2b:null 2c:["$","title",null,{"children":"Bayesian inference for multinomial data | Mattias Villani | Observable"}] 31:T53a,Consider multinomial data ${tex`\boldsymbol{n}=(n_1,n_2,\ldots,n_C)`}, where ${tex`n_c`} is the number of counts in category ${tex`c=1,2,\ldots,C`}. The [multinomial distribution](https://en.wikipedia.org/wiki/Multinomial_distribution) has probability mass function ${tex` p(\boldsymbol{n}) = \frac{n!}{n_1! \cdots n_C!}\theta_1^{n_1} \cdots \theta_C^{n_C} `} where ${tex`\theta_c\geq 0`} is the probability of an observation in category *c* and ${tex`\sum_{c=1}^C \theta_c = 1`}. The most commonly used prior distribution for ${tex`\boldsymbol{\theta}=(\theta_1,\ldots,\theta_C)`} is the [Dirichlet distribution](https://en.wikipedia.org/wiki/Dirichlet_distribution). If ${tex`\boldsymbol{\theta} \sim \mathrm{Dirichlet}(\alpha_1,\ldots,\alpha_C)`} then the density of ${tex`\boldsymbol{\theta}`} is ${tex` p(\boldsymbol{\theta}) = \frac{1}{\mathrm{B}(\alpha_1, \dots, \alpha_C)} \prod_{i=1}^C \theta_i^{\alpha_i - 1}\text{ for } \theta_i \geq 0 \text{ and } \sum_{i=1}^C \theta_i = 1. `} where ${tex`\alpha_i > 0`} and ${tex`\mathrm{B}(\alpha_1, \dots, \alpha_k)`} is the [multivariate Beta function](https://en.wikipedia.org/wiki/Beta_function#Multivariate_beta_function). The posterior distribution after observing the counts ${tex`\boldsymbol{n}=(n_1,n_2,\ldots,n_C)`} is also Dirichlet, and the prior-to-posterior updating is:32:T4b4,projection = ({ width, height, angle = 0 }) => { const sqrt3_2 = Math.sqrt(3) / 2; let scale = 1; let tx = 0; let ty = 0; const ca = Math.cos((angle * Math.PI) / 180); const sa = Math.sin((angle * Math.PI) / 180); const q = (x, y) => [x + y * 0.5, -y * sqrt3_2]; const p = (x, y) => { [x, y] = q(x, y); return [scale * (x * ca + y * sa) + tx, scale * (y * ca - x * sa) + ty]; }; const projection = d3.geoTransform({ point: function (x, y) { this.stream.point(...p(x, y)); }, sphere: function () { this.stream.polygonStart(); this.stream.lineStart(); this.stream.point(...p(0, 0)); this.stream.point(...p(0, 1)); this.stream.point(...p(1, 0)); this.stream.point(...p(0, 0)); this.stream.lineEnd(); this.stream.polygonEnd(); } }); // Fit bounds (with angle!) const [[x1, y1], [x2, y2]] = d3 .geoPath(projection) .bounds({ type: "Sphere" }); const w = width / (x2 - x1); const h = height / (y2 - y1); scale = Math.min(w, h); tx = w > scale ? width / 2 - (scale * (x2 + x1)) / 2 : -x1 * scale; ty = w < scale ? height / 2 - (scale * (y2 + y1)) / 2 : -y1 * scale; return projection; }33:[] 2f:{"id":"136a770293d82a03","slug":"multinomial-dirichlet","title":"Bayesian inference for multinomial data","thumbnail":"ca59aad92107dc9708b33882c4ab14019b146e54dbc9341b729f5fcadc6cff30","access_level":"public","create_time":"$D2023-10-06T07:33:32.172Z","update_time":"$D2023-11-18T22:08:38.846Z","view_time":null,"first_public_time":"$D2023-10-07T00:16:41.801Z","fork":{"id":"f157087519a00163","slug":"ternary-plot","title":"Ternary plot","owner":{"id":"da2a32154638c457","login":"fil","name":"Fil"}},"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":1685,"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 inference for multinomial data","cells":[{"id":1454,"value":"# Bayesian inference for multinomial data\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1429,"value":"$31","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1650,"value":"resultstext = html`

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

Model:	${tex`n_1,\\ldots,n_C \\mid \\boldsymbol{\\theta} \\sim \\operatorname{Multinomial}(\\theta_1,\\ldots,\\theta_C)`}.
Prior:	${tex`\\boldsymbol{\\theta} \\sim \\operatorname{Dirichlet}(\\alpha_1,\\ldots,\\alpha_C)`}
Posterior:	${tex`\\boldsymbol{\\theta} \\mid \\boldsymbol{n} \\sim \\operatorname{Dirichlet}(\\alpha_1 + n_1, \\ldots, \\alpha_C + n_C)`}

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1297,"value":"**Prior**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1064,"value":"viewof prior = Inputs.form([\n Inputs.range([Number.EPSILON, 50], {value: 3, label: tex`\\alpha_1`, step: 0.1}),\n Inputs.range([Number.EPSILON, 50], {value: 3, label: tex`\\alpha_2`, step: 0.1}),\n Inputs.range([Number.EPSILON, 50], {value: 5, label: tex`\\alpha_3`, step: 0.1})\n])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1294,"value":"**Data**","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1292,"value":"viewof data = Inputs.form([\n Inputs.range([0, 100], {value: 5, label: tex`n_1`, step: 1}),\n Inputs.range([0, 100], {value: 10, label: tex`n_2`, step: 1}),\n Inputs.range([0, 100], {value: 3, label: tex`n_3`, step: 1})\n])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1284,"value":" viewof ndraws = Inputs.range([1, 500], {value: 200, label: \"number of draws in plot\", step: 1})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1213,"value":"viewof button = Inputs.button(\"Simulate new draws!\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1264,"value":"\n \n \n \n

Prior distribution

\n ${tex`\\mathbb{E}(\\boldsymbol{\\theta})=[${priormean.map(d => d.toFixed(4))}]`}
\n ${tex`\\mathbb{S}(\\boldsymbol{\\theta})=[${priorstdev.map(d => d.toFixed(4))}]`}\n ${priorplt}\n

Likelihood (normalized)

\n ${tex``}
\n ${tex``}
\n ${likeplt}\n

Posterior distribution

\n ${tex`\\mathbb{E}(\\boldsymbol{\\theta}\\vert \\boldsymbol{n})=[${postmean.map(d => d.toFixed(4))}]`}
\n ${tex`\\mathbb{S}(\\boldsymbol{\\theta}\\vert \\boldsymbol{n})=[${poststdev.map(d => d.toFixed(4))}]`}\n ${postplt}\n

\n \n","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1243,"value":"
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\n\n---\n#### 🙈 Appendix 🙈\n\nHere is just some data, imports and function defintions that you probably don't want to read.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1255,"value":"The ternary plot comes from [@fil](https://observablehq.com/@fil)'s [notebook](https://observablehq.com/@fil/ternary-plot)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":402,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":246,"value":"normalize = ({ a, b, c, ...options } = {}) => {\n if (a == null) throw new Error(\"missing channel a\");\n if (b == null) throw new Error(\"missing channel b\");\n if (c == null) throw new Error(\"missing channel c\");\n const [X, setX] = Plot.column(a); // Using a hides x from the tip!\n const [Y, setY] = Plot.column(b);\n return Plot.transform({ x: X, y: Y, ...options }, (data, facets) => {\n const A = Plot.valueof(data, a);\n const B = Plot.valueof(data, b);\n const C = Plot.valueof(data, c);\n setX(A.map((d, i) => d / (A[i] + B[i] + C[i])));\n setY(B.map((d, i) => d / (A[i] + B[i] + C[i])));\n return { data, facets };\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":68,"value":"graticule = (options) =>\n Plot.link(\n d3.range(0.1, 1, 0.1).flatMap((d) => [\n [d, 0, 0, d],\n [d, 0, d, 1 - d],\n [0, d, 1 - d, d]\n ]),\n {\n x1: \"0\",\n y1: \"1\",\n x2: \"2\",\n y2: \"3\",\n stroke: \"#aaa\",\n strokeWidth: 0.5,\n ...options\n }\n )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":289,"value":"tickLabels = (\n data = d3.range(0.1, 1, 0.1),\n { tickFormat = \".0\", ...options } = {}\n) => {\n const text = d3.format(tickFormat);\n return Plot.marks(\n Plot.text(data, {\n x: (d) => 1 - d,\n y: 0,\n text,\n rotate: 60,\n textAnchor: \"start\",\n dx: 2,\n dy: 5,\n ...options\n }),\n Plot.text(data, {\n x: 0,\n y: (d) => d,\n text,\n rotate: 0,\n textAnchor: \"end\",\n dx: -5,\n dy: 0,\n ...options\n }),\n Plot.text(data, {\n x: (d) => d,\n y: (d) => 1 - d,\n text,\n textAnchor: \"start\",\n rotate: -60,\n dx: 2,\n dy: -5,\n ...options\n })\n );\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":286,"value":"labels = (data, options) =>\n Plot.text(data, {\n x: (_, i) => [1.075, -0.025, -0.025][i],\n y: (_, i) => [-0.05, 1.05, -0.05][i],\n text: Plot.identity,\n ...options\n })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":338,"value":"ternary = ({\n projection,\n graticule,\n normalize,\n sphere: Plot.sphere,\n tickLabels,\n labels\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":172,"value":"axes = [\"θ₁\", \"θ₂\", \"θ₃\"]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":10,"value":"priorplt = Plot.plot({\n width: 500,\n height: 420,\n inset: 45,\n color: { \n legend: true, \n type: \"linear\",\n range: [\"#faf7ee\", \"#806d33\"]\n //scheme: \"YlOrRd\"\n }, \n projection: ternary.projection,\n marks: [\n Plot.sphere(),\n ternary.graticule({ strokeDasharray: \"5 1.5\" }),\n \n Plot.dot(\n priordraws,\n ternary.normalize({\n a: \"x1\",\n b: \"x2\",\n c: \"x3\",\n fill: \"logdensity\",\n stroke: \"white\",\n r: 20/Math.pow(ndraws,0.3),\n channels: {\n label: \"name\",\n x1: \"x1\",\n x2: \"x2\",\n x3: \"x3\"\n },\n tip: true\n })\n ),\n\n ternary.tickLabels(d3.range(0.1, 1, 0.1), {\n fill: \"currentColor\",\n stroke: \"white\",\n fontSize: 12\n }),\n ternary.labels(axes, { fontWeight: \"normal\", fontSize: 18 })\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1509,"value":"likeplt = Plot.plot({\n width: 500,\n height: 420,\n inset: 45,\n color: { \n legend: true, \n type: \"linear\",\n scheme: \"Blues\"\n }, \n projection: ternary.projection,\n marks: [\n Plot.sphere(),\n ternary.graticule({ strokeDasharray: \"5 1.5\" }),\n\n // A blob for each group\n \n Plot.dot(\n likedraws,\n ternary.normalize({\n a: \"x1\",\n b: \"x2\",\n c: \"x3\",\n fill: \"logdensity\",\n stroke: \"white\",\n r: 20/Math.pow(ndraws,0.3),\n channels: {\n label: \"name\",\n x1: \"x1\",\n x2: \"x2\",\n x3: \"x3\"\n },\n tip: true\n })\n ),\n\n ternary.tickLabels(d3.range(0.1, 1, 0.1), {\n fill: \"currentColor\",\n stroke: \"white\",\n fontSize: 12\n }),\n ternary.labels(axes, { fontWeight: \"normal\", fontSize: 18 })\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1273,"value":"postplt = Plot.plot({\n width: 500,\n height: 420,\n inset: 45,\n color: { \n legend: true, \n type: \"linear\",\n range: [\"#f7eded\", \"#b85450\"]\n }, \n projection: ternary.projection,\n marks: [\n Plot.sphere(),\n ternary.graticule({ strokeDasharray: \"5 1.5\" }),\n\n // A blob for each group\n \n Plot.dot(\n postdraws,\n ternary.normalize({\n a: \"x1\",\n b: \"x2\",\n c: \"x3\",\n fill: \"logdensity\",\n stroke: \"white\",\n r: 20/Math.pow(ndraws,0.3),\n channels: {\n label: \"name\",\n x1: \"x1\",\n x2: \"x2\",\n x3: \"x3\"\n },\n tip: true\n })\n ),\n\n ternary.tickLabels(d3.range(0.1, 1, 0.1), {\n fill: \"currentColor\",\n stroke: \"white\",\n fontSize: 12\n }),\n ternary.labels(axes, { fontWeight: \"normal\", fontSize: 18 })\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1112,"value":"alpha = [prior[0], prior[1], prior[2]]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1345,"value":"alphapost = [prior[0] + n[0], prior[1] + n[1] , prior[2] + n[2]]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1513,"value":"alphalike = [ n[0]+0.001, n[1]+0.001 , n[2]+0.001]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1301,"value":"n = [data[0], data[1], data[2]]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1192,"value":"sumalpha = d3.sum(alpha)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1338,"value":"sumalphapost = d3.sum(alphapost)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1516,"value":"sumalphalike = d3.sum(alphalike)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1195,"value":"priormean = [alpha[0]/sumalpha, alpha[1]/sumalpha, alpha[2]/sumalpha]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1343,"value":"postmean = [alphapost[0]/sumalphapost, alphapost[1]/sumalphapost, alphapost[2]/sumalphapost]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1208,"value":"priorstdev = d3.range(3).map(d => Math.sqrt(priormean[d]*(1-priormean[d])/(sumalpha+1)))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1353,"value":"poststdev = d3.range(3).map(d => Math.sqrt(postmean[d]*(1-postmean[d])/(sumalphapost+1)))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1072,"value":"gamma1 = d3.randomGamma(alpha[0], 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1070,"value":"gamma2 = d3.randomGamma(alpha[1], 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1075,"value":"gamma3 = d3.randomGamma(alpha[2], 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1316,"value":"gamma1post = d3.randomGamma(alphapost[0], 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1318,"value":"gamma2post = d3.randomGamma(alphapost[1], 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1320,"value":"gamma3post = d3.randomGamma(alphapost[2], 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1518,"value":"gamma1like = d3.randomGamma(n[0], 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1520,"value":"gamma2like = d3.randomGamma(n[1], 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1522,"value":"gamma3like = d3.randomGamma(n[2], 1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1062,"value":"priordraws = {\n button;\n var draws = [];\n var x1, x2, x3;\n var x1norm, x2norm, x3norm;\n var summa;\n for (var i in d3.range(ndraws)){\n [x1, x2, x3] = [gamma1(),gamma2(),gamma3()]\n summa = x1 + x2 + x3;\n [x1norm, x2norm, x3norm] = [x1/summa, x2/summa, x3/summa]\n draws.push(({x1: x1norm, x2: x2norm, x3: x3norm, density: dirichletDensity([x1norm,x2norm,x3norm], alpha), \n logdensity: dirichletLogDensity([x1norm,x2norm,x3norm], alpha)}))\n }\n return draws\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1308,"value":"postdraws = {\n button;\n var draws = [];\n var x1, x2, x3;\n var x1norm, x2norm, x3norm;\n var summa;\n for (var i in d3.range(ndraws)){\n [x1, x2, x3] = [gamma1post(),gamma2post(),gamma3post()]\n summa = x1 + x2 + x3;\n [x1norm, x2norm, x3norm] = [x1/summa, x2/summa, x3/summa]\n draws.push(({x1: x1norm, x2: x2norm, x3: x3norm, density: dirichletDensity([x1norm,x2norm,x3norm], alphapost),\n logdensity: dirichletLogDensity([x1norm,x2norm,x3norm], alphapost)}))\n }\n return draws\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1525,"value":"likedraws = {\n button;\n var draws = [];\n var x1, x2, x3;\n var x1norm, x2norm, x3norm;\n var summa;\n for (var i in d3.range(ndraws)){\n [x1, x2, x3] = [gamma1like(),gamma2like(),gamma3like()]\n summa = x1 + x2 + x3;\n [x1norm, x2norm, x3norm] = [x1/summa, x2/summa, x3/summa]\n draws.push(({x1: x1norm, x2: x2norm, x3: x3norm, density: dirichletDensity([x1norm,x2norm,x3norm], alphalike),\n logdensity: dirichletLogDensity([x1norm,x2norm,x3norm], alphalike)}))\n }\n return draws\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1046,"value":"mathfn = require('https://bundle.run/mathfn@1.1.0')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1048,"value":"multivariatebeta = (alpha) => {\n let result = 1;\n for(let i = 0; i < alpha.length; i++) {\n result *= mathfn.gamma(alpha[i]);\n }\n result = result / mathfn.gamma(d3.sum(alpha));\n return result;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1466,"value":"logMultivariatebeta = (alpha) => {\n let logMultiBeta = 0;\n for(let i = 0; i < alpha.length; i++) {\n logMultiBeta += mathfn.logGamma(alpha[i]);\n }\n logMultiBeta = logMultiBeta - mathfn.logGamma(d3.sum(alpha));\n return logMultiBeta;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1050,"value":"dirichletDensity = (x, alpha) => {\n let result = 1;\n for(let i = 0; i < x.length; i++) result *= Math.pow(x[i], alpha[i] - 1);\n return result / multivariatebeta(alpha);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1464,"value":"dirichletLogDensity = (x, alpha) => {\n let logdens = 0;\n for(let i = 0; i < x.length; i++) logdens += (alpha[i] - 1)*Math.log(x[i]);\n return logdens - logMultivariatebeta(alpha);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1652,"value":"#### Design","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1662,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1654,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 30:"$2f:body"