701 lines
41 KiB
HTML
701 lines
41 KiB
HTML
|
||
<!doctype html>
|
||
<html lang="en" class="no-js">
|
||
<head>
|
||
|
||
<meta charset="utf-8">
|
||
<meta name="viewport" content="width=device-width,initial-scale=1">
|
||
|
||
|
||
|
||
<link rel="canonical" href="https://rustbas.github.io/blog/maths/baes/">
|
||
|
||
|
||
|
||
|
||
<link rel="icon" href="../../assets/images/favicon.png">
|
||
<meta name="generator" content="mkdocs-1.6.1, mkdocs-material-9.6.14">
|
||
|
||
|
||
|
||
<title>Немного про Байесовскую статистику - Очередные записки очередного гика</title>
|
||
|
||
|
||
|
||
<link rel="stylesheet" href="../../assets/stylesheets/main.342714a4.min.css">
|
||
|
||
|
||
<link rel="stylesheet" href="../../assets/stylesheets/palette.06af60db.min.css">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
|
||
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Roboto:300,300i,400,400i,700,700i%7CJetBrains+Mono:400,400i,700,700i&display=fallback">
|
||
<style>:root{--md-text-font:"Roboto";--md-code-font:"JetBrains Mono"}</style>
|
||
|
||
|
||
|
||
<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce(((e,_)=>(e<<5)-e+_.charCodeAt(0)),0),__md_get=(e,_=localStorage,t=__md_scope)=>JSON.parse(_.getItem(t.pathname+"."+e)),__md_set=(e,_,t=localStorage,a=__md_scope)=>{try{t.setItem(a.pathname+"."+e,JSON.stringify(_))}catch(e){}}</script>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
</head>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<body dir="ltr" data-md-color-scheme="default" data-md-color-primary="green" data-md-color-accent="indigo">
|
||
|
||
|
||
<input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
|
||
<input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
|
||
<label class="md-overlay" for="__drawer"></label>
|
||
<div data-md-component="skip">
|
||
|
||
|
||
<a href="#_1" class="md-skip">
|
||
Skip to content
|
||
</a>
|
||
|
||
</div>
|
||
<div data-md-component="announce">
|
||
|
||
</div>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<header class="md-header md-header--shadow" data-md-component="header">
|
||
<nav class="md-header__inner md-grid" aria-label="Header">
|
||
<a href="../.." title="Очередные записки очередного гика" class="md-header__button md-logo" aria-label="Очередные записки очередного гика" data-md-component="logo">
|
||
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 8a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3 3 3 0 0 0 3 3m0 3.54C9.64 9.35 6.5 8 3 8v11c3.5 0 6.64 1.35 9 3.54 2.36-2.19 5.5-3.54 9-3.54V8c-3.5 0-6.64 1.35-9 3.54"/></svg>
|
||
|
||
</a>
|
||
<label class="md-header__button md-icon" for="__drawer">
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3zm0 5h18v2H3zm0 5h18v2H3z"/></svg>
|
||
</label>
|
||
<div class="md-header__title" data-md-component="header-title">
|
||
<div class="md-header__ellipsis">
|
||
<div class="md-header__topic">
|
||
<span class="md-ellipsis">
|
||
Очередные записки очередного гика
|
||
</span>
|
||
</div>
|
||
<div class="md-header__topic" data-md-component="header-topic">
|
||
<span class="md-ellipsis">
|
||
|
||
Немного про Байесовскую статистику
|
||
|
||
</span>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<form class="md-header__option" data-md-component="palette">
|
||
|
||
|
||
|
||
|
||
<input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="green" data-md-color-accent="indigo" aria-label="Switch to dark mode" type="radio" name="__palette" id="__palette_0">
|
||
|
||
<label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7H7a5 5 0 0 0-5 5 5 5 0 0 0 5 5h10a5 5 0 0 0 5-5 5 5 0 0 0-5-5m0 8a3 3 0 0 1-3-3 3 3 0 0 1 3-3 3 3 0 0 1 3 3 3 3 0 0 1-3 3"/></svg>
|
||
</label>
|
||
|
||
|
||
|
||
|
||
|
||
<input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="teal" data-md-color-accent="indigo" aria-label="Switch to light mode" type="radio" name="__palette" id="__palette_1">
|
||
|
||
<label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 6H7c-3.31 0-6 2.69-6 6s2.69 6 6 6h10c3.31 0 6-2.69 6-6s-2.69-6-6-6m0 10H7c-2.21 0-4-1.79-4-4s1.79-4 4-4h10c2.21 0 4 1.79 4 4s-1.79 4-4 4M7 9c-1.66 0-3 1.34-3 3s1.34 3 3 3 3-1.34 3-3-1.34-3-3-3"/></svg>
|
||
</label>
|
||
|
||
|
||
</form>
|
||
|
||
|
||
|
||
<script>var palette=__md_get("__palette");if(palette&&palette.color){if("(prefers-color-scheme)"===palette.color.media){var media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']");palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent")}for(var[key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
|
||
|
||
|
||
|
||
|
||
|
||
<label class="md-header__button md-icon" for="__search">
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
|
||
</label>
|
||
<div class="md-search" data-md-component="search" role="dialog">
|
||
<label class="md-search__overlay" for="__search"></label>
|
||
<div class="md-search__inner" role="search">
|
||
<form class="md-search__form" name="search">
|
||
<input type="text" class="md-search__input" name="query" aria-label="Search" placeholder="Search" autocapitalize="off" autocorrect="off" autocomplete="off" spellcheck="false" data-md-component="search-query" required>
|
||
<label class="md-search__icon md-icon" for="__search">
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11z"/></svg>
|
||
</label>
|
||
<nav class="md-search__options" aria-label="Search">
|
||
|
||
<button type="reset" class="md-search__icon md-icon" title="Clear" aria-label="Clear" tabindex="-1">
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 6.41 17.59 5 12 10.59 6.41 5 5 6.41 10.59 12 5 17.59 6.41 19 12 13.41 17.59 19 19 17.59 13.41 12z"/></svg>
|
||
</button>
|
||
</nav>
|
||
|
||
</form>
|
||
<div class="md-search__output">
|
||
<div class="md-search__scrollwrap" tabindex="0" data-md-scrollfix>
|
||
<div class="md-search-result" data-md-component="search-result">
|
||
<div class="md-search-result__meta">
|
||
Initializing search
|
||
</div>
|
||
<ol class="md-search-result__list" role="presentation"></ol>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div class="md-header__source">
|
||
<a href="https://github.com/rustbas/blog" title="Go to repository" class="md-source" data-md-component="source">
|
||
<div class="md-source__icon md-icon">
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.7.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M216.29 158.39H137C97 147.9 6.51 150.63 6.51 233.18c0 30.09 15 51.23 35 61-25.1 23-37 33.85-37 49.21 0 11 4.47 21.14 17.89 26.81C8.13 383.61 0 393.35 0 411.65c0 32.11 28.05 50.82 101.63 50.82 70.75 0 111.79-26.42 111.79-73.18 0-58.66-45.16-56.5-151.63-63l13.43-21.55c27.27 7.58 118.7 10 118.7-67.89 0-18.7-7.73-31.71-15-41.07l37.41-2.84zm-63.42 241.9c0 32.06-104.89 32.1-104.89 2.43 0-8.14 5.27-15 10.57-21.54 77.71 5.3 94.32 3.37 94.32 19.11m-50.81-134.58c-52.8 0-50.46-71.16 1.2-71.16 49.54 0 50.82 71.16-1.2 71.16m133.3 100.51v-32.1c26.75-3.66 27.24-2 27.24-11V203.61c0-8.5-2.05-7.38-27.24-16.26l4.47-32.92H324v168.71c0 6.51.4 7.32 6.51 8.14l20.73 2.84v32.1zm52.45-244.31c-23.17 0-36.59-13.43-36.59-36.61s13.42-35.77 36.59-35.77c23.58 0 37 12.62 37 35.77s-13.42 36.61-37 36.61M512 350.46c-17.49 8.53-43.1 16.26-66.28 16.26-48.38 0-66.67-19.5-66.67-65.46V194.75c0-5.42 1.05-4.06-31.71-4.06V154.5c35.78-4.07 50-22 54.47-66.27h38.63c0 65.83-1.34 61.81 3.26 61.81H501v40.65h-60.56v97.15c0 6.92-4.92 51.41 60.57 26.84z"/></svg>
|
||
</div>
|
||
<div class="md-source__repository">
|
||
GitHub
|
||
</div>
|
||
</a>
|
||
</div>
|
||
|
||
</nav>
|
||
|
||
</header>
|
||
|
||
<div class="md-container" data-md-component="container">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<main class="md-main" data-md-component="main">
|
||
<div class="md-main__inner md-grid">
|
||
|
||
|
||
|
||
<div class="md-sidebar md-sidebar--primary" data-md-component="sidebar" data-md-type="navigation" >
|
||
<div class="md-sidebar__scrollwrap">
|
||
<div class="md-sidebar__inner">
|
||
|
||
|
||
|
||
|
||
<nav class="md-nav md-nav--primary" aria-label="Navigation" data-md-level="0">
|
||
<label class="md-nav__title" for="__drawer">
|
||
<a href="../.." title="Очередные записки очередного гика" class="md-nav__button md-logo" aria-label="Очередные записки очередного гика" data-md-component="logo">
|
||
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 8a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3 3 3 0 0 0 3 3m0 3.54C9.64 9.35 6.5 8 3 8v11c3.5 0 6.64 1.35 9 3.54 2.36-2.19 5.5-3.54 9-3.54V8c-3.5 0-6.64 1.35-9 3.54"/></svg>
|
||
|
||
</a>
|
||
Очередные записки очередного гика
|
||
</label>
|
||
|
||
<div class="md-nav__source">
|
||
<a href="https://github.com/rustbas/blog" title="Go to repository" class="md-source" data-md-component="source">
|
||
<div class="md-source__icon md-icon">
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.7.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M216.29 158.39H137C97 147.9 6.51 150.63 6.51 233.18c0 30.09 15 51.23 35 61-25.1 23-37 33.85-37 49.21 0 11 4.47 21.14 17.89 26.81C8.13 383.61 0 393.35 0 411.65c0 32.11 28.05 50.82 101.63 50.82 70.75 0 111.79-26.42 111.79-73.18 0-58.66-45.16-56.5-151.63-63l13.43-21.55c27.27 7.58 118.7 10 118.7-67.89 0-18.7-7.73-31.71-15-41.07l37.41-2.84zm-63.42 241.9c0 32.06-104.89 32.1-104.89 2.43 0-8.14 5.27-15 10.57-21.54 77.71 5.3 94.32 3.37 94.32 19.11m-50.81-134.58c-52.8 0-50.46-71.16 1.2-71.16 49.54 0 50.82 71.16-1.2 71.16m133.3 100.51v-32.1c26.75-3.66 27.24-2 27.24-11V203.61c0-8.5-2.05-7.38-27.24-16.26l4.47-32.92H324v168.71c0 6.51.4 7.32 6.51 8.14l20.73 2.84v32.1zm52.45-244.31c-23.17 0-36.59-13.43-36.59-36.61s13.42-35.77 36.59-35.77c23.58 0 37 12.62 37 35.77s-13.42 36.61-37 36.61M512 350.46c-17.49 8.53-43.1 16.26-66.28 16.26-48.38 0-66.67-19.5-66.67-65.46V194.75c0-5.42 1.05-4.06-31.71-4.06V154.5c35.78-4.07 50-22 54.47-66.27h38.63c0 65.83-1.34 61.81 3.26 61.81H501v40.65h-60.56v97.15c0 6.92-4.92 51.41 60.57 26.84z"/></svg>
|
||
</div>
|
||
<div class="md-source__repository">
|
||
GitHub
|
||
</div>
|
||
</a>
|
||
</div>
|
||
|
||
<ul class="md-nav__list" data-md-scrollfix>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
Математика
|
||
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../linux/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
Linux
|
||
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="md-nav__item">
|
||
<a href="../../common/" class="md-nav__link">
|
||
|
||
|
||
|
||
<span class="md-ellipsis">
|
||
Общее
|
||
|
||
</span>
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
</ul>
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
|
||
<div class="md-sidebar__scrollwrap">
|
||
<div class="md-sidebar__inner">
|
||
|
||
|
||
<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<label class="md-nav__title" for="__toc">
|
||
<span class="md-nav__icon md-icon"></span>
|
||
Table of contents
|
||
</label>
|
||
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_2" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
Задача
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_3" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
Решение
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label="Решение">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_4" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
Библиотеки
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_5" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
Константы и функции
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_6" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
Моделирование
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_7" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
График
|
||
</span>
|
||
</a>
|
||
|
||
<nav class="md-nav" aria-label="График">
|
||
<ul class="md-nav__list">
|
||
|
||
<li class="md-nav__item">
|
||
<a href="#_8" class="md-nav__link">
|
||
<span class="md-ellipsis">
|
||
Мода
|
||
</span>
|
||
</a>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
</nav>
|
||
|
||
</li>
|
||
|
||
</ul>
|
||
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div class="md-content" data-md-component="content">
|
||
<article class="md-content__inner md-typeset">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<h1 id="_1">Немного про Байесовскую статистику</h1>
|
||
<h2 id="_2">Задача</h2>
|
||
<p>Представим, что у нас есть монета, честность которой нам неизвестна (если быть
|
||
точным, мы не знаем с каким шансом выпадает орёл, с каким — решка). Поэтому мы
|
||
бы хотели оценить вероятность <span class="arithmatex">\(p\)</span> — шанс выпадения орла<sup id="fnref:1"><a class="footnote-ref" href="#fn:1">1</a></sup>.</p>
|
||
<p>В <a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0">классической статистике</a>
|
||
эта задача бы решалась бы так:</p>
|
||
<ul>
|
||
<li>Монета подбрасывается <span class="arithmatex">\(n\)</span> раз.</li>
|
||
<li>Из них <span class="arithmatex">\(m\)</span> — количество выпавших орлов.</li>
|
||
<li>Отношение <span class="arithmatex">\(\frac{m}{n}\)</span> будет оценкой <span class="arithmatex">\(p\)</span>.</li>
|
||
</ul>
|
||
<p>В <a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B0%D0%B9%D0%B5%D1%81%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0">Байесовской
|
||
статистике</a>
|
||
подход иной:</p>
|
||
<ol>
|
||
<li>Обозначим монету как <a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%91%D0%B5%D1%80%D0%BD%D1%83%D0%BB%D0%BB%D0%B8">бернуллевскую случайную
|
||
величину</a>
|
||
<span class="arithmatex">\(\xi\)</span> с параметром <span class="arithmatex">\(\theta\)</span>, у которой <span class="arithmatex">\(1\)</span> — это выпадение орла, <span class="arithmatex">\(0\)</span> —
|
||
решки. </li>
|
||
<li>Предполагается априорное распределение <span class="arithmatex">\(\pi(\theta)\)</span> (т.е. распределение,
|
||
которое мы предполагаем, исходя из того, что нам известно о параметре
|
||
<span class="arithmatex">\(\theta\)</span>), как правило, это <a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D1%80%D1%8B%D0%B2%D0%BD%D0%BE%D0%B5_%D1%80%D0%B0%D0%B2%D0%BD%D0%BE%D0%BC%D0%B5%D1%80%D0%BD%D0%BE%D0%B5_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5">равномерное
|
||
распределение</a>
|
||
<span class="arithmatex">\(U \left(0,1\right)\)</span>. </li>
|
||
<li>Монета подбрасывается. </li>
|
||
<li>Распределение <span class="arithmatex">\(\theta\)</span> уточняется по формуле<sup id="fnref:2"><a class="footnote-ref" href="#fn:2">2</a></sup>:
|
||
$$\Large
|
||
\pi(\theta | \xi) = \frac{p(\xi|\theta)p(\theta)}
|
||
{\int\limits_{\Theta}p(\xi|\theta)p(\theta)d\theta}
|
||
$$</li>
|
||
<li>Повторить пункты 2-4, предполагая <span class="arithmatex">\(\pi(\theta) = \pi(\theta|\xi)\)</span></li>
|
||
</ol>
|
||
<p>Тогда оценкой <span class="arithmatex">\(\theta\)</span> будет:
|
||
$$\Large
|
||
\hat \theta = \arg \max_\theta \pi ( \theta | \xi )
|
||
$$</p>
|
||
<p>Иначе говоря,
|
||
<a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D0%B0_%28%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0%29">мода</a>
|
||
апостериорного распределения.</p>
|
||
<p>Мы, в качестве априорного распределения, взяли <a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B5%D1%82%D0%B0-%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5">Бета-распределения</a> <span class="arithmatex">\(Beta(2,2)\)</span>. В
|
||
таком случае, если на <span class="arithmatex">\(i\)</span>-ом шаге мы имеем:
|
||
$$\Large
|
||
\theta \sim Beta \left( \alpha_i, \beta_i \right)
|
||
$$</p>
|
||
<p>то апостериорное распределение будет:</p>
|
||
<div class="arithmatex">\[\Large
|
||
Beta \left( \alpha_i, \beta_i \right) =
|
||
\begin{cases}
|
||
Beta \left( \alpha_i + 1, \beta_i \right), &\xi_i = 1 \\
|
||
Beta \left( \alpha_i, \beta_i + 1 \right), &\xi_i = 0 \\
|
||
\end{cases}
|
||
\]</div>
|
||
<p>Тогда не нужно интегрировать на каждом шаге, что существенно упрощает вычисления.</p>
|
||
<h2 id="_3">Решение</h2>
|
||
<p>Приведем решение на <code>python</code>.</p>
|
||
<h3 id="_4">Библиотеки</h3>
|
||
<p>Сначала нужно импортировать и настроить библиотеки:</p>
|
||
<div class="highlight"><pre><span></span><code><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
|
||
<span class="kn">import</span><span class="w"> </span><span class="nn">pandas</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">pd</span>
|
||
<span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<span class="kn">import</span><span class="w"> </span><span class="nn">seaborn</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sns</span>
|
||
|
||
<span class="kn">from</span><span class="w"> </span><span class="nn">tqdm</span><span class="w"> </span><span class="kn">import</span> <span class="n">tqdm</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="s1">'font.size'</span><span class="p">:</span> <span class="mi">14</span><span class="p">})</span>
|
||
</code></pre></div>
|
||
<h3 id="_5">Константы и функции</h3>
|
||
<p>Обозначим константы:</p>
|
||
<ul>
|
||
<li><code>p</code> — истинная вероятность выпадения орла.</li>
|
||
<li><code>NMODEL</code> — количество бросков монеты.</li>
|
||
</ul>
|
||
<div class="highlight"><pre><span></span><code><span class="n">p</span> <span class="o">=</span> <span class="mf">0.523</span>
|
||
<span class="n">NMODEL</span> <span class="o">=</span> <span class="mi">7501</span>
|
||
</code></pre></div>
|
||
<p>Введем функции:
|
||
- <code>beta</code> — объект для Бета-распределения
|
||
- <code>coin</code> — функция одиночного броска монеты</p>
|
||
<div class="highlight"><pre><span></span><code><span class="kn">from</span><span class="w"> </span><span class="nn">scipy.stats</span><span class="w"> </span><span class="kn">import</span> <span class="n">beta</span>
|
||
<span class="n">coin</span> <span class="o">=</span> <span class="k">lambda</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="n">p</span><span class="p">)</span>
|
||
</code></pre></div>
|
||
<h3 id="_6">Моделирование</h3>
|
||
<p>В качестве априорного распределения тут используется Бета-распределение
|
||
<span class="arithmatex">\(Beta(2,2)\)</span>. Это сделано потому, что расчет моды накладывает ограничение, что оба
|
||
параметра должны быть строго больше 1.</p>
|
||
<p>Мода расчитывается по формуле:</p>
|
||
<div class="arithmatex">\[\Large
|
||
\text{Mode} = \frac{\alpha - 1}{\alpha + \beta - 2}
|
||
\]</div>
|
||
<div class="highlight"><pre><span></span><code><span class="n">alphas</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">NMODEL</span><span class="p">)</span>
|
||
<span class="n">betas</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">NMODEL</span><span class="p">)</span>
|
||
<span class="n">alphas</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">betas</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span><span class="mi">2</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">tqdm</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">NMODEL</span><span class="p">)):</span>
|
||
<span class="n">A</span> <span class="o">=</span> <span class="n">coin</span><span class="p">()</span>
|
||
<span class="k">if</span> <span class="n">A</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||
<span class="n">betas</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="n">alphas</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
|
||
|
||
<span class="n">betas</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">betas</span><span class="p">)</span>
|
||
<span class="n">alphas</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">alphas</span><span class="p">)</span>
|
||
|
||
<span class="n">Es</span> <span class="o">=</span> <span class="p">(</span><span class="n">alphas</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">alphas</span> <span class="o">+</span> <span class="n">betas</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span>
|
||
</code></pre></div>
|
||
<h3 id="_7">График</h3>
|
||
<h4 id="_8">Мода</h4>
|
||
<p>Построим график зависимости моды от количества бросков.</p>
|
||
<div class="highlight"><pre><span></span><code><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">8</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">80</span><span class="p">)</span>
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">lineplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">NMODEL</span><span class="p">),</span>
|
||
<span class="n">y</span><span class="o">=</span><span class="n">Es</span><span class="p">,</span>
|
||
<span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span>
|
||
<span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"arg$\max_p \; f(p|\mathbb</span><span class="si">{X}</span><span class="s2">)$"</span><span class="p">,</span> <span class="p">)</span>
|
||
<span class="n">sns</span><span class="o">.</span><span class="n">lineplot</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">NMODEL</span><span class="p">),</span>
|
||
<span class="n">y</span><span class="o">=</span><span class="n">p</span><span class="p">,</span>
|
||
<span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span>
|
||
<span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"Истинное $p = </span><span class="si">{:.3f}</span><span class="s2">$"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">ylabel</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$E(p|\mathbb</span><span class="si">{X}</span><span class="s2">)$"</span><span class="p">,</span> <span class="n">xlabel</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$i$"</span><span class="p">);</span>
|
||
</code></pre></div>
|
||
<p><img alt="mode.jpg" src="../assets/baes/mode.jpg" /></p>
|
||
<p>Гифка</p>
|
||
<p>Сделаем гифку на тему. Сначала нужно импортировать библиотеку <code>imageio</code> и зададим
|
||
шаг <code>STEP</code> (так как, потом нужно будет сохранять каждый график в опертивной
|
||
памяти, что проблематично при большом <code>NMODEL</code>):</p>
|
||
<div class="highlight"><pre><span></span><code><span class="kn">import</span><span class="w"> </span><span class="nn">imageio</span>
|
||
<span class="n">STEP</span> <span class="o">=</span> <span class="n">NMODEL</span><span class="o">//</span><span class="mi">150</span>
|
||
</code></pre></div>
|
||
<p>После, создадим список функций плотности вероятности:</p>
|
||
<div class="highlight"><pre><span></span><code><span class="n">pi</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">tqdm</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">NMODEL</span><span class="p">)):</span>
|
||
<span class="n">pi</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">beta</span><span class="p">(</span><span class="n">alphas</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">betas</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
|
||
</code></pre></div>
|
||
<p>И по ним сделаем gif-изображение:</p>
|
||
<div class="highlight"><pre><span></span><code><span class="k">def</span><span class="w"> </span><span class="nf">create_frame</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
|
||
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">8</span><span class="p">),</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">80</span><span class="p">)</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">500</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">pi</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">i_max</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||
<span class="n">x_max</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i_max</span><span class="p">]</span>
|
||
<span class="n">y_max</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">i_max</span><span class="p">]</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$f(p|\mathbb</span><span class="si">{X}</span><span class="s2">)$"</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">x</span> <span class="o">=</span> <span class="n">p</span><span class="p">,</span> <span class="n">color</span> <span class="o">=</span> <span class="s1">'red'</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"Истинное $p=</span><span class="si">{</span><span class="n">p</span><span class="si">}</span><span class="s2">$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">([</span><span class="n">x_max</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="n">y_max</span><span class="p">],</span>
|
||
<span class="n">color</span><span class="o">=</span><span class="s2">"green"</span><span class="p">,</span>
|
||
<span class="n">marker</span><span class="o">=</span><span class="s2">"o"</span><span class="p">,</span>
|
||
<span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"Мода $f(p|\mathbb</span><span class="si">{X}</span><span class="s2">)$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">14</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">100</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s1">'$f(p|\mathbb</span><span class="si">{X}</span><span class="s1">)$'</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">14</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">r</span><span class="s1">'Распределение p, $i=</span><span class="si">{}</span><span class="s1">$'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span>
|
||
<span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="sa">f</span><span class="s1">'./img/img_</span><span class="si">{</span><span class="n">i</span><span class="si">:</span><span class="s1">04d</span><span class="si">}</span><span class="s1">.png'</span><span class="p">,</span>
|
||
<span class="n">transparent</span> <span class="o">=</span> <span class="kc">False</span><span class="p">,</span>
|
||
<span class="n">facecolor</span> <span class="o">=</span> <span class="s1">'white'</span>
|
||
<span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
|
||
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">tqdm</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">NMODEL</span><span class="p">,</span> <span class="n">STEP</span><span class="p">)):</span>
|
||
<span class="n">create_frame</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
|
||
<span class="n">frames</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">tqdm</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">NMODEL</span><span class="p">,</span> <span class="n">STEP</span><span class="p">)):</span>
|
||
<span class="n">image</span> <span class="o">=</span> <span class="n">imageio</span><span class="o">.</span><span class="n">v2</span><span class="o">.</span><span class="n">imread</span><span class="p">(</span><span class="sa">f</span><span class="s1">'./img/img_</span><span class="si">{</span><span class="n">i</span><span class="si">:</span><span class="s1">04d</span><span class="si">}</span><span class="s1">.png'</span><span class="p">)</span>
|
||
<span class="n">frames</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">image</span><span class="p">)</span>
|
||
<span class="n">imageio</span><span class="o">.</span><span class="n">mimsave</span><span class="p">(</span><span class="s1">'./distributions.gif'</span><span class="p">,</span>
|
||
<span class="n">frames</span><span class="p">,</span>
|
||
<span class="n">fps</span> <span class="o">=</span> <span class="mi">30</span><span class="p">)</span>
|
||
</code></pre></div>
|
||
<p>Результат:</p>
|
||
<p><img alt="gif" src="../assets/baes/video.gif" /></p>
|
||
<div class="footnote">
|
||
<hr />
|
||
<ol>
|
||
<li id="fn:1">
|
||
<p>Понятно, что вероятность выпадения решки равна <span class="arithmatex">\(1 - p\)</span> <a class="footnote-backref" href="#fnref:1" title="Jump back to footnote 1 in the text">↩</a></p>
|
||
</li>
|
||
<li id="fn:2">
|
||
<p><span class="arithmatex">\(p(\theta|\xi)\)</span> -- это функция правдоподобия. <a class="footnote-backref" href="#fnref:2" title="Jump back to footnote 2 in the text">↩</a></p>
|
||
</li>
|
||
</ol>
|
||
</div>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<aside class="md-source-file">
|
||
|
||
|
||
<span class="md-source-file__fact">
|
||
<span class="md-icon" title="Last update">
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M21 13.1c-.1 0-.3.1-.4.2l-1 1 2.1 2.1 1-1c.2-.2.2-.6 0-.8l-1.3-1.3c-.1-.1-.2-.2-.4-.2m-1.9 1.8-6.1 6V23h2.1l6.1-6.1zM12.5 7v5.2l4 2.4-1 1L11 13V7zM11 21.9c-5.1-.5-9-4.8-9-9.9C2 6.5 6.5 2 12 2c5.3 0 9.6 4.1 10 9.3-.3-.1-.6-.2-1-.2s-.7.1-1 .2C19.6 7.2 16.2 4 12 4c-4.4 0-8 3.6-8 8 0 4.1 3.1 7.5 7.1 7.9l-.1.2z"/></svg>
|
||
</span>
|
||
2025-06-02
|
||
</span>
|
||
|
||
|
||
|
||
|
||
|
||
</aside>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
</article>
|
||
</div>
|
||
|
||
|
||
<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
|
||
</div>
|
||
|
||
</main>
|
||
|
||
<footer class="md-footer">
|
||
|
||
<div class="md-footer-meta md-typeset">
|
||
<div class="md-footer-meta__inner md-grid">
|
||
<div class="md-copyright">
|
||
|
||
|
||
Made with
|
||
<a href="https://squidfunk.github.io/mkdocs-material/" target="_blank" rel="noopener">
|
||
Material for MkDocs
|
||
</a>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
</div>
|
||
</footer>
|
||
|
||
</div>
|
||
<div class="md-dialog" data-md-component="dialog">
|
||
<div class="md-dialog__inner md-typeset"></div>
|
||
</div>
|
||
|
||
<div class="md-progress" data-md-component="progress" role="progressbar"></div>
|
||
|
||
|
||
|
||
|
||
<script id="__config" type="application/json">{"base": "../..", "features": ["navigation.instant", "navigation.instant.progress", "header.autohide"], "search": "../../assets/javascripts/workers/search.d50fe291.min.js", "tags": null, "translations": {"clipboard.copied": "Copied to clipboard", "clipboard.copy": "Copy to clipboard", "search.result.more.one": "1 more on this page", "search.result.more.other": "# more on this page", "search.result.none": "No matching documents", "search.result.one": "1 matching document", "search.result.other": "# matching documents", "search.result.placeholder": "Type to start searching", "search.result.term.missing": "Missing", "select.version": "Select version"}, "version": null}</script>
|
||
|
||
|
||
<script src="../../assets/javascripts/bundle.13a4f30d.min.js"></script>
|
||
|
||
<script src="../../javascripts/mathjax.js"></script>
|
||
|
||
<script src="https://unpkg.com/mathjax@3/es5/tex-mml-chtml.js"></script>
|
||
|
||
|
||
</body>
|
||
</html> |