概率统计中篇.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 数理统计"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "数理统计基本知识\n",
    "\n",
    "* 概率论：随机变量，分布已知，如满足二项分布、高斯分布等\n",
    "* 数理统计：随机变量，分布未知，通过观察值，对分布推断 重复试验的样本数据，但是不知道其整体满足的分布情况 用观测数据取推断其分布\n",
    "* 一个总体对应于一个随机变量$X$，$X_1,\\dots,X_n$随机样本，与$X$独立同分布(分布函数$F$)，$x_1,\\dots,x_n$样本值(观察值)，注意通常情况$y_1,\\dots,y_n ≠ x_1,\\dots,x_n$\n",
    "* 理解：总体 $\\to$ 样本 $\\to$ 样本观察值 $\\to$ 理论分布\n",
    "* 统计量：例如样本平均值$\\bar{X} = \\frac{1}{n}\\sum_{i=1}^nX_i$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最大似然估计\n",
    "\n",
    "* 总体$X$的分布函数已知，但是一个或多个参数未知，我们借助样本来估计总体未知参数值\n",
    "\n",
    "* 最大似然主要思想：对于$P(A \\mid θ)$，在$θ$的可能取值范围内尽量选取使得$P(A \\mid θ)$最大的$\\widehat{θ}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：使得取得的这组观察值最大可能的θ值，也就是说在这个最大的θ下的这组观测值被取到了\n",
    "# 很多情况，我们只有数据，或者通过做试验来获取数据，我们要通过研究这些数据，看其分别满足什么样的分布，以此来求得分布的参数\n",
    "# 引入随机变量，我们就能够研究数据满足某些条件的函数关系，有了函数，再进一步预测其它数据的结果也就可行了\n",
    "# 因为观测数据是来自总体样本，通过观测数据获得的理论分布，同样适用于总体分布\n",
    "# 均值：用来衡量数据的集中程度，我们把它叫做偏差，有算数平均数、几何平均数、加权平均数等\n",
    "# 参考：https://baike.baidu.com/item/平均数?fromtitle=均值&fromid=5922988\n",
    "# 一般有极差：最大值与最小值之差 残差：真实值与估计值 方差：真实值与均值差的平方和的均值 标准差：方差的平方根\n",
    "# 使用何种方法去估计，感觉上有其合理性"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最大似然估计(离散)\n",
    "\n",
    "* 总体$X$，$X_1,\\dots,X_n$来自$X$样本，独立同分布，相应的观测值为$x_1,\\dots,x_N$，参数取值未知。利用已知观测值$x_1,\\dots,x_n$(常数)对$θ$进行点估计\n",
    "\n",
    "* 离散情况：实际中假定一$θ$，利用离散联合分布率定义，$X_1,\\dots,X_n$取到观察值为$x_1,\\dots,x_n$的概率为$L(θ)=\\prod_{i=1}^np(x_i;θ)$。这一概率随着$θ$变化，称为样本似然函数\n",
    "\n",
    "* 由于已知确认取到观测值，可认为取到这一样本的概率$L(θ)$比较大。显然，肯定不会去找让那些不能使样本$x_1,\\dots,x_N$出现的$θ$作为估计值，如果$\\Theta$里面有个能让$L(θ)$取到最大的$\\widehat{θ}$，自然认为$\\widehat{θ}$就是$θ$的估计值"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：\n",
    "# 似然函数是取得所有观测值的概率，自然是一个常数，取到每个样本点的概率积\n",
    "# 要求满足取到这组观测值的最大的θ\n",
    "# 似然函数在离散数据时是联合分布率，在连续时联合概率密度并不是概率密度"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 独立同分布的意思：因采集样本都是来自X，固然满足同分布，而独立，表示各个随机变量之间没有关联，互不影响\n",
    "# 再比如一个班级的男女身高满足正太分布，自然是某个身高下的有多个同学，随机采集一部分样本，构成这个观测值得样本值自然也同样满足正太分布\n",
    "# 通过这个观测值取求得整体的参数值，这就是似然估计的方法\n",
    "# 如果说可以取多组观测值求得相应的θ，自然存在一个最大的θ，也自然这个最大的θ才是最合适的"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最大似然估计(连续)\n",
    "\n",
    "* 连续情况下联合概率密度为$\\prod_{i=1}^nf(x_i;θ)$，由于连续变量在某一点概率为0，考虑随机变量$X$，$X_1,\\dots,X_N$落在点$(x_1,x_2,\\dots,x_n)$周围一个很小区域内(一维下就是求面积)的概率近似为$\\prod_{i=1}^nf(x_i;θ)dx_i$，因此类似我们选取$θ$让$\\prod_{i=1}^nf(x_i;θ)dx_i$($dx_i$是宽度)最大，但是由于$θ$和$dx_i$没关系，因此只需要$θ$让似然函数$\\prod_{i=1}^nf(x_i;θ) = L(θ)$最大即可\n",
    "\n",
    "* 注意：由于$(x_1,x_2,\\dots,x_n)$已知，$L(θ)$只和$θ$有关，是个标准的函数，既不是概率，也不是条件概率密度"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 连续数据的联合概率密度，其实就是离散数据的一种近似\n",
    "# 注意连续数据的似然函数不是概率，也不是条件概率密度，是一个标准的函数\n",
    "# ∏ni=1f(xi;θ)=L(θ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 连续的点更是一个个小的区间"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最大似然估计举例\n",
    "\n",
    "* 设$X \\sim \\mathcal{N}(μ,\\sigma^2)$，但$μ,\\sigma^2$未知。$x_1,\\dots,x_n$来自$X$的一个样本值，求$μ,\\sigma^2$最大似然估计\n",
    "\n",
    "* 已知$f(x;μ,\\sigma^2) = \\frac{1}{\\sqrt{2\\pi}\\sigma}exp\\left[-\\frac{1}{2\\sigma^2}(x-μ)^2\\right]$，似然函数\n",
    "\n",
    "$$L(μ,\\sigma^2) = \\prod_{i=1}^n\\frac{1}{\\sqrt{2\\pi}\\sigma}exp\\left[-\\frac{1}{2\\sigma^2}(x_i-μ)^2\\right]$$\n",
    "\n",
    "* 联立求解$\\frac{\\partial lnL}{\\partial μ}$ = 0，和$\\frac{\\partial lnL}{\\partial \\sigma^2}$ = 0，可得\n",
    "\n",
    "$$\\widehat{μ} = \\frac{1}{n}\\sum_{i=1}^nx_i = \\bar{x}$$\n",
    "\n",
    "$$\\widehat{\\sigma^2} = \\frac{1}{n}\\sum_{i=1}^n(x_i-\\bar{x})^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "推导：\n",
    "\n",
    "1、对似然函数两边取对数，得：\n",
    "\n",
    "$$\\begin{align*}\n",
    "\\ln L(μ,\\sigma^2) &= \\ln \\prod_{i=1}^n\\frac{1}{\\sqrt{2\\pi}\\sigma}exp\\left[-\\frac{1}{2\\sigma^2}(x_i-μ)^2\\right] \\\\\n",
    "&= \\sum_{i=1}^n\\ln \\frac{1}{\\sqrt{2\\pi}\\sigma}exp\\left[-\\frac{1}{2\\sigma^2}(x_i-μ)^2\\right] \\\\\n",
    "&= \\frac{1}{\\sqrt{2\\pi}}\\sum_{i=1}^n\\left[\\ln e^{-\\frac{1}{2\\sigma^2}(x_i-μ)^2} - \\ln \\sigma\\right] \\\\\n",
    "&= -\\frac{1}{\\sqrt{2\\pi}}\\left[\\sum_{i=1}^n\\frac{(x_i-μ)^2}{2\\sigma^2} + n\\ln \\sigma\\right] \\\\\n",
    "\\end{align*}$$\n",
    "\n",
    "2、对$μ$和$\\sigma^2$分别求偏导，有：\n",
    "\n",
    "$$\\begin{align*}\n",
    "\\frac{\\partial lnL}{\\partial μ} &= -\\frac{1}{\\sqrt{2\\pi}}\\sum_{i=1}^n\\frac{2(x_i-μ)(-1)}{2\\sigma^2} \\\\\n",
    "&= \\frac{1}{\\sqrt{2\\pi}}\\sum_{i=1}^n\\frac{x_i-μ}{\\sigma^2} \\\\\n",
    "&= 0\n",
    "\\end{align*}$$\n",
    "\n",
    "$$\\Downarrow$$\n",
    "\n",
    "$$\\sum_{i=1}^nx_i-μ = 0 \\Rightarrow \\widehat{μ} = \\frac{1}{n}\\sum_{i=1}^nx_i = \\bar{x}$$\n",
    "\n",
    "$$和$$\n",
    "\n",
    "$$\\begin{align*}\n",
    "\\frac{\\partial lnL}{\\partial \\sigma^2} &= -\\frac{1}{\\sqrt{2\\pi}}\\sum_{i=1}^n\\frac{(x_i-μ)^2}{-2(\\sigma^2)^2} - \\frac{n}{2\\sigma^2} \\\\\n",
    "&= 0\n",
    "\\end{align*}$$\n",
    "\n",
    "$$\\Downarrow$$\n",
    "\n",
    "$$\\frac{\\sum_{i=1}^n(x_i-μ)^2}{2(\\sigma^2)^2} = \\frac{n}{2\\sigma^2} \\Rightarrow \\widehat{\\sigma^2} = \\frac{1}{n}\\sum_{i=1}^n(x_i-\\bar{x})^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最大似然估计 \n",
    "\n",
    "* 多元高斯似然函数\n",
    "\n",
    "$$\\ln p(\\mathbf{X \\mid μ,Σ}) = -\\frac{ND}{2}\\ln(2\\pi) - \\frac{N}{2}\\ln|\\mathbf{Σ}| - \\frac{1}{2}\\sum_{n=1}^N(\\mathbf{(x_n-μ)^\\mathsf{T}\\mathbf{Σ}^{-1}(\\mathbf{x_n-μ}})$$\n",
    "\n",
    "* 仿照一元的例子，可得\n",
    "\n",
    "$$\\mathbf{μ}_{ML} = \\frac{1}{N}\\sum_{n=1}^N\\mathbf{x_n}$$\n",
    "$$\\mathbf{Σ}_{ML} = \\frac{1}{N}\\sum_{n=1}^N(\\mathbf{x_n-μ_{ML}})(\\mathbf{x_n-μ_{ML}})^\\mathsf{T}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 比例一个班级的男女生身高都服从各自的高斯分布，就是一个二元高斯分布"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "0-1分布\n",
    "\n",
    "离散\n",
    "\n",
    "* (0-1)分布\n",
    "    * $p(x=1 \\mid μ) = μ$\n",
    "    * $p(x=0 \\mid μ) = 1-μ$\n",
    "    * Bern$(x \\mid μ) = μ^x(1-μ)^{1-x}$\n",
    "    * $E(x) = μ$\n",
    "    * $var[x] = μ(1-μ)$\n",
    "    \n",
    "    推导：\n",
    "    \n",
    "    $$E(x) = \\sum_{k=0,1}x_kμ_k = 1 \\times μ + 0 \\times (1-μ) = μ$$\n",
    "    \n",
    "    $$D(x) = \\sum_{k=0,1}[x_k-E(x)]^2μ_k = (1-μ)^2 \\times μ + (0-μ)^2 \\times (1-μ) = μ(1-μ)$$\n",
    "\n",
    "\n",
    "* 观测数据集$\\mathcal{D} = \\{x_1,\\dots,x_N\\}$，每次独立从$p(x \\mid μ)$中抽取\n",
    "\n",
    "* 似然函数\n",
    "\n",
    "$$p(\\mathcal{D} \\mid μ) = \\prod_{n=1}^Np(x_n \\mid μ) = \\prod_{n=1}^Nμ^{x_n}(1-μ)^{1-x_n}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 显然，0-1分布就两类，一对数据样本中，要么属于0，要么属于1，属于离散型，多次重复0-1试验，也就是符合0-1分布，可用于二分类问题\n",
    "# 而正太分布是连续型的，观察数据是落在一区间内的样本，多元高斯分布适用于多分类问题"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最大似然\n",
    "\n",
    "* 对数似然函数\n",
    "\n",
    "$$\\ln p(\\mathcal{D} \\mid μ) = \\sum_{n=1}^N \\ln p(x_n \\mid μ) = \\sum_{n=1}^N\\bigr\\{x_n \\ln μ + (1-x_n) \\ln (1-μ)\\bigr\\}$$\n",
    "\n",
    "* 求导，可得\n",
    "\n",
    "$$μ_{ML} = \\frac{1}{N}\\sum_{n=1}^Nx_n = \\frac{m}{N}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 对于多重0-1分布，我们通过最大似然的方法，得到样本的概率，类似与频率的解释"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 线性回归与逻辑回归"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "再看线性回归\n",
    "\n",
    "* 使用$N$个数据点作为训练集，我们希望得到这个一个模型：给定一个新的输入$\\widehat{x}$，预测它对应的输出$\\widehat{y}$\n",
    "\n",
    "* 模型\n",
    "\n",
    "$$y^{(i)} = \\mathbf{θ^\\mathsf{T}}\\mathbf{x}^{(i)} + \\epsilon^{(i)}$$\n",
    "\n",
    "* 假设$\\epsilon^{(i)}$是i.i.d高斯，$\\epsilon^{(i)} \\sim \\mathcal{N}(0,\\sigma^2)$，则有$\\epsilon^{(i)} \\sim \\mathcal{N}(\\mathbf{θ^\\mathsf{T}}\\mathbf{x}^{(i)},\\sigma^2)$，即\n",
    "\n",
    "$$f(y^{(i)} \\mid \\mathbf{x}^{(i)},\\mathbf{θ}) = \\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\tfrac{(y^{(i)} - \\mathbf{θ^\\mathsf{T}}\\mathbf{x}^{(i)})^2}{2\\sigma^2}}$$\n",
    "\n",
    "* 似然函数\n",
    "\n",
    "$$L(θ) = \\prod_{i=1}^Nf(y^{(i)} \\mid \\mathbf{x}^{(i)},\\mathbf{θ}) = \\prod_{i=1}^N\\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\tfrac{(y^{(i)} - \\mathbf{θ^\\mathsf{T}}\\mathbf{x}^{(i)})^2}{2\\sigma^2}}$$\n",
    "\n",
    "* 对数似然函数\n",
    "\n",
    "$$\\begin{align*}\n",
    "l(θ) &= \\ln L(θ) = \\sum_{i=1}^N \\ln \\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\tfrac{(y^{(i)} - \\mathbf{θ^\\mathsf{T}}\\mathbf{x}^{(i)})^2}{2\\sigma^2}} \\\\\n",
    "&= N \\ln \\frac{1}{\\sigma\\sqrt{2\\pi}} - \\frac{1}{2\\sigma^2}\\sum_{i=1}^N\\left(y^{(i)} - \\mathbf{θ^\\mathsf{T}}\\mathbf{x}^{(i)}\\right)^2\n",
    "\\end{align*}$$\n",
    "\n",
    "* 与之前定义的损失函数一样，最小二乘等价于最大似然估计"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：\n",
    "# 误差项ϵ服从标准高斯，则真实数据也符合高斯分布\n",
    "# 最大化似然函数，相当于最小化损失函数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：最大化l(θ)，自然是最小化∑i=1N(y^(i)−θ^𝖳x^(i))\n",
    "# 等价于最小均方误差，是样本实际值与预测值之间的误差"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "逻辑回归\n",
    "\n",
    "分类问题\n",
    "\n",
    "* $y = \\mathbf{w}^T\\mathbf{x} + b$\n",
    "\n",
    "* 非线性映射1\n",
    "\n",
    "$$z = \n",
    "\\begin{cases}\n",
    "    0 & y < 0 \\\\\n",
    "    0.5 & y = 0 \\\\\n",
    "    1 & y>0\n",
    "\\end{cases}$$\n",
    "\n",
    "* 非线性映射2\n",
    "\n",
    "$$z = \\frac{1}{1+e^{-y}}$$\n",
    "\n",
    "* 假定$h(\\mathbf{x}) = g(\\mathbf{θ}^T\\mathbf{x}) = \\frac{1}{1+e^{-\\mathbf{θ}^T\\mathbf{x}}}$\n",
    "\n",
    "* 假定\n",
    "\n",
    "$$P(y=1 \\mid \\mathbf{x,θ}) = h(\\mathbf{x})$$\n",
    "$$P(y=0 \\mid \\mathbf{x,θ}) = 1 - h(\\mathbf{x})$$\n",
    "\n",
    "* 于是有：\n",
    "\n",
    "$$P(y \\mid \\mathbf{x,θ}) = (h(\\mathbf{x}))^y(1 - h(\\mathbf{x}))^{1-y}$$\n",
    "\n",
    "* 似然函数：\n",
    "\n",
    "$$L(θ) = \\prod_{i=1}^N\\bigr(h\\bigr(\\mathbf{x^{(i)}}\\bigr)\\bigr)^{y^{(i)}}\\bigr(1 - h\\bigr(\\mathbf{x^{(i)}}\\bigr)\\bigr)^{1-y^{(i)}}$$\n",
    "\n",
    "* 对数似然函数\n",
    "\n",
    "$$\\begin{align*}\n",
    "l(θ) &= \\log L(θ) \\\\\n",
    "&= \\sum_{i=1}^Ny^{(i)} \\log h\\bigr(\\mathbf{x^{(i)}}\\bigr) + \\bigr(1 - y^{(i)}\\bigr) \\log \\bigr(1 - h\\bigr(\\mathbf{x^{(i)}}\\bigr)\\bigr)\n",
    "\\end{align*}$$\n",
    "\n",
    "* 等价：凸函数和交叉熵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明最大化l(θ)，自然是最小化∑i=1N-y(i)logh(x(i))-(1−y(i))log((1−h(x(i))就是逻辑回归的损失函数\n",
    "# 存在全局最优解"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "证明：逻辑回归的损失函数是凸函数，思路证明其二阶导Hessian半正定\n",
    "\n",
    "再利用凸函数的性质，多个凸函数的累加和仍然是凸函数，所以只要证明$- \\log h(\\mathbf{x})$和$- \\log \\bigr(1 - h(\\mathbf{x})\\bigr)$是凸函数即可\n",
    "\n",
    "已知$h(\\mathbf{x}) = \\frac{1}{1+e^{-\\mathbf{θ^Tx}}}$\n",
    "\n",
    "已知$hθ(x)(1−hθ(x))$是一常数\n",
    "\n",
    "先证明$- \\log h(\\mathbf{x})$是凸函数，由对数性质，转换为 $\\log(1 + e^{-θ^Tx})$\n",
    "\n",
    "对$θ$求导，得：\n",
    "\n",
    "$$\\begin{align*}\n",
    "    \\nabla_θ\\left[\\log(1 + e^{-θ^Tx})\\right] &= \\bigr(\\frac{-e^{-θ^Tx}}{1 + e^{-θ^Tx}}\\bigr)x \\\\\n",
    "    &= \\bigr(\\frac{1}{1 + e^{-θ^Tx}} -1 \\bigr)x \\\\\n",
    "    &= \\bigr(h(x) -1\\bigr)x\n",
    "  \\end{align*}$$\n",
    "  \n",
    "再对$θ$求一次导，得\n",
    "\n",
    "$$\\nabla_θ^2 = h_θ(\\mathbf{x})[1-h_θ(\\mathbf{x})]\\mathbf{xx^T}$$\n",
    "\n",
    "利用Hessian矩阵的定义，在矩阵两边分别乘以$\\mathbf{z^T}$和$\\mathbf{z}$，得\n",
    "\n",
    "$$\\mathbf{z^T}\\left[h_θ(\\mathbf{x})(1-h_θ(\\mathbf{x}))\\mathbf{xx^T}\\right]\\mathbf{z} = h_θ(\\mathbf{x})(1-h_θ(\\mathbf{x}))(\\mathbf{x^Tz})^2 \\ge 0$$\n",
    "\n",
    "继而，得正$- \\log h(\\mathbf{x})$是凸函数，采用同样思路证明$- \\log \\bigr(1 - h(\\mathbf{x})\\bigr)$是凸函数\n",
    "\n",
    "最终证明逻辑回归的损失函数是凸函数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 贝叶斯的观点\n",
    "# 由于先验的信息，后验的分布也会发生变化，不再是原来那个分布了\n",
    "# 最大似然的点估计到后验分布"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "再看贝叶斯公式\n",
    "\n",
    "* 贝叶斯\n",
    "\n",
    "$$p(θ \\mid \\mathcal{D}) = \\frac{p(\\mathcal{D} \\mid θ)p(θ)}{p(\\mathcal{D})}$$\n",
    "\n",
    "* 上式中\n",
    "\n",
    "    * $p(\\mathcal{D} \\mid θ)$ 似然函数\n",
    "    * $p(θ)$ 先验分布\n",
    "    * $p(θ \\mid \\mathcal{D})$ 后验分布\n",
    "    * 后验$\\propto$似然$\\times$先验\n",
    "    \n",
    "* 共轭先验"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 先验概率在似然函数的作用下发生了变化，也就是我取到这组观测数据，存在这么一个θ使得后验概率最大"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：从直观上去理解，要求得后验概率，必然是在已知参数的情况下，也就是已知先验，并可知这个参数是使得取出样本最答可能的参数\n",
    "# 共轭：一对相匹配的量成对出现\n",
    "# 共轭分布：在贝叶斯统计中，如果先验分布和后验分布属于同类，则称先验分布和后验分布是共轭分布\n",
    "# 共轭先验：先验分布在似然函数的作用下，使得先验分布和后验分布独立同分布，我们说先验分布是似然函数是共轭先验\n",
    "# 一般有的共轭先验分布有：Bernoulli 分布的共轭分布是Beta，Multinomial 的共轭是 Dirichlet\n",
    "# 共轭先验的作用：能够简化模型计算量，同时能够观察出先验分布在似然函数的作用下得到后验分布的\n",
    "# 参考：https://zh.wikipedia.org/wiki/共轭先验"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "二项分布\n",
    "\n",
    "* 二项分布\n",
    "\n",
    "$$Bin(m\\mid N,μ) =\n",
    "\\begin{pmatrix}\n",
    "    N \\\\\n",
    "    m\n",
    "\\end{pmatrix}μ^m(1-μ)^{N-m}$$\n",
    "\n",
    "* 均值：$\\mathbb{E}[m] = Nμ$\n",
    "* 方差：$var[m] = Nμ(1-μ)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Beta分布\n",
    "\n",
    "* 先验分布选择为Beta分布，概率密度\n",
    "\n",
    "$$\\begin{align*}\n",
    "Beta(μ|α,β) &= \\frac{μ^{α-1}(1-μ)^{β-1}}{\\int_0^1μ^{α-1}(1-μ)^{β-1}du} = \\frac{\\Gamma{(α+β)}}{\\Gamma{(α)}\\Gamma{(β)}}μ^{α-1}(1-μ)^{β-1} \\\\\n",
    "&= \\frac{1}{B(α,β)}μ^{α-1}(1-μ)^{β-1}\n",
    "\\end{align*}$$\n",
    "\n",
    "* 均值：$\\mathbb{E}[μ] = \\frac{α}{α+β}$\n",
    "* 方差：$var[μ] = \\frac{αβ}{(α+β)^2(α+β+1)}$\n",
    "\n",
    "* 后验分布，$N$次试验，$m$次正面朝上，$l$次反面朝上，$N = m + l$\n",
    "\n",
    "$$p(μ|m,l,α,β) \\propto Likelihood \\times Beta(μ|α,β)$$\n",
    "\n",
    "$$p(μ|m,l,α,β) \\propto μ^{m + α -1}(1-μ)^{l + β -1}$$\n",
    "\n",
    "$$\\begin{align*}\n",
    "p(μ|m,l,α,β) &= \\frac{\\Gamma(m + α + l + β)}{\\Gamma(m + α)\\Gamma(l + β)}μ^{m + α -1}(1-μ)^{l + β -1} \\\\\n",
    "&= Beta(μ|m + α, l + β)\n",
    "\\end{align*}$$\n",
    "\n",
    "* 均值：$\\mathbb{E}[μ|m,l,α,β] = \\frac{m + α}{m + α + l + β}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：B(α,β)是beta函数\n",
    "# beta函数与gamma函数之间的关系：B(α,β)=Γ(α)Γ(β)/Γ(α+β)\n",
    "# Beta(μ|α,β)：beta分布\n",
    "# 似然函数是二项分布μm(1−μ)N−m\n",
    "# beta分布描述的概率的概率分布，能够反应非等可能事件的概率情况，一般生活中很多事件可能是非等可能的\n",
    "# beta分布中的α和β参数描述了这种等可能性，如果两值相等，值的大小反应了哪个发生的可能性更大\n",
    "# 描述的是前后两事件之间的关系"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：beta分布从某种程度上说明了某件事件发生的概率的描述，比如说我们抛硬币，\n",
    "# 如果说硬币是均匀的，也就是对应了等可能事件，概率是相同的，而大部分情况，每个\n",
    "# 事件发生的概率是不相同的，固然beta分布可以描述这种事件发生的可能性，也就是\n",
    "# 概率，而二项分布中，每个每次试验产生的样本点的概率就是用beta分布来表示，再对\n",
    "# 概率的所有情况进行一个积分，概率范围从[0,1]，也就是有了要求满足某种情况的概率，\n",
    "# 比如说我们要求x=1正面朝上的概率\n",
    "# 再比如，我们的硬币是均匀的，也就是α=β=1，就有了均值概率"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：beta分布可以看做是衡量概率的概率分布，也就是其可能性\n",
    "# beta分布和二项分布是共轭先验，其后验分布也服从beta分布\n",
    "# 在知道先验信息的情况，这边的先验信息是指beta分布，其后验分布也能够通过贝叶斯公式计算出\n",
    "# 为什么要使用这个beta分布？\n",
    "# 比如说生活中如果根据频率派的观点来计算一个人击球的击中率为击中次数/总次数，而有种情况是其\n",
    "# 只打了一次，并且击中，按照频率统计，击中率即为100%，显然这个不合理的，当然可以说多打几次，\n",
    "# 以频率观点统计其击中率也是可行的。然而根据贝叶斯观点，我们之前击球产生的信息，都能够对后\n",
    "# 面击中的概率评估提供先验信息，而这个beta分布就是假设是我们先验的信息，拿这边的击球率来说，\n",
    "# 一般根据往常经验，这个值应该在0.27~0.35之间，根据前一轮的情况，统计这一轮的击球率，显得\n",
    "# 更加合理，从击球的结果，也可以看出贝叶斯的计算出的击球率与频率计算出的击球率还是有些误差的\n",
    "# 参考：https://www.zhihu.com/question/30269898\n",
    "# https://en.wikipedia.org/wiki/Beta_distribution\n",
    "# 简述：知道前一轮的集中率，可以用来统计这一轮的击球率\n",
    "# beta分布的α和β参数是使得模型满足先验信息的，要知道概率的分布区间\n",
    "# 参数α和β也体现了一种均匀性，比如说硬币的均匀性，α=β=1表示硬币均匀的𝔼[μ]=1/2\n",
    "# 如果是α=100，β=200，𝔼[μ]=100/300=1/3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# gamma函数：可以看做阶乘在实数域上扩展，如2.5!\n",
    "# gamma函数的性质：\n",
    "# https://baike.baidu.com/item/伽玛函数/3540177?fr=aladdin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "贝叶斯估计举例\n",
    "\n",
    "* 抛100次硬币，都正面朝上，抛101次正面的概率？\n",
    "\n",
    "* 最大似然？\n",
    "\n",
    "* 引入先验分布，Beta分布\n",
    "\n",
    "* 预测\n",
    "\n",
    "$$\\begin{align*}\n",
    "p(x=1|\\mathcal{D}) &= \\int_0^1p(x=1|μ)p(μ|\\mathcal{D})du \\\\\n",
    "&= \\int_0^1μp(μ|\\mathcal{D})du = \\mathbb{E}[μ|\\mathcal{D}] \\\\\n",
    "&= \\frac{m + α}{m + α + l + β} = \\frac{100 + α}{100 + α + β}\n",
    "\\end{align*}$$\n",
    "\n",
    "* $α = β = 1$\n",
    "* $α = β = 100$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：\n",
    "# 用最大似然，求得100次正面的概率为μ=1，而似然公式μ^x(1-μ)^(1-x)，而101次为正面的概率为1^1(1-1)^(1-1)=1，这就存在问题了，对于这种情况我们使用\n",
    "# beta分布后验概率来计算其概率\n",
    "# 其解释为在知道了前100次都为正面的情况下，能够为101次计算正面概率提供信息\n",
    "# 当α=β=1时，我们可认为硬币是均匀的，则有概率100+1/100+1+1=101/102接近于1，而α=β=100时，100+100/100+200=2/3很明显概率小了很多\n",
    "# beta分布是一个概率的概率分布，自然是连续值，所以求期望使用积分求解"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 0-1分布中，使用最大似然求出的μ就是为1的概率，等于正面的概率数/总数，那抛100次正面朝上的概率为1\n",
    "# 而第101，若正面朝上其概率根据公式得出其概率还是为1，但其实101次的μ是已经发生了变化，因为知道其\n",
    "# 100次情况，自然101是不一样的 这种情况就是最大似然的过拟合\n",
    "# 上述公式说明：为什么要使用边缘概率密度，因需要统计落在[0,1]区间内的概率情况，根据边缘概率公式，\n",
    "# 涉及到联合概率，而联合概率可以根据条件概率密度公式求得，自然就有了上述公式推导\n",
    "# 解释：\n",
    "# p(x=1|D)：是条件概率密度，预测样本集中x=1的概率 是后验概率\n",
    "# p(x=1|μ)：x=1的概率为μ 是先验概率\n",
    "# p(μ|D)：是似然函数 通过前面已知的数据可以计算出模型参数\n",
    "# p(x=1|μ)p(μ|D)：联合概率密度，其实是前面已知数据的概率提供的信息\n",
    "# 连续值的期望是积分形式，而∫10μp(μ|D)du恰好就是给定D对μ求期望，而μ的概率为[0-1]所以对μ积分也是0,1\n",
    "# 机器学习中大部分情况还是使用的最大似然估计，当最大似然存在问题时，可以使用贝叶斯的最大后验估计MAP"
   ]
  },
  {
   "attachments": {
    "Beta%20distribution.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Beta%20distribution.png](attachment:Beta%20distribution.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 信息论概述"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "量化信息\n",
    "\n",
    "* 非常可能发生的事件信息量要比较少，并且极端情况下，确保能够发生的事件应该没有信息量\n",
    "\n",
    "* 较不可能发生的事件具有更高的信息量。\n",
    "\n",
    "* 独立事件应具有增量的信息。例如，投掷的硬币两次正面朝上传递的(有点类似先验$\\to$后验)\n",
    "\n",
    "* 信息量，应该是投掷一次硬币正面朝上的信息量的两倍。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "离散\n",
    "\n",
    "* 自信息\n",
    "\n",
    "$$h(x) = -logp(x)$$\n",
    "\n",
    "* 熵\n",
    "\n",
    "$$H(x) = -\\sum_{x}p(x)log_2p(x)$$\n",
    "\n",
    "* 0-1分布\n",
    "\n",
    "$$H(x) = -μlog_2μ - (1-μ)log_2(1-μ ) $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 解释：\n",
    "# 自信息：信息本体，衡量某事件发生时所包含的信息量的多寡\n",
    "# 熵：对信息的量化度量 决策树中用到信息增益\n",
    "# 0-1分布：交叉熵"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "连续\n",
    "\n",
    "* 微分熵(一元)\n",
    "\n",
    "$$-\\lim_{\\Delta \\to 0}\\left\\{\\sum_{i}p(x_i)\\Delta \\ln(p(x_i))\\right\\} = - \\int p(x)\\ln p(x)dx$$\n",
    "\n",
    "* 微分熵(多元)\n",
    "\n",
    "$$H[\\mathbf{x}] = -\\int p(\\mathbf{x}) \\ln p(\\mathbf{x})d\\mathbf{x}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "KL散度(相对熵)\n",
    "\n",
    "* 衡量两个分布的差异\n",
    "\n",
    "$$\\begin{align*}\n",
    "KL(p\\|q) &=  -\\int p(\\mathbf{x}) \\ln q(\\mathbf{x})d\\mathbf{x} - \\left(-\\int p(\\mathbf{x}) \\ln p(\\mathbf{x})d\\mathbf{x}\\right) \\\\\n",
    "&= -\\int p(\\mathbf{x}) \\ln \\left(\\frac{q(\\mathbf{x})}{p(\\mathbf{x})}\\right)d\\mathbf{x}\n",
    "\\end{align*}$$\n",
    "\n",
    "* 非对称性\n",
    "\n",
    "$$KL(p\\|q) ≠ KL(q\\|p)$$\n",
    "\n",
    "* 非负性\n",
    "\n",
    "$$KL(p\\|q) \\ge 0$$\n",
    "\n",
    "* EM算法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 说明：当p=q时，KL(p∥q)=0"
   ]
  }
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