2b:null 2c:["$","title",null,{"children":"Sampling distribution and likelihood function - Bernoulli model | Mattias Villani | Observable"}] 32:[] 2f:{"id":"b5f194fd5340f567","slug":"sampling-distribution-and-likelihood-function-bern","title":"Sampling distribution and likelihood function - Bernoulli model","thumbnail":"566679ecb567f4a61d3e4708a4c6e8d202ec97a62a91618aa2b4f23986eae586","access_level":"public","create_time":"$D2024-07-28T11:37:17.330Z","update_time":"$D2024-08-16T10:45:50.173Z","view_time":null,"first_public_time":"$D2024-07-28T12:10:20.099Z","fork":{"id":"9cc83e4830769ba8","slug":"sampling-distribution-and-likelihood-function-normal","title":"Sampling distribution and likelihood function - Normal model","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":639,"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":2,"collection":{"id":"3247547e8812c274","slug":"statistics","title":"Basic Statistics","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Sampling distribution and likelihood function - Bernoulli model","cells":[{"id":0,"value":"# Sampling distribution and likelihood function - Bernoulli model","pinned":false,"hidden":false,"mode":"md"},{"id":179,"value":"Assume random sample of size *n* from a Bernoulli population with success probability ${tex`\\theta`}. \n\n${tex.block`X_1,X_2,\\ldots,X_n \\overset{\\mathrm{iid}}{\\sim} \\operatorname{Bernoulli}(\\theta)`}\n\nThe sampling distribution for the number of successes, ${tex`p(S \\vert \\theta)`}, can be looked at from two viewpoints:\n\n- **For a given ${tex`\\theta`}** we obtain the sampling distribution\n\n${tex.block`S \\sim \\operatorname{Binomial}(n,\\theta)`}\n\n- **For a given observed number of successes ${tex`S = s`}** the likelihood function is the probability of the observed data, ${tex`\\operatorname{Binomial}(s \\vert n,\\theta)`}, as function of ${tex`\\theta.`}\n\nHere ${tex`\\operatorname{Binomial}(s \\vert n,\\theta)`} denotes the probability function of the ${tex`\\operatorname{Binomial}(n,\\theta)`} distribution evaluated at ${tex`s`}.\n\nThe widget below allows you to inspect the sampling distribution and the likelihood function for different ${tex`\\theta`} and observed ${tex`S = s`}.","pinned":false,"hidden":false,"mode":"md"},{"id":125,"value":"jointheat = Plot.plot({\n title: \"Sampling distribution\",\n width: 500, // Set the width\n height: 500, // Set the height equal to the width for a square aspect\n x: {label: \"θ\", domain: [0,1]},\n y: {label: \"s\", domain: [-1/2,n+1/2], ticks: d3.range(n+1)},\n color: {\n legend: true,\n scheme: \"Blues\",\n //type: \"identity\",\n label: \"probability\",\n //symmetric: false,\n domain: [0,sampDistMax]\n //transform: d => Math.exp(d - maxloglikIntercept)\n },\n marks: [\n Plot.contour(sampDist, {x: \"theta\", y: \"s\", fill: \"prob\", blur: 0, thresholds: 100}),\n Plot.tip(sampDist, Plot.pointerX({x: \"theta\", y: \"s\", stroke: \"prob\"})),\n Plot.ruleY([-0.5]),\n Plot.ruleX([0]),\n Plot.ruleY([n+0.5]),\n Plot.ruleX([1]),\n Plot.ruleY([s],{stroke: \"#5F9EA0\", strokeWidth: 3}),\n Plot.ruleX([theta],{stroke: \"#8B0000\", strokeWidth: 3})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":347,"value":"viewof settings = Inputs.form({\n theta: Inputs.range([0, 1], {value: 0.5, label: tex`\\theta`, step: 0.01}),\n n: Inputs.range([1, 100], {value: 10, label: tex`n`, step: 1}),\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":547,"value":"viewof s = Inputs.range([0, n], {value: 1, label: tex`s`, step: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":466,"value":"bothplots = html`\n \n \n \n

\n ${plt_samp}\n

\n ${plt_like}\n

\n \n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n---\n#### 🙈 Appendix 🙈\n\nHere is just some code and details that you probably don't want to see.","pinned":false,"hidden":false,"mode":"md"},{"id":608,"value":"#### Style","pinned":false,"hidden":false,"mode":"md"},{"id":606,"value":"html`