selva86.github.io/Bayesian-Statistics-in-R.html at master · selva86/selva86.github.io · GitHub

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<!DOCTYPE html>
<html lang="en">
  <head>
    <title>Bayesian Statistics in R: Build Genuine Intuition Before Opening Stan or brms</title>
    <meta charset="utf-8">
    <meta name="Description" content="Bayesian statistics in R from scratch: simulate the prior-likelihood-posterior update, watch a wrong prior get washed out by data, then bridge to MCMC.">
    <meta name="Keywords" content="Bayesian statistics R, prior likelihood posterior, Beta-Binomial conjugate, posterior simulation, credible interval, Bayesian inference, prior distribution, posterior distribution, grid approximation, MCMC introduction">
    <meta name="Distribution" content="Global">
    <meta name="Author" content="Selva Prabhakaran">
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      html.dark .masthead-search input:focus{border-color:#92a4d8;box-shadow:0 0 0 3px rgba(146,164,216,0.15)}
      html.dark .masthead-icon-btn{color:#b0b5bf}
      html.dark .masthead-icon-btn:hover{background:#1a1d22;color:#e8eaee}
      hr{height:0;box-sizing:content-box;border:0;border-top:1px solid #eee}
      .img-zoomable{cursor:zoom-in;transition:opacity .15s}
      .img-lightbox{display:none;position:fixed;top:0;left:0;right:0;bottom:0;z-index:10001;background:rgba(0,0,0,.85);cursor:zoom-out;align-items:center;justify-content:center}
      .img-lightbox.open{display:flex}
      .img-lightbox img{max-width:95vw;max-height:95vh;border-radius:4px;box-shadow:0 4px 24px rgba(0,0,0,.4)}
      /* CLS prevention: sidebar component styles (match main.css final state) */
      .sidebar-menu{margin:0;padding:0}
      .sidebar-tabs{display:flex;gap:0;margin:4px 0 0;border-bottom:1px solid #d8dce2}
      .sidebar-tab{flex:1;padding:9px 12px 8px;border:none;background:transparent;font:600 13px 'IBM Plex Sans',sans-serif;color:#757a87;cursor:pointer;border-bottom:2px solid transparent;margin-bottom:-1px;transition:color 0.15s,border-color 0.15s}
      .sidebar-tab.active{color:#1d3158;border-bottom-color:#1d3158}
      .sidebar-panel{display:none;padding-top:10px}
      .sidebar-panel.active{display:block}
      .sidebar-tools-list{margin:0;padding:0;list-style:none}
      .sidebar-tools-list li a{display:block;padding:5px 10px 5px 24px;font-size:14px;color:#495057;text-decoration:none;border-left:2px solid transparent;border-radius:0 4px 4px 0}
      .sidebar-section-header{padding:7px 12px;font-size:15px;font-weight:700;color:#212529;cursor:pointer;border-radius:4px}
      .sidebar-chevron{display:inline-block;font-size:10px;margin-right:4px;color:#757575}
      .sidebar-section-items{display:none;padding-left:0;margin:2px 0 6px}
      .sidebar-section.expanded .sidebar-section-items{display:block}
      .sidebar-section-items li{line-height:1.5}
      .sidebar-section-items li a{display:block;padding:4px 10px 4px 32px;font-size:14px;color:#495057;text-decoration:none;border-left:2px solid transparent;border-radius:0 4px 4px 0}
      .sidebar-divider{margin:12px 0 2px;padding:0 12px 4px 24px;font-size:11px;font-weight:700;letter-spacing:0.06em;text-transform:uppercase;color:#9ca3af;border-bottom:1px solid #f0f2f5;list-style:none}
      .sidebar-divider:first-child{margin-top:4px}
      .progress-dot{display:inline-block;width:10px;height:10px;border-radius:50%;border:1.5px solid #d1d5db;margin-right:7px;vertical-align:middle;flex-shrink:0;position:relative}
      /* Empty progress-dot (no status class) reads as an unchecked todo /
         disabled radio next to every sidebar item — visual noise that doesn't
         belong in a nav. Reserve the layout slot but hide the circle until
         the item actually has progress. visibility:hidden (not display:none)
         keeps the width stable when state flips. */
      .progress-dot:not(.started):not(.visited):not(.completed){visibility:hidden}
      /* CLS prevention: TOC sidebar matches main.css final state */
      #toc-wrapper{position:sticky;top:20px;max-height:calc(100vh - 40px);overflow-y:auto;padding-left:14px;border-left:1px solid #e9ecef}
      .toc-title{font-size:11px;font-weight:600;text-transform:uppercase;letter-spacing:0.8px;color:#6c757d;margin-bottom:10px;padding-bottom:0}
      #toc{margin:0;padding:0}
      #toc li{line-height:1.45;margin-bottom:1px}
      #toc li a{font-size:13.5px;color:#6c757d;text-decoration:none;display:block;padding:3px 0;border-left:2px solid transparent;padding-left:8px;margin-left:-15px}
      #toc li a.toc-active{color:#1d3158;font-weight:600;border-left-color:#1d3158}
      /* CLS prevention: engagement strip layout (matches engagement.css) */
      .engagement-header{display:flex;flex-wrap:wrap;align-items:center;gap:6px 10px;font-family:'Inter',-apple-system,BlinkMacSystemFont,'Segoe UI',sans-serif;font-size:12px;color:#64748b;letter-spacing:0.01em;margin:0.4em 0 1em;min-height:22px}
      .engagement-header:empty{display:none}
      /* CLS prevention: progress bar space (matches engagement.css) */
      .engagement-progress{min-height:32px;margin:1.2em 0 1.6em}
      /* CLS prevention: WebR editor min-height so the static→textarea swap doesn't shift layout */
      .webr-editor{min-height:60px}
      /* CLS prevention: images in content always reserve aspect ratio */
      #content img{max-width:100%;height:auto}

      /* =====================================================
         LAYOUT V2 (opt-in via body.layout-v2). Retry 2026-05-29.
         Differences from the first attempt:
           1. Sidebar columns use clamp(180px, 16vw, 260px) so they
              SHRINK on narrow viewports (first attempt used minmax
              which pinned both sides at 260px, squeezing content on
              typical laptops to ~460px).
           2. Sidebar links get overflow-wrap so long tutorial titles
              don't escape the narrower column.
           3. min-width:0 on every grid child so they can shrink.
           4. Right TOC breakpoint raised from 1100px -> 1280px so it
              only shows when the viewport can actually afford it.
           5. The inline-code background restyle is dropped - kept
              current syntax-highlighter behavior.
         All rules scoped under body.layout-v2 so pages without it
         render identically.
         ===================================================== */
      body.layout-v2{
        --lv2-bg:#f8f9fb;--lv2-card:#fff;--lv2-ink:#0d1117;
        --lv2-mut:#4b5260;--lv2-faint:#5b6573;
        --lv2-navy:#1c2c4f;--lv2-navy-h:#2d4173;--lv2-navy-soft:#eef1f7;
        --lv2-accent:#2056d2;--lv2-accent-soft:#e9efff;
        --lv2-line:#e4e7ee;--lv2-border:#d4d9e3;
        --lv2-green:#15803d;
        background:var(--lv2-bg);color:var(--lv2-ink);
      }
      html.dark body.layout-v2{
        --lv2-bg:#0a0f1c;--lv2-card:#0f1525;--lv2-ink:#eef2fa;
        --lv2-mut:#a5afc4;--lv2-faint:#9aa6c0;
        --lv2-navy:#5170b3;--lv2-navy-h:#6587c4;--lv2-navy-soft:#192240;
        --lv2-accent:#7faaff;--lv2-accent-soft:#1a2748;
        --lv2-line:#1f2a48;--lv2-border:#2c3a62;
      }

      /* Wider container; clamp-based grid that scales with viewport.
         width:auto is REQUIRED because the base critical CSS has
         @media(min-width:1200px){.container{width:1170px}} which would
         otherwise pin width to 1170 and ignore max-width:1340. */
      body.layout-v2 .container{max-width:1340px;width:auto;padding:0 22px}
      body.layout-v2 .site-masthead-inner{max-width:1340px}
      /* The legacy .site-masthead has margin:0 -15px to cancel the
         Bootstrap .container padding of 15px. Under layout-v2 the
         container padding is 22px, so the negative margin leaves a
         7px inset. Zero it so the masthead aligns flush with the
         article shell. */
      body.layout-v2 .site-masthead{margin-left:0;margin-right:0}
      body.layout-v2 .row{
        display:grid;
        grid-template-columns:
          clamp(180px, 16vw, 260px)
          minmax(0, 1fr)
          clamp(180px, 16vw, 260px);
        gap:0;margin-left:0;margin-right:0;
      }
      /* Bootstrap inserts .row::before AND .row::after pseudo-elements
         (content:" "; display:table) as a clearfix. Under display:grid
         these pseudos count as grid items. ::before grabs col 1, pushing
         every real child one column to the right and wrapping the last
         child to a new row. The 2026-05-29 first attempt only disabled
         ::after, resulting in nav in col 2, content in col 3, and toc
         wrapping to a new row at col 1 (the bug the user reported as
         "sidebar wide, content narrow"). BOTH pseudos must be killed. */
      body.layout-v2 .row::before,
      body.layout-v2 .row::after{display:none;content:none}
      /* min-width:0 lets grid children shrink below their content's
         intrinsic min-width; without this, long unbreakable text in any
         child forces the column wider than the grid template intended. */
      /* Drop Bootstrap's float and width on the legacy .col-sm-* hooks
         so the parent .row's CSS grid owns sizing. DO NOT zero horizontal
         padding here — per-column padding lives in the dedicated rules
         below (sidebar, main#content, #toc-sidebar) and gives the article
         the 44px breathing room the mock has. */
      body.layout-v2 #nav.col-sm-3,
      body.layout-v2 main#content.col-sm-7,
      body.layout-v2 #toc-sidebar.col-sm-2{
        width:auto!important;float:none!important;
        min-width:0;min-height:0;
      }
      body.layout-v2 #nav.col-sm-3{
        padding:14px 10px 40px;border-right:1px solid var(--lv2-line);
        position:sticky;top:60px;height:calc(100vh - 60px);overflow-y:auto;
      }
      /* Inner #sidebar-nav had its own sticky + overflow from the legacy critical
         CSS; under layout-v2 the outer #nav owns scrolling, so flatten the inner
         one to avoid a double scrollbar. */
      body.layout-v2 #sidebar-nav{
        position:static;max-height:none;overflow:visible;padding:0;
      }
      /* Wrap long tutorial titles in the sidebar so they never overflow */
      body.layout-v2 #nav a,
      body.layout-v2 #nav span,
      body.layout-v2 #sidebar-nav a,
      body.layout-v2 #sidebar-nav li{
        overflow-wrap:break-word;word-break:break-word;
      }
      body.layout-v2 main#content{padding:24px 44px 60px}

      /* S3 — Code-block bottom spacing.
         Default .webr-container margin is 22/22; the dark block visually
         carries a lot of weight, so the next paragraph reads as cramped.
         Lift bottom to 30px under layout-v2. */
      body.layout-v2 #content .webr-container{margin-bottom:30px}

      /* S7 — Further Reading: convert auto-injected <ul> to a 2-column
         card grid that matches the .rt-grid pattern used elsewhere.
         No HTML/auto_link.py changes needed — pure CSS. */
      body.layout-v2 #content #auto-further-reading .further-reading-list{
        list-style:none;padding:0;margin:14px 0 0;
        display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:12px;
      }
      body.layout-v2 #content #auto-further-reading .further-reading-list li{margin:0}
      body.layout-v2 #content #auto-further-reading .further-reading-list li a{
        display:block;padding:14px 16px;
        background:var(--lv2-card);border:1px solid var(--lv2-border);
        border-radius:8px;color:var(--lv2-ink);
        font-weight:500;font-size:15px;line-height:1.4;
        text-decoration:none;
        transition:border-color .15s,background .15s,transform .15s;
      }
      body.layout-v2 #content #auto-further-reading .further-reading-list li a:hover{
        border-color:var(--lv2-accent);background:var(--lv2-accent-soft);
        text-decoration:none;transform:translateY(-1px);
      }
      @media(max-width:820px){
        body.layout-v2 #content #auto-further-reading .further-reading-list{
          grid-template-columns:1fr;
        }
      }
      body.layout-v2 #toc-sidebar{padding:24px 18px;position:sticky;top:74px;align-self:start}
      @media(max-width:1280px){
        body.layout-v2 .row{grid-template-columns:clamp(180px, 18vw, 260px) minmax(0,1fr)}
        body.layout-v2 #toc-sidebar{display:none}
      }
      @media(max-width:820px){
        body.layout-v2 .row{grid-template-columns:1fr}
        body.layout-v2 #nav{display:none}
        body.layout-v2 main#content{padding:20px}
      }

      /* Typography refresh - sized to mock spec */
      body.layout-v2 #content{
        font-size:16.5px;line-height:1.7;color:var(--lv2-ink);
        font-feature-settings:"tnum","ss01","cv11";text-rendering:optimizeLegibility;
      }
      body.layout-v2 #content p{line-height:1.7;margin:13px 0;color:var(--lv2-ink);text-wrap:pretty}
      body.layout-v2 #content h1{
        font-size:38px;font-weight:700;line-height:1.2;letter-spacing:-0.018em;
        margin:10px 0 13px;text-wrap:balance;color:var(--lv2-ink);
      }
      /* H2: hairline divider + breathing room above. Mock breaks each
         section with a top rule and ~48px space; staging used 34px and no
         divider, leaving sections to blur together. :first-of-type keeps
         the very first H2 flush with the actionbar (no double divider). */
      body.layout-v2 #content h2{
        font-size:25px;font-weight:600;line-height:1.25;letter-spacing:-0.018em;
        margin:48px 0 12px;padding-top:32px;border-top:1px solid var(--lv2-line);
        text-wrap:balance;color:var(--lv2-ink);
      }
      body.layout-v2 #content h2:first-of-type{
        margin-top:24px;padding-top:0;border-top:none;
      }
      body.layout-v2 #content h3,body.layout-v2 #content h4{
        font-size:18px;font-weight:600;line-height:1.3;margin:24px 0 8px;text-wrap:balance;color:var(--lv2-ink);
      }
      /* Specificity bump: `body.layout-v2 #content p` (1 id + 1 class
         + 2 elements) was beating `body.layout-v2 p.lead` (2 classes +
         2 elements) and forcing the dek to var(--lv2-ink) instead of
         var(--lv2-mut). Adding #content here matches that specificity. */
      body.layout-v2 #content p.lead{
        font-size:18.5px;line-height:1.6;color:var(--lv2-mut);
        border-left:3px solid var(--lv2-navy);padding-left:17px;margin:-2px 0 20px;font-weight:400;
      }

      /* Body link color: legacy a{color:#1d3158} and Bootstrap .text-muted
         leave inline links almost indistinguishable from body text. Use
         the mock accent so links are obviously clickable. Excludes
         buttons, cards, prev/next nav, and TOC items which have their
         own palette. */
      body.layout-v2 #content a:not(.btn):not(.rt-card):not(.pn-link):not(.lv2-anchor):not([class*="toc"]){
        color:var(--lv2-accent);
      }
      body.layout-v2 #content a:not(.btn):not(.rt-card):not(.pn-link):not(.lv2-anchor):not([class*="toc"]):hover{
        text-decoration:underline;
      }

      /* Inline <code> palette: match the mock's navy-soft tint instead
         of the legacy purple-blue (#eef vs #eef1f7). */
      body.layout-v2 #content :not(pre) > code{
        background:var(--lv2-navy-soft);color:var(--lv2-ink);
        border-radius:4px;padding:2px 5px;font-size:0.92em;
      }

      /* TOC right rail */
      body.layout-v2 #toc-sidebar .toc-title{font-size:10.5px;font-weight:700;text-transform:uppercase;letter-spacing:.08em;color:var(--lv2-faint);margin-bottom:9px}
      body.layout-v2 #toc{font-size:13px}
      body.layout-v2 #toc li{margin:0;line-height:1.45}
      body.layout-v2 #toc li a{
        display:block;padding:5px 0 5px 11px;
        border-left:2px solid var(--lv2-line);
        color:var(--lv2-faint);font-size:13px;
        transition:color .15s,border-color .15s;
      }
      body.layout-v2 #toc li a:hover{color:var(--lv2-ink)}
      body.layout-v2 #toc li a.toc-active{
        border-left-color:var(--lv2-navy);color:var(--lv2-ink);font-weight:600;
      }

      /* Reading progress bar - uses transform:scaleX so we don't fight any
         containing-block / overflow quirks that can pin a width-% bar to 0. */
      .lv2-progress{
        position:fixed;top:0;left:0;width:100vw;height:3px;
        background:var(--lv2-accent,#2056d2);z-index:90;
        transform:scaleX(0);transform-origin:left center;
        transition:transform .1s ease-out;pointer-events:none;
      }

      /* Heading anchor links (hover-reveal) */
      body.layout-v2 #content h2,
      body.layout-v2 #content h3,
      body.layout-v2 #content h4{position:relative}
      body.layout-v2 .lv2-anchor{
        position:absolute;left:-26px;top:50%;transform:translateY(-50%);
        font-size:.55em;color:var(--lv2-faint);text-decoration:none;
        opacity:0;transition:opacity .15s,color .15s;
        cursor:pointer;padding:4px 6px;line-height:1;font-weight:400;
      }
      body.layout-v2 #content h2:hover .lv2-anchor,
      body.layout-v2 #content h3:hover .lv2-anchor,
      body.layout-v2 #content h4:hover .lv2-anchor,
      body.layout-v2 .lv2-anchor:focus-visible{opacity:1}
      body.layout-v2 .lv2-anchor:hover{color:var(--lv2-accent)}
      @media(max-width:820px){body.layout-v2 .lv2-anchor{left:0;top:-22px;transform:none}}

      /* Focus ring refresh */
      body.layout-v2 a:focus-visible,
      body.layout-v2 button:focus-visible,
      body.layout-v2 input:focus-visible,
      body.layout-v2 textarea:focus-visible{
        outline:none;
        box-shadow:0 0 0 3px rgba(32,86,210,.22);
        border-radius:4px;
      }
      html.dark body.layout-v2 a:focus-visible,
      html.dark body.layout-v2 button:focus-visible,
      html.dark body.layout-v2 input:focus-visible{box-shadow:0 0 0 3px rgba(127,170,255,.22)}

      /* Reduced motion: shut everything down for users who request it */
      @media (prefers-reduced-motion: reduce){
        body.layout-v2 *,body.layout-v2 *::before,body.layout-v2 *::after{
          animation-duration:0.01ms!important;
          animation-iteration-count:1!important;
          transition-duration:0.01ms!important;
          scroll-behavior:auto!important;
        }
      }
    </style>
    <!-- Full Bootstrap deferred (non-render-blocking) -->
    <link href="www/bootstrap.min.css?h=dae0a568" rel="stylesheet" media="print" onload="this.media='all'">
    <noscript><link href="www/bootstrap.min.css?h=dae0a568" rel="stylesheet"></noscript>
    <link href="www/highlight.min.css?h=f103014a" rel="stylesheet" media="print" onload="this.media='all'">
    <link href="css/main.min.css?h=5ca8bbcb" rel="stylesheet" media="print" onload="this.media='all'">
    <noscript><link href="css/main.min.css?h=5ca8bbcb" rel="stylesheet"></noscript>
    <style>
      a{color:#1d3158}a:hover{text-decoration:underline}
      li{line-height:1.65}
      .MathJax_Display{margin:0}
      blockquote p{line-height:1.75;color:#717171}
      /* engagement.js renders a meta strip into .engagement-header that
         duplicates the byline (difficulty + time). Hide it; saved-posts-
         button.js reads the data-* attrs directly for the byline. */
      .engagement-header{display:none!important}
      /* Byline (saved-posts widget rebuilds this after H1+lead, mock-style) */
      .byline{display:flex;flex-wrap:wrap;align-items:center;gap:8px;
        font-size:13.5px;color:#4b5260;margin:6px 0 16px;line-height:1.5}
      /* P1 — bump avatar 26→30 to match mock visual weight. */
      .byline .av-sm{width:30px;height:30px;border-radius:50%;
        background:linear-gradient(135deg,#1c2c4f,#2d4173);color:#fff;
        display:inline-flex;align-items:center;justify-content:center;
        font-weight:700;font-size:12px;flex:none;letter-spacing:.02em}
      .byline .who{color:#0d1117;font-weight:600}
      .byline .dot{color:#c0c5cf}
      .byline .rev{color:#137a3e;font-weight:600;display:inline-flex;align-items:center;gap:4px}
      .byline .time{display:inline-flex;align-items:center;gap:4px;color:#6b7280}
      .byline .diff{font-size:10.5px;font-weight:700;text-transform:uppercase;
        letter-spacing:.05em;background:#eef1f7;color:#4b5260;
        padding:3px 8px;border-radius:999px;margin-left:2px}

      /* Action bar (saved-posts widget injects this after byline) */
      .actionbar{display:flex;align-items:center;gap:9px;margin:10px 0 26px;padding:12px 0;
        border-top:1px solid #e4e7ee;border-bottom:1px solid #e4e7ee;flex-wrap:wrap}
      .actionbar .act{display:inline-flex;align-items:center;gap:6px;border:1px solid #d4d9e3;
        background:#fff;border-radius:9px;padding:7px 12px;font-size:13px;font-weight:600;
        color:#4b5260;cursor:pointer;font-family:inherit;
        transition:border-color .15s,color .15s,background .15s}
      .actionbar .act:hover{border-color:#0a0d14;color:#0a0d14}
      .actionbar .act.bookmark-btn[data-saved="1"]{background:#fef3c7;border-color:#f59e0b;color:#92400e}
      .actionbar .act.bookmark-btn[data-saved="1"] svg{fill:#f59e0b}
      .actionbar .act:disabled{opacity:.55;cursor:wait}
      .actionbar-sync{margin-left:auto;font-size:12px;color:#6b7280}
      .actionbar-sync a{color:#2056d2;font-weight:600;text-decoration:none}
      .actionbar-sync a:hover{text-decoration:underline}

      /* Dark-mode overrides for byline + actionbar. The hard-coded #0d1117
         on .who renders invisible on the dark background; the action-bar
         borders and surfaces also need to flip. Tokens (--lv2-ink, --lv2-mut,
         --lv2-line, --lv2-card) already swap under html.dark body.layout-v2;
         we just need to point the hard-coded rules at them under dark. */
      html.dark body.layout-v2 .byline{color:var(--lv2-mut)}
      html.dark body.layout-v2 .byline .who{color:var(--lv2-ink)}
      html.dark body.layout-v2 .byline .dot{color:var(--lv2-faint)}
      html.dark body.layout-v2 .byline .time{color:var(--lv2-mut)}
      html.dark body.layout-v2 .byline .diff{background:var(--lv2-navy-soft);color:var(--lv2-mut)}
      html.dark body.layout-v2 .actionbar{border-top-color:var(--lv2-line);border-bottom-color:var(--lv2-line)}
      html.dark body.layout-v2 .actionbar .act{background:var(--lv2-card);border-color:var(--lv2-border);color:var(--lv2-mut)}
      html.dark body.layout-v2 .actionbar .act:hover{border-color:var(--lv2-ink);color:var(--lv2-ink)}
      html.dark body.layout-v2 .actionbar-sync{color:var(--lv2-mut)}

      /* Toast (Share button "Link copied" feedback) */
      .rs-toast{position:fixed;bottom:24px;left:50%;transform:translateX(-50%) translateY(8px);
        background:#0a0d14;color:#fff;padding:9px 18px;border-radius:8px;font-size:13px;
        font-family:'IBM Plex Sans',sans-serif;z-index:1000;opacity:0;pointer-events:none;
        transition:opacity .2s,transform .2s;box-shadow:0 8px 24px rgba(10,13,20,.25)}
      .rs-toast.show{opacity:1;transform:translateX(-50%) translateY(0)}
      /* Auth-state visibility helpers (used by masthead variants) */
      .auth-anon,.auth-user{display:none}
      body.state-anon .auth-anon{display:inline-flex;align-items:center}
      body.state-pro .auth-user{display:inline-flex;align-items:center;gap:8px}
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data-subkey="sec2sub3"><a href="/Violin-Plot-in-R.html"><span class="progress-dot"></span>Violin Plot</a></li><li data-subkey="sec2sub3"><a href="/Ridgeline-Plot-in-R.html"><span class="progress-dot"></span>Ridgeline Plot</a></li><li data-subkey="sec2sub3"><a href="/Lollipop-Chart-in-R.html"><span class="progress-dot"></span>Lollipop Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub4" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Relationships</li><li data-subkey="sec2sub4"><a href="/Bubble-Chart-in-R.html"><span class="progress-dot"></span>Bubble Chart</a></li><li data-subkey="sec2sub4"><a href="/Heatmap-in-R.html"><span class="progress-dot"></span>Heatmap in R</a></li><li data-subkey="sec2sub4"><a href="/Correlation-Matrix-Plot-in-R.html"><span class="progress-dot"></span>Correlation Matrix</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub5" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Advanced Charts</li><li data-subkey="sec2sub5"><a href="/Pie-Donut-Chart-in-R.html"><span class="progress-dot"></span>Pie &amp; Donut Chart</a></li><li data-subkey="sec2sub5"><a href="/Treemap-in-R.html"><span class="progress-dot"></span>Treemap</a></li><li data-subkey="sec2sub5"><a href="/Waffle-Chart-in-R.html"><span class="progress-dot"></span>Waffle Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub6" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Exploratory Analysis</li><li data-subkey="sec2sub6"><a href="/Exploratory-Data-Analysis-in-R.html"><span class="progress-dot"></span>EDA (7-Step Framework)</a></li><li data-subkey="sec2sub6"><a href="/Univariate-EDA-in-R.html"><span class="progress-dot"></span>Univariate EDA</a></li><li data-subkey="sec2sub6"><a href="/Bivariate-EDA-in-R.html"><span class="progress-dot"></span>Bivariate EDA</a></li><li data-subkey="sec2sub6"><a href="/Descriptive-Statistics-in-R.html"><span class="progress-dot"></span>Descriptive Statistics</a></li><li data-subkey="sec2sub6"><a href="/Correlation-Analysis-in-R.html"><span class="progress-dot"></span>Correlation Analysis</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub7" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Interactive &amp; Maps</li><li data-subkey="sec2sub7"><a href="/Combining-ggplot2-with-plotly.html"><span class="progress-dot"></span>ggplot2 + plotly Interactive</a></li><li data-subkey="sec2sub7"><a href="/Interactive-Maps-in-R-with-leaflet.html"><span class="progress-dot"></span>Leaflet Interactive Maps</a></li><li data-subkey="sec2sub7"><a href="/Spatial-Data-in-R-with-sf.html"><span class="progress-dot"></span>Spatial Data (sf)</a></li><li data-subkey="sec2sub7"><a href="/Choropleth-Maps-in-R.html"><span class="progress-dot"></span>Choropleth Maps (sf)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub8" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Customization &amp; Reference</li><li data-subkey="sec2sub8"><a href="/ggplot2-Legends-in-R.html"><span class="progress-dot"></span>ggplot2 Legends</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-Secondary-Axis.html"><span class="progress-dot"></span>Secondary Axis</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-Log-Scale.html"><span class="progress-dot"></span>Log Scale</a></li><li data-subkey="sec2sub8"><a href="/patchwork-Package.html"><span class="progress-dot"></span>patchwork (Combine Plots)</a></li><li data-subkey="sec2sub8"><a href="/Publication-Quality-Figures-in-R.html"><span class="progress-dot"></span>Publication-Ready Figures</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-cheatsheet.html"><span class="progress-dot"></span>ggplot2 Quickref</a></li></ul></li><li class="sidebar-section expanded"><div class="sidebar-section-header"><span class="sidebar-chevron">&#9656;</span> Statistics<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub1" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> EDA &amp; Data Quality</li><li data-subkey="sec3sub1"><a href="/Automated-EDA-in-R.html"><span class="progress-dot"></span>Automated EDA</a></li><li data-subkey="sec3sub1"><a href="/Missing-Data-Visualization-in-R-naniar.html"><span class="progress-dot"></span>Missing Data Viz (naniar)</a></li><li data-subkey="sec3sub1"><a href="/Outlier-Detection-in-R.html"><span class="progress-dot"></span>Outlier Detection</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub2" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Probability</li><li data-subkey="sec3sub2"><a href="/Sample-Spaces-Events-and-Probability-Axioms-in-R-With-Monte-Carlo-Proof.html"><span class="progress-dot"></span>Probability Axioms</a></li><li data-subkey="sec3sub2"><a href="/Conditional-Probability-in-R.html"><span class="progress-dot"></span>Conditional Probability</a></li><li data-subkey="sec3sub2"><a href="/Random-Variables-in-R.html"><span class="progress-dot"></span>Random Variables</a></li><li data-subkey="sec3sub2"><a href="/Binomial-and-Poisson-Distributions-in-R.html"><span class="progress-dot"></span>Binomial vs Poisson</a></li><li data-subkey="sec3sub2"><a href="/Normal-t-F-and-Chi-Squared-Distributions-in-R.html"><span class="progress-dot"></span>Normal, t, F, Chi-Squared</a></li><li data-subkey="sec3sub2"><a href="/Central-Limit-Theorem-in-R.html"><span class="progress-dot"></span>Central Limit Theorem</a></li><li data-subkey="sec3sub2"><a href="/Sampling-Distributions-in-R.html"><span class="progress-dot"></span>Sampling Distributions</a></li><li data-subkey="sec3sub2"><a href="/Law-of-Large-Numbers-vs-CLT-in-R.html"><span class="progress-dot"></span>LLN vs CLT</a></li><li data-subkey="sec3sub2"><a href="/What-Is-Probability-Simulation-First-Intuition-in-R-Before-the-Formulas.html"><span class="progress-dot"></span>Probability (Simulation-First)</a></li><li data-subkey="sec3sub2"><a href="/Expected-Value-and-Variance-in-R.html"><span class="progress-dot"></span>Expected Value and Variance</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub3" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Inference &amp; Estimation</li><li data-subkey="sec3sub3"><a href="/Maximum-Likelihood-Estimation-in-R.html"><span class="progress-dot"></span>Maximum Likelihood Estimation</a></li><li data-subkey="sec3sub3"><a href="/Hypothesis-Testing-in-R.html"><span class="progress-dot"></span>Hypothesis Testing</a></li><li data-subkey="sec3sub3"><a href="/Sample-Size-Planning-in-R.html"><span class="progress-dot"></span>Sample Size Planning</a></li><li data-subkey="sec3sub3"><a href="/Which-Statistical-Test-in-R.html"><span class="progress-dot"></span>Choosing the Right Test</a></li><li data-subkey="sec3sub3"><a href="/Statistical-Tests-in-R.html"><span class="progress-dot"></span>Statistical Tests</a></li><li data-subkey="sec3sub3"><a href="/Measures-of-Association-in-R.html"><span class="progress-dot"></span>Measures of Association</a></li><li data-subkey="sec3sub3"><a href="/Point-Estimation-in-R.html"><span class="progress-dot"></span>Point Estimation</a></li><li data-subkey="sec3sub3"><a href="/Confidence-Intervals-in-R.html"><span class="progress-dot"></span>Confidence Intervals</a></li><li data-subkey="sec3sub3"><a href="/Type-I-and-Type-II-Errors-in-R.html"><span class="progress-dot"></span>Type I and II Errors</a></li><li data-subkey="sec3sub3"><a href="/Statistical-Power-Analysis-in-R.html"><span class="progress-dot"></span>Power Analysis</a></li><li data-subkey="sec3sub3"><a href="/Effect-Size-in-R.html"><span class="progress-dot"></span>Effect Size</a></li><li data-subkey="sec3sub3"><a href="/t-Tests-in-R.html"><span class="progress-dot"></span>t-Tests</a></li><li data-subkey="sec3sub3"><a href="/Proportion-Tests-in-R.html"><span class="progress-dot"></span>Proportion Tests</a></li><li data-subkey="sec3sub3"><a href="/Normality-and-Variance-Tests-in-R.html"><span class="progress-dot"></span>Normality &amp; Variance Tests</a></li><li data-subkey="sec3sub3"><a href="/Chi-Square-Tests-in-R.html"><span class="progress-dot"></span>Chi-Square Tests</a></li><li data-subkey="sec3sub3"><a href="/Wilcoxon-Mann-Whitney-and-Kruskal-Wallis-in-R.html"><span class="progress-dot"></span>Wilcoxon, Mann-Whitney &amp; Kruskal-Wallis</a></li><li data-subkey="sec3sub3"><a href="/Multiple-Comparisons-in-R.html"><span class="progress-dot"></span>Multiple Testing Correction</a></li><li data-subkey="sec3sub3"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Hypothesis Testing Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub4" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Regression</li><li data-subkey="sec3sub4"><a href="/Linear-Regression.html"><span class="progress-dot"></span>Linear Regression</a></li><li data-subkey="sec3sub4"><a href="/Logistic-Regression-With-R.html"><span class="progress-dot"></span>Logistic Regression</a></li><li data-subkey="sec3sub4"><a href="/Variable-Selection-and-Importance-With-R.html"><span class="progress-dot"></span>Feature Selection</a></li><li data-subkey="sec3sub4"><a href="/Model-Selection-in-R.html"><span class="progress-dot"></span>Model Selection</a></li><li data-subkey="sec3sub4"><a href="/Missing-Value-Treatment-With-R.html"><span class="progress-dot"></span>Missing Value Treatment</a></li><li data-subkey="sec3sub4"><a href="/Outlier-Treatment-With-R.html"><span class="progress-dot"></span>Outlier Analysis</a></li><li data-subkey="sec3sub4"><a href="/adv-regression-models.html"><span class="progress-dot"></span>Advanced Regression Models</a></li><li data-subkey="sec3sub4"><a href="/Linear-Regression-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Linear Regression Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub5" data-collapsed="false"><span class="subsec-chevron">&#9660;</span> Reporting</li><li data-subkey="sec3sub5"><a href="/Statistical-Consulting-in-R.html"><span class="progress-dot"></span>Statistical Consulting</a></li><li data-subkey="sec3sub5"><a href="/Statistical-Report-Writing-in-R.html"><span class="progress-dot"></span>Statistical Report Writing</a></li><li data-subkey="sec3sub5"><a href="/Bootstrap-Confidence-Intervals-in-R.html"><span class="progress-dot"></span>Bootstrap Confidence Intervals</a></li><li data-subkey="sec3sub5"><a href="/Reporting-Statistics-in-R.html"><span class="progress-dot"></span>Reporting Statistics</a></li><li data-subkey="sec3sub5"><a href="/Correlation-in-R.html"><span class="progress-dot"></span>Correlation (Pearson, Spearman, Kendall)</a></li><li data-subkey="sec3sub5"><a href="/Linear-Regression-Assumptions-in-R.html"><span class="progress-dot"></span>Linear Regression Assumptions</a></li><li data-subkey="sec3sub5"><a href="/Dummy-Variables-in-R.html"><span class="progress-dot"></span>Dummy Variables in R</a></li><li data-subkey="sec3sub5"><a href="/Interaction-Effects-in-R.html"><span class="progress-dot"></span>Interaction Effects</a></li><li data-subkey="sec3sub5"><a href="/Regression-Diagnostics-in-R.html"><span class="progress-dot"></span>Regression Diagnostics</a></li><li data-subkey="sec3sub5"><a href="/Logistic-Regression-in-R.html"><span class="progress-dot"></span>Logistic Regression (glm + ROC)</a></li><li data-subkey="sec3sub5"><a href="/Variable-Selection-in-R.html"><span class="progress-dot"></span>Variable Selection</a></li><li data-subkey="sec3sub5"><a href="/Poisson-Regression-in-R.html"><span class="progress-dot"></span>Poisson Regression</a></li><li data-subkey="sec3sub5"><a href="/Ridge-and-Lasso-Regression-in-R.html"><span class="progress-dot"></span>Ridge &amp; Lasso Regression</a></li><li data-subkey="sec3sub5"><a href="/Polynomial-and-Spline-Regression-in-R.html"><span class="progress-dot"></span>Polynomial &amp; Splines</a></li><li data-subkey="sec3sub5"><a href="/Regression-Tables-in-R.html"><span class="progress-dot"></span>Regression Tables (3 packages)</a></li><li data-subkey="sec3sub5"><a href="/One-Way-ANOVA-in-R.html"><span class="progress-dot"></span>One-Way ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Post-Hoc-Tests-After-ANOVA.html"><span class="progress-dot"></span>Post-Hoc Tests After ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Two-Way-ANOVA-in-R.html"><span class="progress-dot"></span>Two-Way ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Repeated-Measures-ANOVA-in-R.html"><span class="progress-dot"></span>Repeated Measures ANOVA</a></li><li data-subkey="sec3sub5"><a href="/ANCOVA-in-R.html"><span class="progress-dot"></span>ANCOVA</a></li><li data-subkey="sec3sub5"><a href="/Experimental-Design-Principles-in-R.html"><span class="progress-dot"></span>Experimental Design in R</a></li><li data-subkey="sec3sub5"><a href="/Factorial-Experiments-in-R.html"><span class="progress-dot"></span>Factorial Designs (2^k)</a></li><li data-subkey="sec3sub5"><a href="/AB-Testing-in-R.html"><span class="progress-dot"></span>A/B Testing</a></li><li data-subkey="sec3sub5"><a href="/MANOVA-in-R.html"><span class="progress-dot"></span>MANOVA</a></li><li data-subkey="sec3sub5"><a href="/Mixed-ANOVA-in-R.html"><span class="progress-dot"></span>Mixed ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Multivariate-Statistics-in-R.html"><span class="progress-dot"></span>Multivariate Distances &amp; Hotelling's T²</a></li><li data-subkey="sec3sub5"><a href="/PCA-in-R.html"><span class="progress-dot"></span>PCA with prcomp()</a></li><li data-subkey="sec3sub5"><a href="/Interpreting-PCA-Results-in-R.html"><span class="progress-dot"></span>Interpreting PCA Output</a></li><li data-subkey="sec3sub5"><a href="/Exploratory-Factor-Analysis-in-R.html"><span class="progress-dot"></span>Exploratory Factor Analysis</a></li><li data-subkey="sec3sub5"><a href="/CFA-and-Structural-Equation-Modeling-in-R.html"><span class="progress-dot"></span>SEM and CFA (lavaan)</a></li><li data-subkey="sec3sub5"><a href="/Linear-Discriminant-Analysis-in-R.html"><span class="progress-dot"></span>LDA (Linear Discriminant Analysis)</a></li><li data-subkey="sec3sub5"><a href="/Cluster-Analysis-in-R.html"><span class="progress-dot"></span>Clustering (k-Means / HC / DBSCAN)</a></li><li data-subkey="sec3sub5"><a href="/Correspondence-Analysis-in-R.html"><span class="progress-dot"></span>Correspondence Analysis</a></li><li data-subkey="sec3sub5"><a href="/t-SNE-and-UMAP-in-R.html"><span class="progress-dot"></span>t-SNE and UMAP</a></li><li data-subkey="sec3sub5"><a href="/Simple-Linear-Regression-in-R.html"><span class="progress-dot"></span>Simple Linear Regression</a></li><li data-subkey="sec3sub5"><a href="/Multiple-Regression-in-R.html"><span class="progress-dot"></span>Multiple Regression</a></li><li data-subkey="sec3sub5"><a href="/Robust-Regression-in-R.html"><span class="progress-dot"></span>Robust Regression (rlm)</a></li><li data-subkey="sec3sub5"><a href="/factoextra-and-FactoMineR.html"><span class="progress-dot"></span>factoextra (PCA + Clusters)</a></li><li data-subkey="sec3sub5"><a href="/Categorical-Data-in-R.html"><span class="progress-dot"></span>Categorical Data (Tables &amp; Mosaic)</a></li><li data-subkey="sec3sub5"><a href="/Chi-Square-Test-of-Independence-in-R.html"><span class="progress-dot"></span>Chi-Square Test of Independence</a></li><li data-subkey="sec3sub5"><a href="/Chi-Square-Goodness-of-Fit-Test-in-R.html"><span class="progress-dot"></span>Chi-Square Goodness-of-Fit</a></li><li data-subkey="sec3sub5"><a href="/Fishers-Exact-Test-in-R.html"><span class="progress-dot"></span>Fisher's Exact Test</a></li><li data-subkey="sec3sub5"><a href="/Odds-Ratios-and-Relative-Risk-in-R.html"><span class="progress-dot"></span>Odds Ratios &amp; Relative Risk</a></li><li data-subkey="sec3sub5"><a href="/Logistic-Regression-in-R-2.html"><span class="progress-dot"></span>Logistic Regression (Diagnostics)</a></li><li data-subkey="sec3sub5"><a href="/Poisson-and-Negative-Binomial-Regression.html"><span class="progress-dot"></span>Poisson &amp; Negative Binomial Regression</a></li><li data-subkey="sec3sub5"><a href="/Multinomial-and-Ordinal-Logistic-Regression-in-R.html"><span class="progress-dot"></span>Multinomial &amp; Ordinal Logistic Regression</a></li><li data-subkey="sec3sub5"><a href="/When-to-Use-Nonparametric-Tests-in-R.html"><span class="progress-dot"></span>When to Use Nonparametric Tests</a></li><li data-subkey="sec3sub5"><a href="/Wilcoxon-Signed-Rank-Test-in-R.html"><span class="progress-dot"></span>Wilcoxon Signed-Rank Test</a></li><li data-subkey="sec3sub5"><a href="/Mann-Whitney-U-Test-in-R.html"><span class="progress-dot"></span>Mann-Whitney U Test</a></li><li data-subkey="sec3sub5"><a href="/Kruskal-Wallis-Test-in-R-2.html"><span class="progress-dot"></span>Kruskal-Wallis Test</a></li><li data-subkey="sec3sub5"><a href="/Friedman-Test-in-R.html"><span class="progress-dot"></span>Friedman Test</a></li><li data-subkey="sec3sub5"><a href="/Spearman-and-Kendall-Correlation-in-R.html"><span class="progress-dot"></span>Spearman &amp; Kendall Correlation</a></li><li data-subkey="sec3sub5"><a href="/Bootstrap-in-R.html"><span class="progress-dot"></span>Bootstrap (boot package)</a></li><li data-subkey="sec3sub5"><a href="/Quantile-Regression-in-R-2.html"><span class="progress-dot"></span>Quantile Regression</a></li><li data-subkey="sec3sub5"><a href="/Matrix-Operations-in-R.html"><span class="progress-dot"></span>Matrix Operations in R</a></li><li data-subkey="sec3sub5"><a href="/Solving-Linear-Systems-in-R.html"><span class="progress-dot"></span>Solving Linear Systems in R</a></li><li data-subkey="sec3sub5"><a href="/Eigenvalues-and-Eigenvectors-in-R.html"><span class="progress-dot"></span>Eigenvalues &amp; Eigenvectors in R</a></li><li data-subkey="sec3sub5"><a href="/Singular-Value-Decomposition-in-R.html"><span class="progress-dot"></span>Singular Value Decomposition in R</a></li><li data-subkey="sec3sub5"><a href="/Projections-and-the-Hat-Matrix-in-R.html"><span class="progress-dot"></span>Projections &amp; the Hat Matrix</a></li><li data-subkey="sec3sub5"><a href="/QR-Decomposition-in-R.html"><span class="progress-dot"></span>QR Decomposition in R</a></li><li data-subkey="sec3sub5"><a href="/Quadratic-Forms-in-R.html"><span class="progress-dot"></span>Quadratic Forms</a></li><li data-subkey="sec3sub5"><a 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          <nav class="breadcrumb-nav" aria-label="Breadcrumb"><a href="/">Home</a> <span class="breadcrumb-sep">&rsaquo;</span> <span>Statistics</span> <span class="breadcrumb-sep">&rsaquo;</span> <span class="breadcrumb-current">Bayesian Statistics in R: Build Genuine Intuition Before Opening Stan or brms</span></nav>
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<h1>Bayesian Statistics in R: Build Genuine Intuition Before Opening Stan or brms</h1>

<p class="lead">Bayesian statistics in R updates a prior belief about an unknown parameter with observed data, producing a <a class="auto-link" href="Bayesian-Statistics-Exercises-in-R.html" title="Bayesian Statistics Exercises in R: 20 Practice Problems">posterior distribution</a> you can plot, integrate, and reason about. Unlike frequentist methods that return a single point estimate plus a confidence interval whose interpretation trips up most students, the Bayesian workflow gives you the full probability curve over plausible parameter values, ready for decision-making.</p>
<div class="post-byline" style="color:#6b7280;font-size:14px;margin:2px 0 18px 0;line-height:1.5;">By <strong>Selva Prabhakaran</strong> &nbsp;&middot;&nbsp; Published May 13, 2026 &nbsp;&middot;&nbsp; Last updated May 13, 2026</div>

<div class="engagement-header" data-difficulty="Intermediate" data-time="35" data-exercises="10" data-xp="150"></div>

<h2>How does Bayes' theorem turn data into a posterior?</h2>

<p>Frequentist tools answer "what is the parameter?" with a point estimate and a confidence interval whose interpretation bends most readers' minds. <a class="auto-link" href="Bayesian-Linear-Regression-in-R.html" title="Bayesian Linear Regression in R: Get Uncertainty Estimates lm() Cannot Give You">Bayesian inference</a> flips the question. You start with a prior belief about the parameter, observe data, and end with a posterior distribution: a probability curve over every plausible value. This section shows that update happen in a single line of base R using the Beta-Binomial pair, the simplest example of an analytic posterior.</p>

<p>The math behind every Bayesian update is one line:</p>

<p>$$ p(\theta \mid \text{data}) \;\propto\; p(\text{data} \mid \theta) \cdot p(\theta) $$</p>

<p>Where: $p(\theta)$ is the prior, $p(\text{data} \mid \theta)$ is the likelihood, and $p(\theta \mid \text{data})$ is the posterior. The proportional sign hides a normalizing constant that does not affect the shape of the curve.</p>

<p>Suppose you flip a possibly-biased coin 100 times and see 65 heads. You want to estimate the unknown success probability theta. A Beta(2, 2) prior is mildly skeptical of extreme values, gently centered at 0.5. The posterior comes out in closed form because the Beta family is conjugate to the Binomial likelihood, meaning the posterior stays in the Beta family.</p>

<div class="webr-container" data-block-title="Beta-Binomial posterior from 100 flips">
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    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Beta-Binomial posterior from 100 flips</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl">n <span class="o">&lt;-</span> <span class="m">100</span>                    <span class="c1"># total flips</span></span>
<span class="cl">k <span class="o">&lt;-</span> <span class="m">65</span>                     <span class="c1"># heads observed</span></span>
<span class="cl">alpha_prior <span class="o">&lt;-</span> <span class="m">2</span>            <span class="c1"># prior shape parameters</span></span>
<span class="cl">beta_prior  <span class="o">&lt;-</span> <span class="m">2</span>            <span class="c1"># ... encoding &quot;fair-ish, but uncertain&quot;</span></span>
<span class="cl"></span>
<span class="cl">alpha_post <span class="o">&lt;-</span> alpha_prior <span class="o">+</span> k          <span class="c1"># closed-form posterior shape</span></span>
<span class="cl">beta_post  <span class="o">&lt;-</span> beta_prior  <span class="o">+</span> n <span class="o">-</span> k      <span class="c1"># ... thanks to Beta-Binomial conjugacy</span></span>
<span class="cl"></span>
<span class="cl">post_mean <span class="o">&lt;-</span> alpha_post <span class="o">/</span> (alpha_post <span class="o">+</span> beta_post)</span>
<span class="cl">cri       <span class="o">&lt;-</span> <span class="nf">qbeta</span>(<span class="nf">c</span>(<span class="m">0.025</span>, <span class="m">0.975</span>), alpha_post, beta_post)</span>
<span class="cl"></span>
<span class="cl">post_mean</span>
<span class="cl"><span class="c1">#&gt; [1] 0.6442308</span></span>
<span class="cl">cri</span>
<span class="cl"><span class="c1">#&gt; [1] 0.5497073 0.7321076</span></span></div>
    <div class="webr-buttons">
      <button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run</button>
      <button class="btn btn-sm btn-default webr-reset-btn" onclick="resetWebR(this)">&#8634; Reset</button>
    </div>
    <pre class="webr-output"></pre>
  </div>
  <div class="webr-plot-output"></div>
</div>

<p>The posterior is Beta(67, 37). Its mean of 0.644 sits between the prior mean of 0.5 and the data proportion of 0.65, gently pulled toward 0.5 by the prior's weight. The 95% credible interval [0.55, 0.73] is the range that contains 95% of the <a class="auto-link" href="Bayes-Theorem-in-R.html" title="Bayes' Theorem in R: Why Medical Tests Mislead You, A Simulation That Shows Why">posterior probability</a> mass. That is the interpretation people incorrectly give a frequentist confidence interval.</p>

<p><img src="screenshots/Bayesian-Statistics-in-R-workflow.webp" alt="The Bayesian update workflow" class="img-responsive img-zoomable" loading="lazy" width="2248" height="380" /></p>

<p><em>Figure 1: The Bayesian update workflow. Prior plus data give a posterior, which you then <a class="auto-link" href="dplyr-group-by-summarise.html" title="dplyr group_by() + summarise(): The Combination That Answers Most Business Questions">summarize</a>.</em></p>

<div class="callout callout-insight"><div class="callout-label">Key Insight</div><div class="callout-body"><strong>The posterior is the prior reweighted by the likelihood.</strong> No exotic computation, just multiplication and renormalization. The Beta-Binomial pair gives you the answer in closed form because the Beta family is conjugate to the Binomial, meaning the posterior stays in the Beta family.</div></div>

<section class="tryit-block">
<p><strong>Try it:</strong> Repeat the calculation with a much tighter prior, Beta(20, 20). What does the posterior mean become and why?</p>
<div class="webr-container" data-block-title="Your turn: tighten the prior">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: tighten the prior</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl">ex_alpha <span class="o">&lt;-</span> <span class="m">20</span>            <span class="c1"># try a tight prior</span></span>
<span class="cl">ex_beta  <span class="o">&lt;-</span> <span class="m">20</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># compute the posterior shape parameters and posterior mean here</span></span>
<span class="cl"><span class="c1"># ex_alpha_post &lt;- ?</span></span>
<span class="cl"><span class="c1"># ex_beta_post  &lt;- ?</span></span>
<span class="cl"><span class="c1"># ex_post_mean  &lt;- ?</span></span>
<span class="cl"><span class="c1">#&gt; Expected: posterior mean closer to 0.5 than 0.65</span></span></div>
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<details><summary>Click to reveal solution</summary>

<div class="webr-container" data-block-title="Tighter prior solution">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Tighter prior solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl">ex_alpha_post <span class="o">&lt;-</span> ex_alpha <span class="o">+</span> k</span>
<span class="cl">ex_beta_post  <span class="o">&lt;-</span> ex_beta  <span class="o">+</span> n <span class="o">-</span> k</span>
<span class="cl">ex_post_mean  <span class="o">&lt;-</span> ex_alpha_post <span class="o">/</span> (ex_alpha_post <span class="o">+</span> ex_beta_post)</span>
<span class="cl">ex_post_mean</span>
<span class="cl"><span class="c1">#&gt; [1] 0.6071429</span></span></div>
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<p>A Beta(20, 20) prior is equivalent to having seen 38 prior flips with 19 heads. Adding the new 100 flips gives a posterior that is pulled noticeably back toward 0.5. A stronger prior carries more weight against the same data, that is the lesson here.</p>

</details>
</section>

<h2>What does a prior actually encode?</h2>

<p>A prior is just a probability distribution over the parameter. Anything you can put on a curve, you can use as a prior. The Beta family is convenient for proportions because it lives on [0, 1] and supports two intuitive shape parameters that act like pseudo-counts of prior successes and failures. Three Beta priors illustrate the spectrum from ignorance to strong belief.</p>

<div class="webr-container" data-block-title="Three priors plotted on one panel">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Three priors plotted on one panel</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl">theta_grid <span class="o">&lt;-</span> <span class="nf">seq</span>(<span class="m">0</span>, <span class="m">1</span>, length.out <span class="o">=</span> <span class="m">200</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">plot</span>(theta_grid, <span class="nf">dbeta</span>(theta_grid, <span class="m">1</span>, <span class="m">1</span>), type <span class="o">=</span> <span class="s">&quot;l&quot;</span>, lwd <span class="o">=</span> <span class="m">2</span>,</span>
<span class="cl">     ylim <span class="o">=</span> <span class="nf">c</span>(<span class="m">0</span>, <span class="m">6</span>), xlab <span class="o">=</span> <span class="nf">expression</span>(theta), ylab <span class="o">=</span> <span class="s">&quot;density&quot;</span>,</span>
<span class="cl">     main <span class="o">=</span> <span class="s">&quot;Three priors over a proportion&quot;</span>)</span>
<span class="cl"><span class="nf">lines</span>(theta_grid, <span class="nf">dbeta</span>(theta_grid, <span class="m">20</span>, <span class="m">20</span>), lwd <span class="o">=</span> <span class="m">2</span>, col <span class="o">=</span> <span class="s">&quot;steelblue&quot;</span>)</span>
<span class="cl"><span class="nf">lines</span>(theta_grid, <span class="nf">dbeta</span>(theta_grid,  <span class="m">2</span>,  <span class="m">5</span>), lwd <span class="o">=</span> <span class="m">2</span>, col <span class="o">=</span> <span class="s">&quot;tomato&quot;</span>)</span>
<span class="cl"><span class="nf">legend</span>(<span class="s">&quot;topright&quot;</span>, lwd <span class="o">=</span> <span class="m">2</span>,</span>
<span class="cl">       col <span class="o">=</span> <span class="nf">c</span>(<span class="s">&quot;black&quot;</span>, <span class="s">&quot;steelblue&quot;</span>, <span class="s">&quot;tomato&quot;</span>),</span>
<span class="cl">       legend <span class="o">=</span> <span class="nf">c</span>(<span class="s">&quot;Beta(1,1) uniform&quot;</span>, <span class="s">&quot;Beta(20,20) tight at 0.5&quot;</span>, <span class="s">&quot;Beta(2,5) skewed low&quot;</span>))</span></div>
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<p>Beta(1, 1) is flat: every value of theta is equally plausible before seeing data. Beta(20, 20) is tight around 0.5: a strong belief that the coin is fair. Beta(2, 5) is skewed low: a belief that small values of theta are more likely. Each shape encodes a different domain assumption, and each will pull the posterior in a different direction.</p>

<p>You often have a substantive belief such as "I think theta is between 0.4 and 0.6 with about 90% probability." That language has a direct Beta translation. Search for shape parameters whose 5th and 95th percentiles match the beliefs. A symmetric, moderately tight Beta(45, 45) lands close.</p>

<div class="webr-container" data-block-title="Quantile-based prior elicitation">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Quantile-based prior elicitation</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl"><span class="nf">qbeta</span>(<span class="nf">c</span>(<span class="m">0.05</span>, <span class="m">0.95</span>), <span class="m">45</span>, <span class="m">45</span>)</span>
<span class="cl"><span class="c1">#&gt; [1] 0.4133497 0.5866503</span></span></div>
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<p>A Beta(45, 45) prior places 90% of its mass between 0.41 and 0.59, a near-perfect match for the stated belief. If you wanted a less tight prior, lower both shape parameters; for a more confident prior, raise them. This is how to translate qualitative belief into a quantitative prior without throwing darts.</p>

<div class="callout callout-tip"><div class="callout-label">Tip</div><div class="callout-body"><strong>Pick a prior whose shape matches your prior belief, document the choice, and plan to report sensitivity.</strong> Beta(2, 2) is mildly skeptical of extreme values; Beta(1, 1) is informationless; Beta(50, 50) is hard to budge. Match the prior to the belief, then check how much the answer changes if you nudge the prior.</div></div>

<section class="tryit-block">
<p><strong>Try it:</strong> Encode the belief "I think theta is around 0.7 with mild uncertainty" as Beta shape parameters. A good answer keeps most mass in the [0.6, 0.8] range.</p>
<div class="webr-container" data-block-title="Your turn: encode a belief at 0.7">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: encode a belief at 0.7</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl">ex_alpha2 <span class="o">&lt;-</span> <span class="m">0</span>   <span class="c1"># replace</span></span>
<span class="cl">ex_beta2  <span class="o">&lt;-</span> <span class="m">0</span>   <span class="c1"># replace</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Verify with: qbeta(c(0.05, 0.95), ex_alpha2, ex_beta2)</span></span>
<span class="cl"><span class="c1">#&gt; Expected: 90% interval roughly [0.6, 0.8]</span></span></div>
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<details><summary>Click to reveal solution</summary>

<div class="webr-container" data-block-title="Belief-at-0.7 solution">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Belief-at-0.7 solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl">ex_alpha2 <span class="o">&lt;-</span> <span class="m">35</span></span>
<span class="cl">ex_beta2  <span class="o">&lt;-</span> <span class="m">15</span></span>
<span class="cl"><span class="nf">qbeta</span>(<span class="nf">c</span>(<span class="m">0.05</span>, <span class="m">0.95</span>), ex_alpha2, ex_beta2)</span>
<span class="cl"><span class="c1">#&gt; [1] 0.5901019 0.8003617</span></span></div>
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<p>Beta(35, 15) has mean 35/50 = 0.7 with 90% of its mass between 0.59 and 0.80. The ratio alpha/(alpha+beta) controls the center; the sum alpha+beta controls how tight the curve is. Combine those two levers to encode any belief on [0,1].</p>

</details>
</section>

<h2>How do prior and likelihood combine into a posterior?</h2>

<p>The likelihood is a function of the parameter, given fixed data. It is not itself a probability distribution over theta, just a curve showing which theta values are most consistent with what you saw. Multiply the likelihood curve by the prior curve, normalize so the area is 1, and you have the posterior. Plotting all three on the same axes makes the arithmetic visual.</p>

<div class="webr-container" data-block-title="Prior, likelihood, posterior on one panel">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Prior, likelihood, posterior on one panel</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl">likelihood_vals <span class="o">&lt;-</span> <span class="nf">dbinom</span>(k, size <span class="o">=</span> n, prob <span class="o">=</span> theta_grid)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">par</span>(mfrow <span class="o">=</span> <span class="nf">c</span>(<span class="m">1</span>, <span class="m">3</span>), mar <span class="o">=</span> <span class="nf">c</span>(<span class="m">4</span>, <span class="m">4</span>, <span class="m">3</span>, <span class="m">1</span>))</span>
<span class="cl"></span>
<span class="cl"><span class="nf">plot</span>(theta_grid, <span class="nf">dbeta</span>(theta_grid, alpha_prior, beta_prior),</span>
<span class="cl">     type <span class="o">=</span> <span class="s">&quot;l&quot;</span>, lwd <span class="o">=</span> <span class="m">2</span>, main <span class="o">=</span> <span class="s">&quot;Prior  Beta(2, 2)&quot;</span>,</span>
<span class="cl">     xlab <span class="o">=</span> <span class="nf">expression</span>(theta), ylab <span class="o">=</span> <span class="s">&quot;density&quot;</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">plot</span>(theta_grid, likelihood_vals, type <span class="o">=</span> <span class="s">&quot;l&quot;</span>, lwd <span class="o">=</span> <span class="m">2</span>, col <span class="o">=</span> <span class="s">&quot;tomato&quot;</span>,</span>
<span class="cl">     main <span class="o">=</span> <span class="s">&quot;Likelihood  Binomial(n=100, k=65)&quot;</span>,</span>
<span class="cl">     xlab <span class="o">=</span> <span class="nf">expression</span>(theta), ylab <span class="o">=</span> <span class="s">&quot;L(theta)&quot;</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">plot</span>(theta_grid, <span class="nf">dbeta</span>(theta_grid, alpha_post, beta_post),</span>
<span class="cl">     type <span class="o">=</span> <span class="s">&quot;l&quot;</span>, lwd <span class="o">=</span> <span class="m">2</span>, col <span class="o">=</span> <span class="s">&quot;steelblue&quot;</span>,</span>
<span class="cl">     main <span class="o">=</span> <span class="s">&quot;Posterior  Beta(67, 37)&quot;</span>,</span>
<span class="cl">     xlab <span class="o">=</span> <span class="nf">expression</span>(theta), ylab <span class="o">=</span> <span class="s">&quot;density&quot;</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">par</span>(mfrow <span class="o">=</span> <span class="nf">c</span>(<span class="m">1</span>, <span class="m">1</span>))</span></div>
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<p>The likelihood peaks at the maximum likelihood estimate, exactly k/n = 0.65. The prior peaks at 0.5. The posterior peaks slightly below 0.65, pulled toward the prior in proportion to the prior's tightness. With a weak prior and large n, the posterior almost coincides with the likelihood.</p>

<p><img src="screenshots/Bayesian-Statistics-in-R-conjugate.webp" alt="Beta-Binomial conjugate update" class="img-responsive img-zoomable" loading="lazy" width="1486" height="690" /></p>

<p><em>Figure 2: <a class="auto-link" href="Conjugate-Priors-in-R.html" title="Conjugate Priors in R: The Shortcut That Gives Exact Posteriors Without MCMC">Beta-Binomial conjugate</a>. Closed-form posterior parameters absorb counts of observed successes and failures.</em></p>

<div class="callout callout-insight"><div class="callout-label">Key Insight</div><div class="callout-body"><strong>The posterior always lies between the prior and the likelihood, weighted by their relative confidence.</strong> A flat prior gives you back the likelihood. A point-mass prior ignores the data entirely. Real priors live in between, and the data nudges the answer accordingly.</div></div>

<section class="tryit-block">
<p><strong>Try it:</strong> Suppose you observed 20 heads in 100 flips instead of 65. Recompute the posterior parameters and the posterior mean.</p>
<div class="webr-container" data-block-title="Your turn: shift the data">
  <div class="webr-code-block">
    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: shift the data</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl">ex_k_new <span class="o">&lt;-</span> <span class="m">20</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># compute new alpha_post, beta_post, and posterior mean</span></span>
<span class="cl"><span class="c1">#&gt; Expected: posterior mean near 0.21</span></span></div>
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<details><summary>Click to reveal solution</summary>

<div class="webr-container" data-block-title="Shifted-data solution">
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    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Shifted-data solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl">ex_alpha_post_new <span class="o">&lt;-</span> alpha_prior <span class="o">+</span> ex_k_new</span>
<span class="cl">ex_beta_post_new  <span class="o">&lt;-</span> beta_prior <span class="o">+</span> n <span class="o">-</span> ex_k_new</span>
<span class="cl">ex_alpha_post_new <span class="o">/</span> (ex_alpha_post_new <span class="o">+</span> ex_beta_post_new)</span>
<span class="cl"><span class="c1">#&gt; [1] 0.2115385</span></span></div>
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<p>The posterior peaks near 0.21, just above the data proportion 0.20, slightly pulled toward 0.5 by the prior. The shape and the location of the curve both follow the data, while the prior modulates the pull.</p>

</details>
</section>

<h2>How does the posterior shift as more data arrives?</h2>

<p>A common worry about Bayesian methods is "what if I pick the wrong prior?" The honest answer: with enough data, the prior gets washed out. Likelihood scales with n, prior does not, so the posterior shifts toward the data as n grows. Showing this with a deliberately bad prior makes the point concrete.</p>

<div class="webr-container" data-block-title="Posterior with n = 10, 100, 1000 against a wrong prior">
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    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Posterior with n = 10, 100, 1000 against a wrong prior</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl">true_theta <span class="o">&lt;-</span> <span class="m">0.7</span>                          <span class="c1"># the truth we are trying to recover</span></span>
<span class="cl">sims <span class="o">&lt;-</span> <span class="nf">list</span>(</span>
<span class="cl">  small  <span class="o">=</span> <span class="nf">list</span>(n <span class="o">=</span>   <span class="m">10</span>, k <span class="o">=</span> <span class="nf">round</span>(<span class="m">0.7</span> <span class="o">*</span>   <span class="m">10</span>)),   <span class="c1"># 7  / 10</span></span>
<span class="cl">  medium <span class="o">=</span> <span class="nf">list</span>(n <span class="o">=</span>  <span class="m">100</span>, k <span class="o">=</span> <span class="nf">round</span>(<span class="m">0.7</span> <span class="o">*</span>  <span class="m">100</span>)),   <span class="c1"># 70 / 100</span></span>
<span class="cl">  large  <span class="o">=</span> <span class="nf">list</span>(n <span class="o">=</span> <span class="m">1000</span>, k <span class="o">=</span> <span class="nf">round</span>(<span class="m">0.7</span> <span class="o">*</span> <span class="m">1000</span>))    <span class="c1"># 700/ 1000</span></span>
<span class="cl">)</span>
<span class="cl">wrong_alpha <span class="o">&lt;-</span> <span class="m">80</span>    <span class="c1"># a stubborn prior centered at 0.8 ...</span></span>
<span class="cl">wrong_beta  <span class="o">&lt;-</span> <span class="m">20</span>    <span class="c1"># ... that disagrees with the truth</span></span>
<span class="cl"></span>
<span class="cl"><span class="nf">plot</span>(theta_grid, <span class="nf">dbeta</span>(theta_grid, wrong_alpha, wrong_beta), type <span class="o">=</span> <span class="s">&quot;l&quot;</span>, lwd <span class="o">=</span> <span class="m">2</span>,</span>
<span class="cl">     ylim <span class="o">=</span> <span class="nf">c</span>(<span class="m">0</span>, <span class="m">30</span>), xlab <span class="o">=</span> <span class="nf">expression</span>(theta), ylab <span class="o">=</span> <span class="s">&quot;density&quot;</span>,</span>
<span class="cl">     main <span class="o">=</span> <span class="s">&quot;Posterior shifts toward the truth as n grows&quot;</span>)</span>
<span class="cl"><span class="nf">abline</span>(v <span class="o">=</span> true_theta, lty <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl">cols <span class="o">&lt;-</span> <span class="nf">c</span>(<span class="s">&quot;tomato&quot;</span>, <span class="s">&quot;orange&quot;</span>, <span class="s">&quot;steelblue&quot;</span>)</span>
<span class="cl"><span class="kr">for</span> (i <span class="kr">in</span> <span class="nf">seq_along</span>(sims)) {</span>
<span class="cl">  s <span class="o">&lt;-</span> sims[[i]]</span>
<span class="cl">  <span class="nf">lines</span>(theta_grid,</span>
<span class="cl">        <span class="nf">dbeta</span>(theta_grid, wrong_alpha <span class="o">+</span> s<span class="o">$</span>k, wrong_beta <span class="o">+</span> s<span class="o">$</span>n <span class="o">-</span> s<span class="o">$</span>k),</span>
<span class="cl">        lwd <span class="o">=</span> <span class="m">2</span>, col <span class="o">=</span> cols[i])</span>
<span class="cl">}</span>
<span class="cl"><span class="nf">legend</span>(<span class="s">&quot;topleft&quot;</span>, lwd <span class="o">=</span> <span class="m">2</span>, col <span class="o">=</span> <span class="nf">c</span>(<span class="s">&quot;black&quot;</span>, cols),</span>
<span class="cl">       legend <span class="o">=</span> <span class="nf">c</span>(<span class="s">&quot;Wrong prior Beta(80,20)&quot;</span>, <span class="s">&quot;n=10&quot;</span>, <span class="s">&quot;n=100&quot;</span>, <span class="s">&quot;n=1000&quot;</span>))</span></div>
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<p>The prior peaks at 0.8 and refuses to budge much for n=10. By n=100 the posterior straddles 0.75. By n=1000 it has converged tightly around the true 0.7. The posterior peak migrates from prior toward truth as the data accumulates.</p>

<p>A subtler property of Bayes' theorem is that updates are order-independent. If you observe one batch, then another, then update sequentially, the result is mathematically identical to combining everything and updating once. The check below verifies that.</p>

<div class="webr-container" data-block-title="Sequential vs one-shot update">
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    <div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Sequential vs one-shot update</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">&#9654; Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
    <div class="webr-editor" data-language="r"><span class="cl">batch1 <span class="o">&lt;-</span> <span class="nf">list</span>(n <span class="o">=</span> <span class="m">50</span>, k <span class="o">=</span> <span class="m">35</span>)</span>
<span class="cl">batch2 <span class="o">&lt;-</span> <span class="nf">list</span>(n <span class="o">=</span> <span class="m">50</span>, k <span class="o">=</span> <span class="m">40</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Sequential: update after each batch</span></span>
<span class="cl">seq_first  <span class="o">&lt;-</span> <span class="nf">c</span>(alpha <span class="o">=</span> <span class="m">1</span> <span class="o">+</span> batch1<span class="o">$</span>k,</span>
<span class="cl">                beta  <span class="o">=</span> <span class="m">1</span> <span class="o">+</span> batch1<span class="o">$</span>n <span class="o">-</span> batch1<span class="o">$</span>k)         <span class="c1"># Beta(36, 16)</span></span>