From dab57bece7c63e66ace10b8d5ece5de38c26d2e1 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= <petrvelicka@tuta.io>
Date: Tue, 8 Apr 2025 12:14:08 +0200
Subject: [PATCH] prednaska 8.4.2025

---
 skripta.pdf           | Bin 301936 -> 314262 bytes
 skripta.tex           |   4 ++
 statisticke-uceni.tex | 107 ++++++++++++++++++++++++++++++++++++++++++
 3 files changed, 111 insertions(+)
 create mode 100644 statisticke-uceni.tex

diff --git a/skripta.pdf b/skripta.pdf
index 6d1566d337dc4f61d88219063f0e649b645a4367..0f669de0f2f06e48f2864b54bdf7412589a42314 100644
GIT binary patch
delta 84351
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diff --git a/skripta.tex b/skripta.tex
index e11b5d2..0b56a69 100644
--- a/skripta.tex
+++ b/skripta.tex
@@ -28,6 +28,9 @@
 \DeclareMathOperator{\Cov}{Cov}
 \DeclareMathOperator{\sd}{sd}
 \DeclareMathOperator{\sgn}{sgn}
+\DeclareMathOperator{\bias}{bias}
+\DeclareMathOperator{\se}{se}
+\DeclareMathOperator{\MSE}{MSE}
 
 \DeclarePairedDelimiter\ceil{\lceil}{\rceil}
 \DeclarePairedDelimiter\floor{\lfloor}{\rfloor}
@@ -50,6 +53,7 @@
 \include{stredni-hodnota}
 \include{stochasticke-nerovnosti}
 \include{stochasticke-konvergence}
+\include{statisticke-uceni}
 \include{ukazkove-pisemky}
 
 \end{document}
diff --git a/statisticke-uceni.tex b/statisticke-uceni.tex
new file mode 100644
index 0000000..9a7fb16
--- /dev/null
+++ b/statisticke-uceni.tex
@@ -0,0 +1,107 @@
+\section{Statistické učení}
+
+V této kapitole se budeme věnovat základům matematické statistiky, což je obor, který bude středobodem naší pozornosti po celý zbytek semestru. Začneme formalizací pojmů týkajících se opakovaného provádění experimentu a charakterizací statistických modelů.
+
+\begin{definition}
+	Pokud jsou $X_1, \dots, X_n$ nezávislé a každá má stejné marginální rozdělení a distribuční funkci $F$, říkáme, že $X_1, \dots, X_n$ jsou IID (nezávislé a stejně rozdělené) a píšeme
+	$$ X_1, \dots, X_n \overset{IID}\sim F. $$
+	Takové $X_1, \dots, X_n$ nazýváme \textit{náhodný výběr} velikosti $n$ z $F$.
+\end{definition}
+
+Obecně si představujeme měřitelná zobrazení $X_1, \dots, X_n$. V praxi však většinou dostaneme pouze reálná čísla $X_i(\omega)$ pro pro jedno konkrétní $\omega \in \Omega$. Možná rozdělení těchto náhodných veličin budeme modelovat pomocí takzvaných parametrických modelů, tedy množin $\mathcal{F}$ rozdělení, jež se dají parametrizovat konečným počtem parametrů.
+
+\begin{example}[Normální model]
+	$$ \mathcal{F} = \left\{ f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left\{- \frac{(x - \mu)^2}{2\sigma^2} \right\}, \mu \in \R, \sigma^2 > 0\right\}. $$
+	Taková data pochází z normálního rozdělení se dvěma parametry $\mu$ a $\sigma^2$.
+\end{example}
+
+Všechny parametrické modely můžeme obecně zapsat ve tvaru
+$$ \mathcal{F} = \{ f(\cdot; \vec \theta) : \vec \theta \in \vec \Theta \subseteq \R^d \}. $$
+V dalším textu budeme využívat následující značení:
+$$ P_\theta[X\in A] := \int_A f(x; \theta) dx; $$
+$$ \E_\theta[g(X)] := \int_\R g(x) f(x; \theta) dx. $$
+
+Velkou skupinou modelů jsou také neparametrické modely, který nemůžeme parametrizovat konečným počtem parametrů. Například, celou funkci hustoty můžeme považovat za nekonečnědimenzionální prostor. Uvedeme si jeden příklad takového neparametrického modelu.
+
+\begin{example}[Model Sobolevova prostoru]
+	$$ \mathcal{F} = \left\{ f: \int_\R (f''(x))^2 dx < \infty \right\}. $$
+	Data pochází z rozdělení s nepříliš ``vlnitou" hustotou.
+\end{example}
+
+\begin{definition}
+	\textit{Bodový odhad} $\hat \theta_n$ parametru $\theta$ je měřitelná funkce $t$ náhodných veličin $X_1, \dots, X_n$:
+	$$ \hat \theta_n = t(X_1, \dots, X_n). $$
+\end{definition}
+
+V této definici předpokládáme, že $\theta$ je pevné ale neznámé reálné číslo (vektor). Avšak získaný odhad $\hat \theta_n$ je sice náhodná veličina, ale umíme ji přesně charakterizovat.
+
+\begin{definition}
+	Odhad $\hat \theta_n$ je \textit{nestranný}, pokud $\E[\hat \theta_n] = \theta$ pro všechna $n \in \N$. \textit{Vychýlení} odhadu definujeme jako $\bias(\hat \theta_n) := \E[\hat \theta_n] - \theta$. Odhad je \textit{konzistentní}, jestliže $\hat \theta_n \overset P \to \theta$ pro $n \to \infty$.
+
+\end{definition}
+
+V dnešní době je díky vývoji výpočetní techniky nestrannost více upozaďována, větší důraz proto klademe na konzistenci modelu.
+
+\begin{definition}
+	Rozdělení odhadu $\hat \theta_n$ nazýváme \textit{výběrové rozdělení}. Standardní odchylku $\hat \theta_n$ nazýváme \textit{standardní chyba} $\se(\hat \theta_n) = \sqrt{\Var \hat \theta_n}$.
+\end{definition}
+
+V těchto případech je standardní chyba $\se$ neznámá veličina (parametr), ale obvykle ji můžeme odhadnout. Takovou odhadnutou standardní chybu značíme $\widehat \se$.
+
+\begin{example}
+	\label{ex-coin-bernoulli}
+	Mějme Bernoulliho náhodný výběr $X_1, \dots, X_n \overset{IID}\sim Be(p)$ a parametr $p \in (0, 1)$. Potom můžeme uvažovat odhad
+	$$ \hat p_n := \frac{1}{n} \sum_{i=1}^n X_i $$
+	a z toho získáme odhad standardní chyby (díky nezávislosti a stejné rozdělenosti máme $\Var(\hat p_n) = \frac{p(1 - p)}{n}$). Jelikož přesná hodnota $p$ je neznámá, musíme tento parametr také odhadnout, proto
+	$$ \widehat \se(\hat p_n) := \sqrt{\frac{\hat p_n(1 - \hat p_n)}{n}}. $$
+\end{example}
+
+\begin{definition}
+	Kvalitu bodového odhadu můžeme posuzovat pomocí \textit{střední kvadratické chyby}
+	$$ \MSE(\hat \theta_n) := \E_\theta [\hat \theta_n - \theta]^2. $$
+\end{definition}
+
+Mějme na paměti, že $\E_\theta$ se v případě nezávislých a stejně rozdělených $X_i$ vztahuje k očekávané hodnotě vzhledem k rozdělení
+$$ f(x_1, \dots, x_n; \theta) = \prod_{i=1}^n f(x_i; \theta). $$
+
+\begin{theorem}[Rozklad střední kvadratické chyby]
+	\label{thm-mse-bias-var}
+	Mějme odhad $\hat \theta_n$. Pak vždy platí  $\MSE(\hat \theta_n) = \bias^2(\hat \theta_n) + \Var(\hat \theta_n)$.
+\end{theorem}
+
+\begin{proof}
+	Rozepsáním definice dostáváme
+	$$ \MSE[\hat \theta_n] = \E_\theta[\hat \theta_n - \theta]^2 = \E_\theta\left\{ [\hat \theta_n - \E\hat \theta_n + \E\hat \theta_n - \theta]^2 \right\} = $$
+	$$ \E_\theta\left\{[\hat \theta_n - \E \hat\theta_n]^2 - 2[\hat\theta_n - \E\hat\theta_n][\E\hat\theta_n - \theta] + [\E\hat\theta_n - \theta]^2 \right\} = $$
+	$$ = \Var(\hat\theta_n) + \bias^2(\hat\theta_n), $$
+	kde poslední rovnost plyne z toho, že druhý sčítanec je nulový, což plyne z linearity střední hodnoty.
+\end{proof}
+
+\begin{theorem}[Postačující podmínka pro konzistenci]
+	\label{thm-consistence-sufficient-condition}
+	Nechť platí $\bias(\hat \theta_n) \to 0$ a $\Var(\hat \theta_n) \to 0$. Potom platí $\hat\theta_n$ je konzistentní.
+\end{theorem}
+
+\begin{proof}
+	Z Věty \ref{thm-mse-bias-var} dostáváme, že $\MSE(\hat\theta_n) = \E_\theta[\hat\theta_n - \theta]^2 \to 0$. Z definice $L_2$ konvergence dostáváme, že $\hat\theta_n \overset{L_2}\to \theta$. Zbytek dostáváme z faktu, že $L_2$ konvergence implikuje konvergenci v pravděpodobnosti.
+\end{proof}
+
+\begin{example}
+	Mějme stejnou situaci jako v Příkladu \ref{ex-coin-bernoulli}. Jelikož náš odhad je nestranný ($\E(\hat p_n) = p$) a $\Var(\hat p_n) = \frac{p(1 - p)}{n} \to 0$ pro $n \to \infty$, dostáváme díky Větě \ref{thm-consistence-sufficient-condition}, že $\hat p_n \overset P \to p$.
+\end{example}
+
+\begin{definition}
+	Odhad $\hat \theta_n$ parametru $\theta$ se nazývá \textit{asymptoticky standardně normální}, jestliže pro $n \to \infty$ platí
+	$$ \frac{\hat \theta_n - \theta}{\se(\hat \theta_n)} \overset D \to N(0, 1). $$
+\end{definition}
+
+\begin{definition}
+	$(1-\alpha)$-\textit{interval spolehlivosti} (konfidenční interval) pro parametr $\theta$ je interval $C_n = (a, b)$, kde $a = a(X_1, \dots, X_n)$ a $b = b(X_1, \dots, X_n)$ jsou měřitelné funkce dat takové, že pro všechna $\theta \in \Theta$
+	$$ P_\theta[\theta \in C_n] = 1 - \alpha. $$
+	\textit{Asymptotický} (přibližný) $(1 - \alpha)$-\textit{interval spolehlivosti} pro parametr $\theta$ je interval $C_n$ takový, že pro všechna $\theta \in \Theta$
+	$$ \lim_{n \to \infty} P_\theta [\theta \in C_n] = 1 - \theta. $$
+\end{definition}
+
+Tato definice říká, že interval $C_n$ zachytí $\theta$ s pravděpodobností (přibližně) $1 - \alpha$. Tento parametr nazýváme \textit{pokrytí} intervalu spolehlivosti (CI). Interval spolehlivosti je náhodná veličina, i přestože $\theta$ je pevné deterministické. Pro vícerozměrné prostory uvažujeme kouli/elipsoid spolehlivosti (ale toto rozšíření je komplikovanější, protože na $\R^d, d>1$ neexistuje vhodné uspořádání).
+
+\hfill \textit{konec 15. přednášky (8.4.2025)}