2a:null 2b:["$","title",null,{"children":"Kullback-Leibler divergence - continuous distributions | Mattias Villani | Observable"}] 31:T4be,reverseKLtext = md` The **Reverse KL** is instead defined with the ${tex`q(x)`} as the judge of the importance of the difference ${tex`x`}-values: ${tex.block`\mathrm{KL}(q \mid \mid p) = \int q(x)\log\frac{q(x)}{p(x)}dx = \mathbb{E}_{q}\log \frac{q(x)}{p(x)}`} When using these two different versions of the KL divergence to find the best approximation ${tex`q(x)`} of ${tex`p(x)`} will often give rise to very different approximating distributions. It may seem strange in the reverse KL to let the approximation itself determine which ${tex`x`}-values are important, and indeed it is. But the reverse KL gives a tractable optimization problem and is therefore the choice in variational approximations of the posterior distribution. The difference in approximation behavior between forward and reverse KL is is best illustrated when using a unimodal distribution ${tex`q(x)`} to approximate a multimodal distribution ${tex`p(x)`}, where: - The *forward KL* tends to give an optimal approximating distribution ${tex`q(x)`} with a large variance to cover all modes of ${tex`p(x)`} - The *reverse KL* tends to give an optimal approximating distribution ${tex`q(x)`} with most of its mass at one of the modes. `32:T6c3,viewof pparams = { var form; if (pdist == 'normal'){ form = Inputs.form([ Inputs.range([-10, 10], {value: 0, step: 0.1, label: tex`\text{mean }\mu`}), Inputs.range([Number.EPSILON, 5], {value: 1, step: 0.1, label: tex`\text{stdev }\sigma`}) ]) } if (pdist == 'beta'){ form = Inputs.form([ Inputs.range([0.1, 20], {value: 3, step: 0.1, label: tex`\text{shape }\alpha`}), Inputs.range([0.1, 20], {value: 5, step: 0.1, label: tex`\text{shape }\beta`}) ]) } else if (pdist == 'studentt'){ form = Inputs.form([ Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\text{location }\mu`}), Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma`}), Inputs.range([1, 50], {value: 4, step: 1, label: tex`\text{df }\nu`}) ]) } else if (pdist == 'chi2'){ form = Inputs.form([ Inputs.range([1, 50], {value: 3, step: 1, label: tex`\text{degrees of freedom }\nu`}) ]) } else if (pdist == 'lognormal'){ form = Inputs.form([ Inputs.range([-5,5], {value: 0, step: 0.1, label: tex`\text{location }\mu`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma`}) ]) } else if (pdist == 'bimodal'){ form = Inputs.form([ Inputs.range([-5,5], {value: -2, step: 0.1, label: tex`\text{location }\mu_1`}), Inputs.range([-5,5], {value: 2, step: 0.1, label: tex`\text{location }\mu_2`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma_1`}), Inputs.range([0.1,5], {value: 0.5, step: 0.1, label: tex`\text{scale }\sigma_2`}), Inputs.range([0,1], {value: 0.3, step: 0.01, label: tex`\text{weight }w_1`}) ]) } return form; }33:T6c5,viewof qparams = { var form; if (qdist == 'normal'){ form = Inputs.form([ Inputs.range([-10, 10], {value: 1, step: 0.1, label: tex`\text{mean }\mu`}), Inputs.range([Number.EPSILON, 5], {value: 0.5, step: 0.1, label: tex`\text{stdev }\sigma`}) ]) } if (qdist == 'beta'){ form = Inputs.form([ Inputs.range([0.1, 20], {value: 3, step: 0.1, label: tex`\text{shape }\alpha`}), Inputs.range([0.1, 20], {value: 5, step: 0.1, label: tex`\text{shape }\beta`}) ]) } else if (qdist == 'studentt'){ form = Inputs.form([ Inputs.range([-5, 5], {value: 0, step: 0.1, label: tex`\text{location }\mu`}), Inputs.range([0.1, 5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma`}), Inputs.range([1, 50], {value: 4, step: 1, label: tex`\text{df }\nu`}) ]) } else if (qdist == 'chi2'){ form = Inputs.form([ Inputs.range([1, 50], {value: 3, step: 0.1, label: tex`\text{degrees of freedom }\nu`}) ]) } else if (qdist == 'lognormal'){ form = Inputs.form([ Inputs.range([-5,5], {value: 0, step: 0.1, label: tex`\text{location }\mu`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma`}) ]) } else if (qdist == 'bimodal'){ form = Inputs.form([ Inputs.range([-5,5], {value: -2, step: 0.1, label: tex`\text{location }\mu_1`}), Inputs.range([-5,5], {value: 2, step: 0.1, label: tex`\text{location }\mu_2`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma_1`}), Inputs.range([0.1,5], {value: 1, step: 0.1, label: tex`\text{scale }\sigma_2`}), Inputs.range([0,1], {value: 0.5, step: 0.01, label: tex`\text{weight }w_1`}) ]) } return form; }34:T4bd,plt_pdf = Plot.plot({ color: { legend: true, domain: ["p-distribution", "q-distribution", "Optimal Forward KL", "Optimal Reverse KL"], range: [colors[0], colors[1], colors[2], colors[3]] }, x: { label: "x", axis: true }, y: { label: yaxislabel, transform: (x) => {if (showlogpdf){return Math.log(x)}else{return x}}, //axis: false, //domain: [0, scaledinvchi2pdf((params[0]*params[1])/(params[0] + 2) , params[0], params[1]) ] }, marks: [ Plot.ruleY([0]), Plot.ruleX([limits[0]]), Plot.line(pdfdata, {x: "x", y: "ppdf", stroke: colors[0], strokeWidth: 2}), Plot.line(pdfdata, {x: "x", y: "qpdf", stroke: colors[1], strokeWidth: 2}), Plot.areaY(optpdfdata, {filter: d => view_kl[0], x: "x", y: "qpdfForward", fill: colors[2], opacity: 0.1}), Plot.line(optpdfdata, {filter: d => view_kl[0], x: "x", y: "qpdfForward", stroke: colors[2], strokeWidth: 2}), Plot.areaY(optpdfdata, {filter: d => view_kl[1], x: "x", y: "qpdfReverse", fill: colors[3], opacity: 0.1}), Plot.line(optpdfdata, {filter: d => view_kl[1], x: "x", y: "qpdfReverse", stroke: colors[3], strokeWidth: 2}), ] })35:Tf19,qopttext = { let disttext = [md``,md``] if (qdist == 'normal'){ if (view_kl[0]){ disttext[0] = md`Optimal Forward KL: Normal(${tex`\mu=`}${optresForward.param[0].toFixed(3)},${tex`\sigma=`}${optresForward.param[1].toFixed(3)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresForward.optimum).toFixed(4)}`}` } if (view_kl[1]){ disttext[1] = md`Optimal Reverse KL: Normal(${tex`\mu=`}${optresReverse.param[0].toFixed(3)},${tex`\sigma=`}${optresReverse.param[1].toFixed(3)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresReverse.optimum).toFixed(4)}`}` } } if (qdist == 'chi2'){ if (view_kl[0]){ disttext[0] = md`Optimal Forward KL: Chi2(${tex`\nu=`}${optresForward.param[0].toFixed(3)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresForward.optimum).toFixed(4)}`}` } if (view_kl[1]){ disttext[1] = md`Optimal Reverse KL: Chi2(${tex`\nu=`}${optresReverse.param[0].toFixed(3)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresReverse.optimum).toFixed(4)}`}` } } if (qdist == 'lognormal'){ if (view_kl[0]){ disttext[0] = md`Optimal Forward KL: LogNormal(${tex`\mu=`}${optresForward.param[0].toFixed(3)},${tex`\sigma=`}${optresForward.param[1].toFixed(3)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresForward.optimum).toFixed(4)}`}` } if (view_kl[1]){ disttext[1] = md`Optimal Reverse KL: LogNormal(${tex`\mu=`}${optresReverse.param[0].toFixed(3)},${tex`\sigma=`}${optresReverse.param[1].toFixed(3)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresReverse.optimum).toFixed(4)}`}` } } if (qdist == 'beta'){ if (view_kl[0]){ disttext[0] = md`Optimal Forward KL: Beta(${tex`\alpha=`}${optresForward.param[0].toFixed(3)},${tex`\beta=`}${optresForward.param[1].toFixed(3)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresForward.optimum).toFixed(4)}`}` } if (view_kl[1]){ disttext[1] = md`Optimal Reverse KL: Beta(${tex`\alpha=`}${optresReverse.param[0].toFixed(3)},${tex`\beta=`}${optresReverse.param[1].toFixed(3)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresReverse.optimum).toFixed(4)}`}` } } if (qdist == 'studentt'){ if (view_kl[0]){ disttext[0] = md`Optimal Forward KL: Student(${tex`\mu=`}${optresForward.param[0].toFixed(3)},${tex`\sigma=`}${optresForward.param[1].toFixed(3)},${tex`\nu=`}${optresForward.param[2].toFixed(3)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresForward.optimum).toFixed(4)}`}` } if (view_kl[1]){ disttext[1] = md`Optimal Reverse KL: Student(${tex`\mu=`}${optresReverse.param[0].toFixed(3)},${tex`\sigma=`}${optresReverse.param[1].toFixed(3)},${tex`\nu=`}${optresReverse.param[2].toFixed(3)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresReverse.optimum).toFixed(4)}`}` } } if (qdist == 'bimodal'){ if (view_kl[0]){ disttext[0] = md`Optimal Forward KL: Bimodal( ${tex`\mu_1=`}${optresForward.param[0].toFixed(3)}, ${tex`\mu_2=`}${optresForward.param[1].toFixed(3)}, ${tex`\sigma_1=`}${optresForward.param[2].toFixed(3)}, ${tex`\sigma_2=`}${optresForward.param[3].toFixed(4)}, ${tex`\omega=`}${optresForward.param[4].toFixed(5)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresForward.optimum).toFixed(4)}`}` } if (view_kl[1]){ disttext[1] = md`Optimal Reverse KL: Bimodal( ${tex`\mu_1=`}${optresReverse.param[0].toFixed(3)}, ${tex`\mu_2=`}${optresReverse.param[1].toFixed(3)}, ${tex`\sigma_1=`}${optresReverse.param[2].toFixed(3)}, ${tex`\sigma_2=`}${optresReverse.param[3].toFixed(4)}, ${tex`\omega=`}${optresReverse.param[4].toFixed(5)}) with ${tex`\mathrm{KL}(p \mid \mid q)=${(optresReverse.optimum).toFixed(4)}`}` } } return disttext }36:[] 2e:{"id":"752c89ad620c4829","slug":"kl-divergence-continuous","title":"Kullback-Leibler divergence - continuous distributions","thumbnail":"6302f35079ba98bcf64b6bb1ea5f2a6cf494339a3fdaf263edc4b427d3328b23","access_level":"public","create_time":"$D2024-05-29T14:17:50.752Z","update_time":"$D2026-05-12T17:05:59.925Z","view_time":null,"first_public_time":"$D2024-05-29T16:01:58.397Z","fork":{"id":"5d58e6a216ca0aed","slug":"inverse-gaussian-distribution","title":"Inverse Gaussian (Wald) distribution","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"stars":1,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":1455,"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"},{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":{"id":"05cb69ead7642f7e","slug":"probability","title":"Probability","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Kullback-Leibler divergence - continuous distributions","cells":[{"id":38,"value":"# Kullback-Leibler divergence - continuous distributions","pinned":false,"hidden":false,"mode":"md"},{"id":662,"value":"KLtext = md`The **Kullback-Leibler (KL) divergence** of the distribution ${tex`q(x)`} from the distribution ${tex`p(x)`} is defined as\n\n${tex.block`\\mathrm{KL}(p \\mid \\mid q) = \\int p(x)\\log\\frac{p(x)}{q(x)}dx = \\mathbb{E}_{p}\\log \\frac{p(x)}{q(x)}`}\n\nThe KL divergence is not a distance because it is general not symmetric in the two distributions \n\n${tex.block`\\mathrm{KL}(p \\mid \\mid q) \\neq \\mathrm{KL}(q \\mid \\mid p)`}\n\nThis asymmetry comes from the fact that the expectation is with respect to the distribution ${tex`p(x)`}. It therefore ${tex`p(x)`} that gets to determine which ${tex`x`} are important for the divergence: only the log-differences ${tex`\\log p(x) - \\log q(x)`} where ${tex`p(x)`} is sizeable will matter for the KL divergence. This particular definition of the KL divergence is sometimes called the **Forward KL**.
\n`\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":1126,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":496,"value":"## Interactive exploration","pinned":false,"hidden":false,"mode":"md"},{"id":1339,"value":"#### Target distribution parameters","pinned":false,"hidden":false,"mode":"md"},{"id":902,"value":"viewof pdist = Inputs.select(['beta', 'bimodal', 'chi2', 'normal', 'lognormal', 'studentt'], \n {value: 'studentt', label: \"Target p(x)\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":916,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":1343,"value":"#### Approximating distribution parameters","pinned":false,"hidden":false,"mode":"md"},{"id":913,"value":"viewof qdist = Inputs.select(['beta', 'bimodal', 'chi2', 'lognormal', 'normal', 'studentt'], \n {value: 'normal', label: \"Approximating q(x)\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":920,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":1248,"value":"viewof checkboxinputs = Inputs.form([ \n Inputs.checkbox([\"forward KL\", \"reverse KL\"], {label: \"plot best approx:\", value: []}),\n Inputs.toggle({label: \"show pdfs in logs\", value: false})\n])\n ","pinned":false,"hidden":false,"mode":"ojs"},{"id":1424,"value":"md`Forward KL for selected ${tex`q`}-density is ${tex`\\mathrm{KL}(p \\mid \\mid q)=${(KLdiv).toFixed(4)}`}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":971,"value":"$34","pinned":false,"hidden":false,"mode":"ojs"},{"id":1363,"value":"qopttext[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1398,"value":"qopttext[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nJust some code and details that you probably want to ignore.","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":1172,"value":"optimjs = require('optimization-js@1.5.0/dist/optimization.js')","pinned":false,"hidden":false,"mode":"ojs"},{"id":973,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":841,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":927,"value":"ppdf = {\n if (pdist == 'normal'){\n return function pdf(x){\n return jstat.normal.pdf(x, pparams[0], pparams[1])\n } \n }\n if (pdist == 'chi2'){\n return function pdf(x){\n return jstat.chisquare.pdf(x,pparams[0])\n }\n }\n if (pdist == 'lognormal'){\n return function pdf(x){\n return jstat.lognormal.pdf(x, pparams[0],pparams[1])\n }\n }\n if (pdist == 'beta'){\n return function pdf(x){\n return jstat.beta.pdf(x, pparams[0],pparams[1])\n }\n }\n if (pdist == \"studentt\"){\n return function pdf(x){\n return jstat.studentt.pdf((x-pparams[0])/pparams[1], pparams[2])/pparams[1]\n }\n }\n if (pdist == \"bimodal\"){\n return function pdf(x){\n return pparams[4]*jstat.normal.pdf(x, pparams[0], pparams[2]) + (1-pparams[4])*jstat.normal.pdf(x, pparams[1], pparams[3])\n }\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":942,"value":"qpdf = {\n if (qdist == 'normal'){\n return function pdf(x){\n return jstat.normal.pdf(x, qparams[0], qparams[1])\n } \n }\n if (qdist == 'chi2'){\n return function pdf(x){\n return jstat.chisquare.pdf(x,qparams[0])\n }\n }\n if (qdist == 'lognormal'){\n return function pdf(x){\n return jstat.lognormal.pdf(x, qparams[0],qparams[1])\n }\n }\n if (qdist == 'beta'){\n return function pdf(x){\n return jstat.beta.pdf(x, qparams[0], qparams[1])\n }\n }\n if (qdist == \"studentt\"){\n return function pdf(x){\n return jstat.studentt.pdf((x-qparams[0])/qparams[1], qparams[2])/qparams[1]\n }\n }\n if (qdist == \"bimodal\"){\n return function pdf(x){\n return qparams[4]*jstat.normal.pdf(x, qparams[0], qparams[2]) + (1-qparams[4])*jstat.normal.pdf(x, qparams[1], qparams[3])\n }\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1150,"value":"qpdfparam = {\n if (qdist == 'normal'){\n return function pdf(x, mu, sigma){\n return jstat.normal.pdf(x, mu, sigma)\n } \n }\n if (qdist == 'chi2'){\n return function pdf(x, nu){\n return jstat.chisquare.pdf(x,nu)\n }\n }\n if (qdist == 'lognormal'){\n return function pdf(x, mu, sigma){\n return jstat.lognormal.pdf(x, mu, sigma)\n }\n }\n if (qdist == 'beta'){\n return function pdf(x, alpha, beta){\n return jstat.beta.pdf(x, alpha, beta)\n }\n }\n if (qdist == \"studentt\"){\n return function pdf(x, mu, sigma, nu){\n return jstat.studentt.pdf((x-mu)/sigma, nu)/sigma\n }\n }\n if (qdist == \"bimodal\"){\n return function pdf(x, mu1, mu2, sigma1, sigma2, w){\n return w*jstat.normal.pdf(x, mu1, sigma1) + (1-w)*jstat.normal.pdf(x, mu2, sigma2)\n }\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1347,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":959,"value":"plimits = {\n if (pdist == 'normal'){\n return [jstat.normal.inv(0.001, pparams[0], pparams[1]),jstat.normal.inv(0.999, pparams[0], pparams[1])]\n }\n if (pdist == 'chi2'){\n return [Number.EPSILON,jstat.chisquare.inv(0.99, pparams[0])]\n }\n if (pdist == 'lognormal'){\n return [Number.EPSILON,jstat.lognormal.inv(0.99, pparams[0], pparams[1])]\n }\n if (pdist == 'beta'){\n return [Number.EPSILON,1-Number.EPSILON]\n }\n if (pdist == 'studentt'){\n return [pparams[0] + pparams[1]*jstat.studentt.inv(0.001, pparams[2]) , pparams[0] + pparams[1]*jstat.studentt.inv(0.999, pparams[2])]\n }\n if (pdist == 'bimodal'){\n return [ Math.min(jstat.normal.inv(0.0001, pparams[0], pparams[2]),jstat.normal.inv(0.0001, pparams[1], pparams[3])),\n Math.max(jstat.normal.inv(0.9999, pparams[0], pparams[2]),jstat.normal.inv(0.9999, pparams[1], pparams[3]))]\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":998,"value":"qlimits = {\n if (qdist == 'normal'){\n return [jstat.normal.inv(0.001, qparams[0], qparams[1]),jstat.normal.inv(0.999, qparams[0], qparams[1])]\n }\n if (qdist == 'chi2'){\n return [Number.EPSILON,jstat.chisquare.inv(0.99, qparams[0])]\n }\n if (qdist == 'lognormal'){\n return [Number.EPSILON,jstat.lognormal.inv(0.99, qparams[0], qparams[1])]\n }\n if (qdist == 'beta'){\n return [Number.EPSILON,1-Number.EPSILON]\n }\n if (qdist == 'studentt'){\n return [qparams[0] + qparams[1]*jstat.studentt.inv(0.001, qparams[2]) , qparams[0] + qparams[1]*jstat.studentt.inv(0.999, qparams[2])]\n }\n if (qdist == 'bimodal'){\n return [ Math.min(jstat.normal.inv(0.001, qparams[0], qparams[2]),jstat.normal.inv(0.001, qparams[1], qparams[3])),\n Math.max(jstat.normal.inv(0.999, qparams[0], qparams[2]),jstat.normal.inv(0.999, qparams[1], qparams[3]))]\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1270,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":1447,"value":"klselector = checkboxinputs[0]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1449,"value":"showlogpdf = checkboxinputs[1]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1437,"value":"yaxislabel = {if (showlogpdf){return \"log pdf\"}else{return \"pdf\"}}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1272,"value":"view_kl = [klselector.includes(\"forward KL\"), klselector.includes('reverse KL')]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1000,"value":"limits = [Math.min(plimits[0],qlimits[0]), Math.max(plimits[1],qlimits[1])]","pinned":false,"hidden":false,"mode":"ojs"},{"id":952,"value":"nbins = 1000","pinned":false,"hidden":false,"mode":"ojs"},{"id":1012,"value":"binwidth = (limits[1]-limits[0])/nbins","pinned":false,"hidden":false,"mode":"ojs"},{"id":1153,"value":"xgrid = d3.range(limits[0], limits[1], (limits[1]-limits[0])/nbins); ","pinned":false,"hidden":false,"mode":"ojs"},{"id":944,"value":"pdfdata = {\n const pdf = xgrid.map(x => ({x: x, ppdf: ppdf(x), qpdf: qpdf(x), logdiff: Math.log(ppdf(x))-Math.log(qpdf(x))}));\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":946,"value":"KLdiv = d3.sum(pdfdata.map(d => d.ppdf*(Math.log(d.ppdf) - Math.log(d.qpdf))*binwidth))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1122,"value":"KLdivRev = d3.sum(pdfdata.map(d => d.qpdf*(Math.log(d.qpdf) - Math.log(d.ppdf))*binwidth))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1147,"value":"function KLForwarddivOpt(qparams){\n return d3.sum(pdfdata.map(d => d.ppdf*( Math.log(d.ppdf) - Math.log(qpdfparam(d.x, ...qparams)) )*binwidth))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1223,"value":"function KLReversedivOpt(qparams){\n return d3.sum(pdfdata.map(d => qpdfparam(d.x, ...qparams)*\n ( Math.log(qpdfparam(d.x, ...qparams)) - Math.log(d.ppdf) )*binwidth))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1177,"value":"optresForward = { \n if (view_kl[0]){\n const optRes = optimjs.minimize_Powell(KLForwarddivOpt, qparams)\n return {optimum: optRes.fncvalue, param: optRes.argument}\n }else{\n return {optimum: 0, param: qparams} \n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1230,"value":"optresReverse = { \n if (view_kl[1]){\n const optRes = optimjs.minimize_Powell(KLReversedivOpt, qparams)\n return {optimum: optRes.fncvalue, param: optRes.argument}\n }else{\n return {optimum: 0, param: qparams}\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1181,"value":"optpdfdata = xgrid.map(x => ({x: x, \n qpdfForward: qpdfparam(x, ...optresForward.param),\n qpdfReverse: qpdfparam(x, ...optresReverse.param)\n })\n )","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W36","stdlib":"1","titling":"auto"}} 2f:"$2e:body"