.studentt.pdf((x-mu)/sigma, nu)/sigma\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1188,"value":"function marginal_pdf(mu_vect, sigma_vect, nu, varnames){\n const p = mu_vect.length;\n var allmarginals = [];\n for (let i = 0; i <= p; i++){\n var mu = mu_vect[i]\n var sigma = sigma_vect[i]\n var x = d3.range(mu - 4*sigma, mu + 4*sigma, 8*sigma/100);\n let marginals = x.map(x => ({x: x, pdf: studentt_pdf(x, mu, sigma, nu), variable: varnames[i]}));\n allmarginals = allmarginals.concat(marginals)\n }\n return allmarginals\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1180,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":193,"value":"ols = least_squares(y, Z)","pinned":false,"hidden":false,"mode":"ojs"},{"id":198,"value":"betahat = ols.betahat._data","pinned":false,"hidden":false,"mode":"ojs"},{"id":204,"value":"stdbetahat = d3.range(Z.size()[1]).map(i => math.sqrt(ols.covbeta._data[i][i]))","pinned":false,"hidden":false,"mode":"ojs"},{"id":667,"value":"tratios = math.dotDivide(betahat, stdbetahat)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1776,"value":"ols_summary = d3.range(betahat.length).map(i => ({variable: varnames[i], beta_mle: (betahat[i]).toPrecision(4), beta_bayes: (mu_vect[i]).toPrecision(4), se_mle: (stdbetahat[i]).toPrecision(4), sigma_bayes: (sigma_vect[i]).toPrecision(4)}))","pinned":false,"hidden":false,"mode":"ojs"},{"id":818,"value":"unitinfo = zellner_prior(settings[1], settings[2], Z)","pinned":false,"hidden":false,"mode":"ojs"},{"id":804,"value":"bres = {\n if (settings[0] == \"Identity\"){\n const p = Z.size()[1]\n const Ω0 = math.multiply(settings[1], math.identity(p))\n var μ0 = math.zeros(p);\n μ0.subset(math.index(0), settings[2])\n return bayeslinreg(y, Z, μ0, Ω0, settings[3], settings[4]**2)\n }else{\n const unitinfo = zellner_prior(settings[1], settings[2], Z)\n return bayeslinreg(y, Z, unitinfo.μ0, unitinfo.Ω0, settings[3], settings[4]**2)\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1224,"value":"mu_vect = d3.range(17).map(i => bres.μn._data[i])","pinned":false,"hidden":false,"mode":"ojs"},{"id":1212,"value":" sigma_vect = d3.range(17).map(i => math.sqrt(math.multiply(bres.σ2n, bres.invΩn)._data[i][i]))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1236,"value":"marginal_pdfs = marginal_pdf(mu_vect, sigma_vect, bres.νn, varnames)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1340,"value":"marginal_pdf_main = marginal_pdfs.filter(d => d.variable === \"intercept\" || d.variable === \"feeltemp\" || \n d.variable === \"hum\" || d.variable === \"wind\" || d.variable === \"holiday\" ||\n d.variable === \"year\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1793,"value":"ols_summary_main = ols_summary.filter(d => d.variable === \"intercept\" || d.variable === \"feeltemp\" || \n d.variable === \"hum\" || d.variable === \"wind\" || d.variable === \"holiday\" ||\n d.variable === \"year\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1355,"value":"marginal_pdfs_season = marginal_pdfs.filter(d => d.variable === \"spring\" || d.variable === \"summer\" || \n d.variable === \"fall\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1800,"value":"ols_summary_season = ols_summary.filter(d => d.variable === \"spring\" || d.variable === \"summer\" || \n d.variable === \"fall\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1373,"value":"marginal_pdfs_weather = marginal_pdfs.filter(d => d.variable === \"mist\" || d.variable === \"rain/snow\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1804,"value":"ols_summary_weather = ols_summary.filter(d => d.variable === \"mist\" || d.variable === \"rain/snow\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1362,"value":"marginal_pdfs_weekday = marginal_pdfs.filter(d => d.variable === \"monday\" || d.variable === \"tuesday\" || \n d.variable === \"wednesday\" || d.variable === \"thursday\" || d.variable === \"friday\" || \n d.variable === \"saturday\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1806,"value":"ols_summary_weekday = ols_summary.filter(d => d.variable === \"monday\" || d.variable === \"tuesday\" || \n d.variable === \"wednesday\" || d.variable === \"thursday\" || d.variable === \"friday\" || \n d.variable === \"saturday\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":1531,"value":"function marginal_pdf_sigma2(ν0, σ20, νn, σ2n){\n var x = d3.range(0.1, 3000^2);\n var sigma2pdfs = x.map(x => ({pdf: scaledinvchi2pdf(x, ν0, σ20), type: \"prior\"}));\n sigma2pdfs = sigma2pdfs.concat( x.map(x => ({pdf: scaledinvchi2pdf(x, νn, σ2n), type: \"posterior\"})) );\n return sigma2pdfs\n} ","pinned":false,"hidden":false,"mode":"ojs"},{"id":1764,"value":"#### Plotting data","pinned":false,"hidden":false,"mode":"md"},{"id":1713,"value":"pltdata1 = Plot.plot({\n color: {\n legend: true,\n type: \"categorical\",\n },\n caption: \"Plot of nrides against feeltemp - colorcoded by season\",\n marks: [\n Plot.dot(bikes, {x: \"feeltemp\", y: \"nrides\", fill: \"season\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1732,"value":"pltdata2 = Plot.plot({\n color: {\n legend: true,\n type: \"categorical\",\n },\n caption: \"Plot of nrides against feeltemp - colorcoded by year\",\n marks: [\n Plot.dot(bikes, {x: \"feeltemp\", y: \"nrides\", fill: \"year\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1748,"value":"pltdata3 = Plot.plot({\n color: {\n legend: true,\n type: \"categorical\",\n },\n caption: \"Plot of nrides against feeltemp - colorcoded by weather\",\n marks: [\n Plot.dot(bikes, {x: \"feeltemp\", y: \"nrides\", fill: \"weathersit\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1751,"value":"pltdata4 = Plot.plot({\n color: {\n legend: true,\n type: \"categorical\",\n },\n caption: \"Plot of nrides against feeltemp - colorcoded by holiday\",\n marks: [\n Plot.dot(bikes, {x: \"feeltemp\", y: \"nrides\", fill: \"holiday\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1766,"value":"#### Plotting posteriors","pinned":false,"hidden":false,"mode":"md"},{"id":1300,"value":"plt1 = Plot.plot({\n color: {\n legend: true\n },\n x: {\n domain: [-4000,6500]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.lineY(marginal_pdf_main, {x: \"x\", y: \"pdf\", stroke: \"variable\", strokeWidth: 2}),\n Plot.dot(ols_summary_main, {x: \"beta_mle\", y: 0, fill: \"variable\", r: 4})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1317,"value":"plt2 = Plot.plot({\n color: {\n legend: true\n },\n x: {\n domain: [-500,2000]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.lineY(marginal_pdfs_season, {x: \"x\", y: \"pdf\", stroke: \"variable\", strokeWidth: 2}),\n Plot.dot(ols_summary_season, {x: \"beta_mle\", y: 0, fill: \"variable\", r: 4})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1364,"value":"plt3 = Plot.plot({\n color: {\n legend: true\n },\n x: {\n domain: [-1000, 1000]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.lineY(marginal_pdfs_weekday, {x: \"x\", y: \"pdf\", stroke: \"variable\", strokeWidth: 2}),\n Plot.dot(ols_summary_weekday, {x: \"beta_mle\", y: 0, fill: \"variable\", r: 4})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1382,"value":"plt4 = Plot.plot({\n color: {\n legend: true\n },\n x: {\n domain: [-2800,1000]\n },\n marks: [\n Plot.ruleY([0]),\n Plot.lineY(marginal_pdfs_weather, {x: \"x\", y: \"pdf\", stroke: \"variable\", strokeWidth: 2}),\n Plot.dot(ols_summary_weather, {x: \"beta_mle\", y: 0, fill: \"variable\", r: 4})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}} 30:"$2f:body"