{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 预估比例\n",
"\n",
"这章主要讲如何选择 prior distributions。"
]
},
{
"cell_type": "code",
"execution_count": 203,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np \n",
"from empiricaldist import Pmf \n",
"import matplotlib.pyplot as plt \n",
"import pandas as pd\n",
"from scipy.stats import binom "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"一枚硬币，随机抛掷 250 次，观测的结果是 140 次正面朝上，110 次背面朝上。将这枚硬币随机抛掷一次，是正面朝上的概率最有可能是多少？\n",
"\n",
"\n",
"\n",
"**上图来自[搜狐](https://www.sohu.com/a/225881799_100071627)**\n",
"\n",
"我们可以用「分布」那部分的例子来思考。我们可以把这个问题转化为：\n",
"\n",
"我们目前有 $n$ 枚硬币，其各有如下特征：\n",
"\n",
"- 随机抛硬币 0，正面朝上的概率是 0/n\n",
"- 随机抛硬币 1，正面朝上的概率是 1/n\n",
"- 随机抛硬币 2，正面朝上的概率是 2/n\\\n",
"...\n",
"- 随机抛硬币 n，正面朝上的概率是 n/n\n",
"\n",
"我们从这些硬币中随机挑了一枚，随机抛掷 250 次，观测的结果是 140 次正面朝上，110 次背面朝上。这枚硬币最有可能是哪枚硬币？\n",
"\n",
"我们先用 uniform prior，也就是说，每一枚硬币是该硬币的概率相同。"
]
},
{
"cell_type": "code",
"execution_count": 204,
"metadata": {},
"outputs": [],
"source": [
"# 稍微修改一下之前的 update_bowls_pmf\n",
"def update_coins_pmf(n, h, t):\n",
" \"\"\"\n",
" n: 总共几枚硬币\n",
" h: 正面朝上\n",
" t: 背面朝上\n",
" \"\"\"\n",
" hypos = np.linspace(0, 1, n+1)\n",
" prior = Pmf(1, hypos)\n",
" likelihood_head = [i/(n-1) for i in range(n+1)]\n",
" likelihood_tail = [1- i for i in likelihood_head]\n",
" likelihood = {\n",
" \"head\": likelihood_head,\n",
" \"tail\": likelihood_tail\n",
" }\n",
" dataset = [\"head\"]*h + [\"tail\"]*t\n",
" posterior = prior.copy()\n",
" for data in dataset:\n",
" posterior *= likelihood[data]\n",
" posterior.normalize()\n",
" return posterior"
]
},
{
"cell_type": "code",
"execution_count": 205,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([0. , 0.11111111, 0.22222222, 0.33333333, 0.44444444,\n",
" 0.55555556, 0.66666667, 0.77777778, 0.88888889, 1. ])"
]
},
"execution_count": 205,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"hypos = np.linspace(0, 1, 10)\n",
"hypos"
]
},
{
"cell_type": "code",
"execution_count": 206,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" probs | \n",
"
\n",
" \n",
" \n",
" \n",
" | 0.000000 | \n",
" 1 | \n",
"
\n",
" \n",
" | 0.111111 | \n",
" 1 | \n",
"
\n",
" \n",
" | 0.222222 | \n",
" 1 | \n",
"
\n",
" \n",
" | 0.333333 | \n",
" 1 | \n",
"
\n",
" \n",
" | 0.444444 | \n",
" 1 | \n",
"
\n",
" \n",
" | 0.555556 | \n",
" 1 | \n",
"
\n",
" \n",
" | 0.666667 | \n",
" 1 | \n",
"
\n",
" \n",
" | 0.777778 | \n",
" 1 | \n",
"
\n",
" \n",
" | 0.888889 | \n",
" 1 | \n",
"
\n",
" \n",
" | 1.000000 | \n",
" 1 | \n",
"
\n",
" \n",
"
\n",
"