2b:null 2c:["$","title",null,{"children":"Multivariate LogitNormal Distribution | Mattias Villani | Observable"}] 31:T40a,md` The [Multivariate LogitNormal distribution](https://en.wikipedia.org/wiki/Logit-normal_distribution#Multivariate_generalization) is a multivariate probability distribution defined over the unit simplex ${tex`\sum_{k=1}^K x_k = 1 \text{ and } x_k \geq 0 \text{ for all } k = 1, \dots, K.`} The distribution is obtained from a drawing a ${tex`K-1`}-dimensional multivariate normal vector ${tex`\boldsymbol{y} =(y_1,\ldots,y_{K-1})^\top \sim N(\boldsymbol{\mu},\boldsymbol{\Sigma})`} and applying a softmax transformation ${tex`x_k = \frac{e^{y_k}}{\sum_{k=1}^{K}e^{y_k}} \text{ for }k=1,\ldots,K,`} where ${tex`y_K = 0`}. The density of the multivariate LogitNormal distribution is given by ${tex` f(\boldsymbol{x}) = \vert 2\pi \boldsymbol{\Sigma} \vert^{-1/2} \Big(\prod_{k=1}^K x_k\Big)^{-1}\exp\Big( -\frac{1}{2}(\tilde \boldsymbol{x}-\boldsymbol{\mu})^\top\boldsymbol{\Sigma}^{-1}(\tilde \boldsymbol{x} - \boldsymbol{\mu}) \Big) `} where ${tex`\tilde \boldsymbol{x} = (\log(x_1/x_K),\ldots,\log(x_{K-1}/x_K))^\top`}. `32:[] 2f:{"id":"14554c2f6979d078","slug":"multivariate-logitnormal-distribution","title":"Multivariate LogitNormal Distribution","thumbnail":"59af026b39e0ffe239a52d9c6a4cb9dc343df1421e9c7259a571ee5b269e9759","access_level":"public","create_time":"$D2023-12-31T08:44:50.636Z","update_time":"$D2026-08-22T18:38:23.335Z","view_time":null,"first_public_time":"$D2023-12-31T10:25:02.059Z","fork":{"id":"3b61b39228e56c0c","slug":"dirichlet-distribution","title":"Dirichlet 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":855,"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,"subscribed":null,"forks":1,"collection":{"id":"40c8ef5af1a51aff","slug":"distributions","title":"Statistical distributions","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Multivariate LogitNormal Distribution","cells":[{"id":706,"value":"# Multivariate LogitNormal Distribution","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":0,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":756,"value":"viewof param = Inputs.form([\n Inputs.range([-1, 1], {label: tex`\\mu_1`, step: 0.1}),\n Inputs.range([-1, 1], {label: tex`\\mu_2`, step: 0.1}),\n Inputs.range([0.1, 2], {label: tex`\\sigma_1`, step: 0.1}),\n Inputs.range([0.1, 2], {label: tex`\\sigma_2`, step: 0.1}),\n Inputs.range([-0.99, 0.99], {label: tex`\\rho`, step: 0.01})\n])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":491,"value":"ternaryDensity(density, densityResolution, { size: 400 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":530,"value":"legend({color: d3.scaleSequential(d3.extent(density.map(d => d.density)), d3.interpolateBlues), title: \"Density\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":716,"value":"

","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":758,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":749,"value":"math = require(\"mathjs\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":828,"value":"disttext = md`

Ternary density plot from **Herb Susmann** at https://observablehq.com/@herbps10`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":485,"value":"import {ternaryDensity} from \"@mattiasvillani/imports\" ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":762,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":764,"value":"mu = [param[0],param[1]]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":767,"value":"sigma1 = param[2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":769,"value":"sigma2 = param[3]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":771,"value":"rho = param[4]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":776,"value":"Sigma = [[sigma1**2,rho*sigma1*sigma2],[rho*sigma1*sigma2,sigma2**2]]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":760,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":747,"value":"function multiLogitNormal_pdf(x, mu, Sigma){\n const p = math.matrix(mu).size()[0]; // Dimension of x is K, dim of mu and Sigma is p = K-1\n \n const logDet = -(p/2)*math.log(2*math.pi) -0.5*math.log(math.det(Sigma));\n const logJacobian = - d3.sum(x.map(d => Math.log(d)))\n //let xtilde = math.multiply( (1/x[p]), x.slice(0, p - 1)) // This is p = K-1 dimensional\n //xtilde = xtilde.map(d => Math.log(d))\n const xtilde = x.slice(0, p).map(d => Math.log(d/x[p]))\n \n const quad = \n -0.5*math.multiply( math.multiply( math.transpose(math.subtract(xtilde,mu)), math.inv(Sigma)), \n math.subtract(xtilde,mu))\n return Math.exp(logDet + logJacobian + quad)\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":533,"value":"density = {\n const density = [];\n for(let i = 1; i < densityResolution; i++) {\n for(let j = 1; j < densityResolution - i; j++) {\n const x1 = i / densityResolution ;\n const x2 = j / densityResolution;\n const x3 = Math.max(0, 1 - x1 - x2);\n const x = [ x1, x2, x3 ];\n\n density.push({ x, density: multiLogitNormal_pdf(x, mu, Sigma)});\n }\n }\n return density;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":535,"value":"densityResolution = 40","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":51,"value":"margin = ({ left: 30, top: 30, right: 30, bottom: 30 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":8,"value":"height = 400","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":4,"value":"d3 = require(\"d3@v6\", \"d3-hexbin@0.2\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":179,"value":"import {Range} from \"@observablehq/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":275,"value":"import {legend} from \"@d3/color-legend\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 30:"$2f:body"