2a:null 2b:["$","title",null,{"children":"Bayesian hypothesis test for a mean in a normal population | Mattias Villani | Observable"}] 31:T4f4,The **prior predictive density** for a sample of *n* observations ${tex`X_1,\ldots,X_n`} from model ${tex`M_k`} is the marginal density ${tex.block`p_k(X_1,\ldots,X_n) =\int p_k(X_1,\ldots,X_n \vert \theta_k)p(\theta_k)d\theta_k `} where we subscript the likelihood, prior and parameter vector with *k* to denote that these quantities are for model ${tex`M_k`}. The prior predictive density ${tex`p_k(X_1,\ldots,X_n)`} therefore describes how probable a sample of observations is when using the model ${tex`p_k(X_1,\ldots,X_n \vert \theta_k)`} with parameters drawn from the prior ${tex`p(\theta_k)`}. The **marginal likelihood** of a model ${tex`M_k`} is the prior predictive density under model ${tex`M_k`} *evaluated at the observed sample* ${tex`x_1,\ldots,x_n`} ${tex.block`p_k(x_1,\ldots,x_n) =\int p_k(x_1,\ldots,x_n \vert \theta_k)p(\theta_k)d\theta_k `} So the marginal likelihood is the *average* likelihood of the sample where the parameters are integrated out with respect to the prior. Note the absolutely crucial point that the marginal likelihood evaluates both the likelihood *and* the prior. > A **Bayesian model** is a **likelihood** ${tex`p_k(x_1,\ldots,x_n \vert \theta_k)`}
with parameters drawn from the **prior** ${tex`p(\theta_k)`}. 32:T5a2,plt = Plot.plot({ color: { legend: true, domain: ["Data density p(x̄) under M₀", "Data density p(x̄) under M₁", "Observed data mean x̄"], range: [colors[1], colors[0], colors[2]] }, x: { label: "sample mean, x̄", axis: true, domain: [0,40] }, y: { axis: true, domain: [0,1.05*max_y], label: "density" }, marks: [ Plot.ruleY([0]), Plot.ruleX([0]), Plot.ruleX([xbar], {stroke: colors[2], strokeWidth: 1.5}), Plot.line(pdfdata, {filter: d => d.dist == "H0", x: "x", y: "pdf", stroke : colors[1], strokeWidth: 3}), Plot.line(pdfdata, {filter: d => d.dist == "H1", x: "x", y: "pdf", stroke : colors[0], strokeWidth: 3}), //Plot.dot([{x: xbar, y: 0}], {x: "x", y: "y", fill: colors[2], symbol: "circle", r: 5}), Plot.dot([{x: xbar, y: jstat.normal.pdf(xbar, theta0, sigma*Math.sqrt(1/n))}], {x: "x", y: "y", fill: colors[1], symbol: "circle", r: 5}), Plot.dot([{x: xbar, y: jstat.normal.pdf(xbar, theta0, sigma*Math.sqrt(1/n + 1/kappa0))}], {x: "x", y: "y", fill: colors[0], symbol: "circle", r: 5}), //Plot.arrow([{x1: xbar, y1: max_y, x2:xbar, y2: 0.01}], {x1: "x1", y1: "y1", x2: "x2", y2: "y2", stroke: colors[2]}), //Plot.text([{x1: xbar, y1: max_y, texter: "hej"}], {x: "x1", y: "y1", text: "texter", fontSize: textfontsize}), ] })33:[] 2e:{"id":"d76fa5195e787b41","slug":"bayesian-test-mean-normal-pop","title":"Bayesian hypothesis test for a mean in a normal population","thumbnail":"5faedc1c9d38f93426968a9385b376e483827dba1a25ae376c3149c15bef60d9","access_level":"public","create_time":"$D2024-10-12T14:27:29.669Z","update_time":"$D2025-10-27T09:01:05.429Z","view_time":null,"first_public_time":"$D2024-10-12T20:36:24.226Z","fork":{"id":"0e0e67cef3873adb","slug":"hypothesis-test-mean","title":"Hypothesis test for a mean in a normal population","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":1496,"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":0,"collection":{"id":"9fdb7c9ba7be6ba8","slug":"bayesian-learning","title":"Bayesian Learning","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Bayesian hypothesis test for a mean in a normal population","cells":[{"id":38,"value":"# Bayesian hypothesis test for a mean in a normal population","pinned":false,"hidden":false,"mode":"md"},{"id":842,"value":"Consider the normal model with known variance ${tex`\\sigma^2`} \n${tex.block`X_1,\\ldots,X_n \\vert \\theta \\overset{\\mathrm{iid}}{\\sim} N(\\theta,\\sigma)\\text{ with }\\sigma\\text{ known}`}\n\nWhere are interested in testing\n${tex.block`M_0: \\theta = \\theta_{0} = ${theta0}\\\\\nM_1: \\theta \\neq \\theta_{0} = ${theta0}`}\nwhere we write ${tex`M_0`} for the null *model* instead of the more common ${tex`H_0`} for the null *hypothesis*. The reason is that Bayesian hypothesis testing is a special case of Bayesian model comparison.","pinned":false,"hidden":false,"mode":"md"},{"id":1128,"value":"We use the conjugate prior with variance scaled the the population variance ${tex`\\sigma^2`}\n${tex.block`\\theta \\sim N \\bigg(\\theta_0,\\frac{\\sigma^2}{\\kappa_0} \\bigg)`}","pinned":false,"hidden":false,"mode":"md"},{"id":1151,"value":"Note that: \n- the prior under the alternative model is centered on the null hypothesis ${tex`\\theta_0`}\n- the prior sample size ${tex`\\kappa_0`} determines the strength of prior information.","pinned":false,"hidden":false,"mode":"md"},{"id":1393,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":1306,"value":"The **Bayes factor** compares a model ${tex`M_0`} against another model ${tex`M_1`} by the ratio of the marginal likelihoods\n${tex.block`\\mathrm{BF}_{01} = \\frac{p_0(x_1,\\ldots,x_n)}{p_1(x_1,\\ldots,x_n)}.`}\n\nIn the case with the iid Normal model with known variance, the sample mean ${tex`\\bar x`} is a sufficient statistic and we only need to evaluate ${tex`p(\\bar x)`} instead of the joint ${tex`p(x_1,\\ldots,x_n)`} distribution of all data points, which is convenient for plotting. The Bayes factor in this model can be shown to be\n ${tex.block`\\mathrm{BF}_{01} = \\frac{\nN\\big(\\bar x \\vert \\theta_0,\\frac{\\sigma^2}{n}\\big)}\n{N\\Big(\\bar x \\vert\\theta_0,\\sigma^2\\big(\\frac{1}{n} + \\frac{1}{\\kappa_0} \\big)\\Big)\n}`}\nwhere ${tex`N(x \\vert\\theta,\\sigma^2)`} denotes the ${tex`N(\\theta,\\sigma^2)`} density evaluated at ${tex`x`}.\n","pinned":false,"hidden":false,"mode":"md"},{"id":1453,"value":"If we assign prior probabilities ${tex`\\mathrm{Pr}(M_k)`} for the model themselves, we can use Bayes' theorem to compute **posterior model probabilities** \n${tex.block`\\mathrm{Pr}(M_k \\vert \\bar x) \\propto p(\\bar x \\vert M_k)\\mathrm{Pr}(M_k),`}\nwhere ${tex`p(\\bar x \\vert M_k)`} is the marginal likelihood under model ${tex`M_k`}.","pinned":false,"hidden":false,"mode":"md"},{"id":1448,"value":"The interactive graph below lets you experiment with different data samples, null values ${tex`\\theta_0`} and different prior hyperparameters. Note in particular a very small value of ${tex`\\kappa_0`} corresponds to a Bayesian model where the parameters are drawn from a rather wild distribution, resulting in a very spread out prior predictive density with low probability for the actually observed data. ","pinned":false,"hidden":false,"mode":"md"},{"id":118,"value":"viewof inputs = Inputs.form({\n n: Inputs.range([2, 100], {value: 5, step: 1, label: tex`\\text{sample size }n`}), \n xbar: Inputs.range([0, 50], {value: 16, step: 1, label: tex`\\text{sample mean }\\bar x`}),\n sigma: Inputs.range([Number.EPSILON, 10], {value: 5, step: 0.1, label: tex`\\text{population stdev }\\sigma`}),\n theta0: Inputs.range([0, 50], {value: 20, step: 0.1, label: tex`\\text{null value }\\theta_0`}),\n kappa0: Inputs.range([0.0001, 1000], {value: 1, step: 0.0001, transform: Math.log, label: tex`\\text{prior sample size }\\kappa_0`}),\n prH0: Inputs.range([0, 1], {value: 0.5, step: 0.01, label: tex`\\mathrm{Pr}(H_0)`}),\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1035,"value":"The **Bayes factor** in favor of ${tex`H_0`} is ${bayesfactor.toPrecision(4)} is *${jeffreystext}*.
\nThe **posterior model probabilities** are: ${tex`\\mathrm{Pr}(H_0 \\vert \\textbf{x}) = `} ${PostProbH0.toPrecision(4)}\nand ${tex`\\mathrm{Pr}(H_1 \\vert \\textbf{x}) = `} ${(1 - PostProbH0).toPrecision(4)}","pinned":false,"hidden":false,"mode":"md"},{"id":305,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"
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\n\n---\n## 🙈 Appendix 🙈\n\nHere is only code and details that you probably don't want to read.","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":1202,"value":"import {colors} from \"@mattiasvillani/statistics-tools\"","pinned":true,"hidden":false,"mode":"ojs"},{"id":972,"value":"#### Settings","pinned":false,"hidden":false,"mode":"md"},{"id":974,"value":"textfontsize = 12","pinned":false,"hidden":false,"mode":"ojs"},{"id":1141,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md"},{"id":1143,"value":"function logmarglike(xBar, n, σ, μ0, κ0) {\n return -(1/2) * Math.log(κ0/(κ0 + n)) - ((n * Math.pow(xBar - μ0, 2))/(2 * Math.pow(σ, 2))) * (n/(κ0 + n));\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":223,"value":"#### Data","pinned":false,"hidden":false,"mode":"md"},{"id":1352,"value":"jeffreystext = {\n if (bayesfactor > 1) {\n if (bayesfactor <= 3){\n var jeffreys = \"barely worth mentioning\"}\n else{\n if (bayesfactor <= 20){\n var jeffreys = \"positive\"\n }\n else{\n if(bayesfactor <= 150){\n var jeffreys = \"strong\"\n }\n else{\n var jeffreys = \"very strong\"\n }\n }\n }\n }else{\n if (1/bayesfactor <= 3){\n var jeffreys = \"barely worth mentioning\"}\n else{\n if (1/bayesfactor <= 20){\n var jeffreys = \"positive\"\n }\n else{\n if(1/bayesfactor <= 150){\n var jeffreys = \"strong\"\n }\n else{\n var jeffreys = \"very strong\"\n }\n }\n }\n }\n return jeffreys \n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":573,"value":"n = inputs.n","pinned":false,"hidden":false,"mode":"ojs"},{"id":558,"value":"xbar = inputs.xbar","pinned":false,"hidden":false,"mode":"ojs"},{"id":559,"value":"sigma = inputs.sigma","pinned":false,"hidden":false,"mode":"ojs"},{"id":564,"value":"theta0 = inputs.theta0","pinned":false,"hidden":false,"mode":"ojs"},{"id":1200,"value":"kappa0 = inputs.kappa0","pinned":false,"hidden":false,"mode":"ojs"},{"id":1225,"value":"bayesfactor = jstat.normal.pdf(xbar, theta0, sigma*(1/Math.sqrt(n)))/jstat.normal.pdf(xbar, theta0, sigma*Math.sqrt(1/n + 1/kappa0))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1243,"value":"PostProbH0 = bayesfactor*inputs.prH0/(bayesfactor*inputs.prH0 + (1 - inputs.prH0))","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"pdfdata = {\n const x = d3.range(-20, 50, 0.01);\n var data = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, theta0, sigma*(1/Math.sqrt(n))), dist: \"H0\", type: \"pdf\"}));\n const pdfH1 = x.map(x => ({x: x, pdf: jstat.normal.pdf(x, theta0, sigma*Math.sqrt(1/n + 1/kappa0)), dist: \"H1\", type: \"pdf\"}));\n data = data.concat(pdfH1)\n return data\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":682,"value":"max_y = 1.05*d3.max(pdfdata.map(d => d.pdf))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1015,"value":"conclusioncolor = {\n if(pvaluestudentt < alpha){\n return \"red\"\n }else{\n return \"steelblue\"\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":930,"value":"nudge = 1.1","pinned":false,"hidden":false,"mode":"ojs"},{"id":984,"value":"bignudge = 1.3","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":202,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":58,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}} 2f:"$2e:body"