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zeQd?be4cPzUBHMeTc&Hz=b9FjP%;5<-{U;#1__n;ssfUbbEY$;tszkpHU+!A=f#KB zMaD_C7jUR0yHvF_Et1abo-M&Ivvx5uTe=Z}CI7yGHzCOpIa-EAiQJEklSTc&P|N>SE* Q3HN^r`_^Ux2LNRM3*H|y#{d8T literal 0 HcmV?d00001 diff --git a/docs/Математика/baes.md b/docs/Математика/baes.md index a207e90..b3be0b4 100644 --- a/docs/Математика/baes.md +++ b/docs/Математика/baes.md @@ -17,13 +17,19 @@ - Из них $m$ — количество выпавших орлов. - Отношение $\frac{m}{n}$ будет оценкой $p$. -В Байесовской статистике подход иной: +В [Байесовской +статистике](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) +подход иной: -1. Обозначим монету как бернуллевскую случайную величину $\xi$ с параметром $\theta$, у - которой $1$ — это выпадение орла, $0$ — решки. -2. Предполагается априорное распределение $\pi(\theta)$ (т.е. распределение, которое мы - предполагаем, исходя из того, что нам известно о параметре $\theta$), как правило, - это равномерное распределение $U \left(0,1\right)$. +1. Обозначим монету как [бернуллевскую случайную + величину](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) + $\xi$ с параметром $\theta$, у которой $1$ — это выпадение орла, $0$ — + решки. +2. Предполагается априорное распределение $\pi(\theta)$ (т.е. распределение, + которое мы предполагаем, исходя из того, что нам известно о параметре + $\theta$), как правило, это [равномерное + распределение](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) + $U \left(0,1\right)$. 3. Монета подбрасывается. 4. Распределение $\theta$ уточняется по формуле: $$\Large @@ -37,6 +43,184 @@ 5. Повторить пункты 2-4, предполагая $\pi(\theta) = \pi(\theta|\xi)$ Тогда оценкой $\theta$ будет: -$$ +$$\Large \hat \theta = \arg \max_\theta \pi ( \theta | \xi ) $$ + +Иначе говоря, +[мода](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) +апостериорного распределения. + +Мы, в качестве априорного распределения, взяли [Бета-распределения](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) $Beta(2,2)$. В +таком случае, если на $i$-ом шаге мы имеем: +$$\Large +\theta \sim Beta \left( \alpha_i, \beta_i \right) +$$ + +то апостериорное распределение будет: + +$$\Large +Beta \left( \alpha_i, \beta_i \right) = +\begin{cases} +Beta \left( \alpha_i + 1, \beta_i \right), &\xi_i = 1 \\ +Beta \left( \alpha_i, \beta_i + 1 \right), &\xi_i = 0 \\ +\end{cases} +$$ + +Тогда не нужно интегрировать на каждом шаге, что существенно упрощает вычисления. + +## Решение + +Приведем решение на `python`. + +### Библиотеки + +Сначала нужно импортировать и настроить библиотеки: + +```python +import numpy as np +import pandas as pd +import matplotlib.pyplot as plt +import seaborn as sns + +from tqdm import tqdm + +plt.rcParams.update({'font.size': 14}) +``` + +### Константы и функции + +Обозначим константы: + +- `p` — истинная вероятность выпадения орла. +- `NMODEL` — количество бросков монеты. + +```python +p = 0.523 +NMODEL = 7501 +``` + +Введем функции: +- `beta` — объект для Бета-распределения +- `coin` — функция одиночного броска монеты + +```python +from scipy.stats import beta +coin = lambda: np.random.binomial(n=1, p=p) +``` + +### Моделирование + +В качестве априорного распределения тут используется Бета-распределение +$Beta(2,2)$. Это сделано потому, что расчет моды накладывает ограничение, что оба +параметра должны быть строго больше 1. + +Мода расчитывается по формуле: + +$$\Large +\text{Mode} = \frac{\alpha - 1}{\alpha + \beta - 2} +$$ + +```python +alphas = np.zeros(NMODEL) +betas = np.zeros(NMODEL) +alphas[0], betas[0] = 2,2 +for i in tqdm(range(1, NMODEL)): + A = coin() + if A == 0: + betas[i] = 1 + else: + alphas[i] = 1 + +betas = np.cumsum(betas) +alphas = np.cumsum(alphas) + +Es = (alphas - 1) / (alphas + betas - 2) +``` + +### График + +#### Мода + +Построим график зависимости моды от количества бросков. + +```python +fig, ax = plt.subplots(1, 1, figsize=(10,8), dpi=80) +sns.lineplot(x=np.arange(NMODEL), + y=Es, + ax=ax, + label=r"arg$\max_p \; f(p|\mathbb{X})$", ) +sns.lineplot(x=np.arange(NMODEL), + y=p, + ax=ax, + label=r"Истинное $p = {:.3f}$".format(p)) +ax.grid() +ax.set(ylabel=r"$E(p|\mathbb{X})$", xlabel=r"$i$"); +``` + +![mode.jpg](assets/baes/mode.jpg) + +Гифка + +Сделаем гифку на тему. Сначала нужно импортировать библиотеку `imageio` и зададим +шаг `STEP` (так как, потом нужно будет сохранять каждый график в опертивной +памяти, что проблематично при большом `NMODEL`): + +```python +import imageio +STEP = NMODEL//150 +``` + +После, создадим список функций плотности вероятности: + +```python +pi = [] +for i in tqdm(range(0, NMODEL)): + pi.append(beta(alphas[i], betas[i])) +``` + +И по ним сделаем gif-изображение: + +```python +def create_frame(i): + fig, ax = plt.subplots(1, 1, figsize=(10,8), dpi=80) + x = np.linspace(0,1,500) + y = pi[i].pdf(x) + i_max = np.argmax(y) + x_max = x[i_max] + y_max = y[i_max] + ax.grid() + ax.plot(x,y, label=r"$f(p|\mathbb{X})$") + ax.axvline(x = p, color = 'red', label = f"Истинное $p={p}$") + plt.scatter([x_max], + [y_max], + color="green", + marker="o", + label=r"Мода $f(p|\mathbb{X})$") + plt.xlim([0,1]) + plt.xlabel('x', fontsize = 14) + plt.ylim([0,100]) + plt.ylabel(r'$f(p|\mathbb{X})$', fontsize = 14) + plt.title(r'Распределение p, $i={}$'.format(i+1), + fontsize=14) + ax.legend() + plt.savefig(f'./img/img_{i:04d}.png', + transparent = False, + facecolor = 'white' + ) + plt.close() + +for i in tqdm(range(0, NMODEL, STEP)): + create_frame(i) +frames = [] +for i in tqdm(range(0, NMODEL, STEP)): + image = imageio.v2.imread(f'./img/img_{i:04d}.png') + frames.append(image) +imageio.mimsave('./distributions.gif', + frames, + fps = 30) +``` + +Результат: + +![gif](assets/baes/video.gif)