{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "前两讲分别抛了硬币和骰子，这一节抛啥？还是抛硬币，而且要可劲儿的抛，那抛完硬币看啥？看硬币正面出现的概率，即$\\frac{m}{N}$，这里$N$表示抛硬币的次数，$m=X_1+X_2+\\cdots+X_N$，表示正面出现的次数，$X_i=1$表示正面，$X_i=0$表示反面，下面演示一下" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "#首先定义抛一次硬币的过程\n", "def toss_a_coin():\n", " return np.random.rand()<0.5 #假设正面的概率为0.5" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "#定义次数\n", "N=[10,500,1000,5000]\n", "plt.figure(figsize = (18,4))\n", "for index,num in enumerate(N):\n", " p=[]\n", " for _ in range(0,200):\n", " c=0\n", " for _ in range(0,num):\n", " c+=toss_a_coin() \n", " p.append(c/num)\n", " plt.subplot(1,4,index+1)\n", " plt.hist(p,normed=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "可以发现统计量$\\frac{m}{N}$有服从正态分布的趋势，均值稳定在0.5，而方差越来越小（峰越来越尖）" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 一.正态分布的定义\n", "对于一元变量的情况，正态分布可以写作： \n", "\n", "$$\n", "N(x\\mid\\mu,\\sigma^2)=\\frac{1}{(2\\pi\\sigma^2)^{\\frac{1}{2}}}exp\\{-\\frac{1}{2\\sigma^2}(x-\\mu)^2\\}\n", "$$ \n", "\n", "其中，$\\mu$是均值，$\\sigma^2$是方差，对于$D$维向量$x$，它的高斯分布写作： \n", "\n", "$$\n", "N(x\\mid\\mu,\\Sigma)=\\frac{1}{(2\\pi)^{\\frac{D}{2}}}\\frac{1}{|\\Sigma|^{\\frac{1}{2}}}exp\\{-\\frac{1}{2}(x-\\mu)^T\\Sigma^{-1}(x-\\mu)\\}\n", "$$ \n", "\n", "其中，$\\mu$是一个$D$维的均值向量，$\\Sigma$是一个$D\\times D$的协方差矩阵，$|\\Sigma|$是$\\Sigma$的行列式\n", "\n", "#### 均值、协方差\n", "下面直接写一下均值和协方差...\n", "$$\n", "E[x]=\\mu\\\\\n", "var[x]=E[(x-E[x])(x-E[x])^T]=\\Sigma\n", "$$ \n", "\n", "#### 极大似然估计\n", "$$\n", "\\mu_{ML}=\\frac{1}{N}\\sum_{i=1}^Nx_i\\\\\n", "\\Sigma_{ML}=\\frac{1}{N}\\sum_{i=1}^N(x_i-\\mu_{ML})(x_i-\\mu_{ML})^T\n", "$$ \n", "\n", "这里需要注意下协方差的极大似然估计是有偏的，即 \n", "\n", "$$\n", "E[\\Sigma_{ML}]=\\frac{N-1}{N}\\Sigma\n", "$$ \n", "\n", "#### 条件概率分布以及边缘概率分布\n", "\n", "对于多元高斯分布，它的条件概率分布以及边缘概率分布也是一个高斯分布，我们不妨将随机变量拆为两部分： \n", "\n", "$$\n", "x=\\binom{x_a}{x_b}\n", "$$ \n", "\n", "那么，对应的均值和协方差可以写作： \n", "\n", "$$\n", "\\mu=\\binom{\\mu_a}{\\mu_b}\\\\\n", "\\Sigma=\\begin{pmatrix}\n", "\\Sigma_{aa} & \\Sigma_{ab}\\\\ \n", "\\Sigma_{ba} & \\Sigma_{bb}\n", "\\end{pmatrix}\n", "$$ \n", "\n", "有时为了方便表示，我们会直接用到协方差矩阵的逆，称为精度，可以写作： \n", "\n", "$$\n", "\\Lambda = \\Sigma^{-1}=\\begin{pmatrix}\n", "\\Lambda_{aa} & \\Lambda_{ab}\\\\ \n", "\\Lambda_{ba} & \\Lambda_{bb}\n", "\\end{pmatrix}\n", "$$ \n", "\n", "那么**条件概率分布**公式： \n", "\n", "$$\n", "p(x_a\\mid x_b)=N(x_a\\mid \\mu_{a\\mid b},\\Lambda_{aa}^{-1})\n", "$$ \n", "\n", "这里，$\\mu_{a\\mid b}=\\mu_a-\\Lambda_{aa}^{-1}\\Lambda_{ab}(x_b-\\mu_b)$，接下来，**边缘概率分布**的公式： \n", "\n", "$$\n", "p(x_a)=N(x_a\\mid\\mu_a,\\Sigma_{aa})\n", "$$ \n", "\n", "如下图，左侧是两个变量上的高斯联合概率分布$p(x_a,x_b)$的轮廓线（绿色），右侧是边缘概率分布$p(x_a)$（蓝色）和$x_b=0.7$的条件概率分布$p(x_a\\mid x_b)$（红色曲线）\n", "![avatar](./source/12_高斯条件概率与边缘概率.png)\n", "\n", "接下来，让我们继续看看正态分布的共轭先验..." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 二.共轭先验\n", "\n", "这一节就只推导一维高斯分布的共轭先验，我们首先假设$\\sigma^2$是已知的情况下，推导$\\mu$的共轭先验，然后再假设$\\mu$已知的情况下，推导$\\sigma^2$的共轭先验，然后再假设$\\mu,\\sigma^2$均未知的情况下，推导其共轭先验\n", "\n", "#### $\\sigma^2$已知的情况\n", "假设$\\sigma^2$已知，我们有一组观测$X=\\{x_1,x_2,...,x_N\\}$，假设均值为$\\mu$，那么此时的似然函数可以看做是关于$\\mu$的函数： \n", "\n", "$$\n", "p(X\\mid\\mu)=\\prod_{i=1}^Np(x_n\\mid\\mu)=\\frac{1}{(2\\pi\\sigma^2)^{\\frac{N}{2}}}exp\\{-\\frac{1}{2\\sigma^2}\\sum_{n=1}^N(x_n-\\mu)^2\\}\n", "$$ \n", "\n", "接下来需要找到一个$p(u)$，让它与$p(X\\mid\\mu)$相乘后具有与$p(u)$相同的形式，显然$p(\\mu)$同样选择一个高斯分布就可以满足，因为都只有指数部分含有$\\mu$，而且是关于$\\mu$的二次函数，这样就可以将他们整合在一起了，我们不妨假设先验概率分布如下： \n", "\n", "$$\n", "p(\\mu)=N(\\mu\\mid\\mu_0,\\sigma_0^2)\n", "$$ \n", "\n", "从而后验概率： \n", "\n", "$$\n", "p(\\mu\\mid X)\\propto p(X\\mid\\mu)p(\\mu)\n", "$$ \n", "\n", "对指数部分进行配方整理后，可以得到后验概率分布的形式为： \n", "\n", "$$\n", "p(\\mu\\mid X)=N(\\mu\\mid\\mu_N,\\sigma_N^2)\n", "$$ \n", "\n", "其中： \n", "\n", "$$\n", "\\mu_N=\\frac{\\sigma^2}{N\\sigma_0^2+\\sigma^2}\\mu_0+\\frac{N\\sigma_0^2}{N\\sigma_0^2+\\sigma^2}\\mu_{ML}\\\\\n", "\\frac{1}{\\sigma_N^2}=\\frac{1}{\\sigma_0^2}+\\frac{N}{\\sigma^2}\n", "$$ \n", "\n", "这里$\\mu_{ML}$是$\\mu$的最大似然解，即： \n", "\n", "$$\n", "\\mu_{ML}=\\frac{1}{N}\\sum_{n=1}^Nx_n\n", "$$ \n", "\n", "通过上面的公式，我们可以得到一些有意思的结论： \n", "\n", "（1）当$N=0$时，后概率分布等于先验概率分布，这在我们意料之中； \n", "（2）当$N\\rightarrow\\infty$时，后验均值等于最大似然的均值，后验方差趋近于0，说明后验概率分布会在$\\mu_{ML}$处形成一个尖峰； \n", "（3）当$N$固定，若$\\sigma_0^2\\rightarrow\\infty$时，后验均值就变成了$\\mu_{ML}$，这个容易理解，$\\sigma_0^2\\rightarrow\\infty$时，先验分布很平，几乎不能提供有用的先验信息； \n", "\n", "下图演示了后验概率分布随着样本量$N$增加时的变化，其中$N=0$表示先验概率分布： \n", "![avatar](./source/12_高斯分布_均值的贝叶斯推断.png)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### $\\mu$已知的情况\n", "当$\\mu$已知时，我们的似然函数可以写作： \n", "\n", "$$\n", "p(X\\mid\\lambda)=\\prod_{n=1}^NN(x_n\\mid\\mu,\\lambda^{-1})\\propto\\lambda^{\\frac{N}{2}}exp[-\\frac{\\lambda}{2}\\sum_{n=1}^N(x_n-\\mu)^2]\n", "$$ \n", "\n", "其中，$\\lambda=\\frac{1}{\\sigma^2}$，这里用精度表示，后续的推导会更方便，从上面的形式可以看出，对应的共轭先验分布应该满足：（1）正比于$\\lambda$的幂指数；（2）同时正比于$\\lambda$的线性函数的指数；这样的分布有的，那就是Gamma分布： \n", "\n", "$$\n", "Gam(\\lambda\\mid a,b)=\\frac{1}{\\Gamma(a)}b^a\\lambda^{a-1}exp(-b\\lambda)\n", "$$\n", "\n", "不同$a,b$取值下的Gamma分布如下图： \n", "![avatar](./source/12_gamma分布.png)\n", "直接说一下Gamma分布的均值为$E[\\lambda]=\\frac{a}{b}$，方差为$var[\\lambda]=\\frac{a}{b^2}$，接下来考虑一个后验分布的形式，假设我们已经定义了一个先验分布$Gam(\\lambda\\mid a_0,b_0)$，然后乘以上面的似然函数： \n", "\n", "$$\n", "p(\\lambda\\mid X)\\propto \\lambda^{a_0-1}\\lambda^{\\frac{N}{2}}exp[-b_0\\lambda-\\frac{\\lambda}{2}\\sum_{n=1}^N(x_n-\\mu)^2]\n", "$$ \n", "\n", "这显然也是一个Gamma分布的形式，不妨记为$Gam(\\lambda\\mid a_N,b_N)$，其中： \n", "$$\n", "a_N=a_0+\\frac{N}{2}\\\\\n", "b_N=b_0+\\frac{1}{2}\\sum_{n=1}^N(x_n-\\mu)^2=b_0+\\frac{N}{2}\\sigma_{ML}^2\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### $\\mu$和$\\sigma^2$均未知的情况\n", "\n", "大家可能已经能猜想到，这种情况下的共轭先验应该既与高斯分布相关，又与Gamma分布有关，下面推导一下： \n", "\n", "$$\n", "p(X\\mid\\mu,\\lambda)=\\prod_{n=1}^N(\\frac{\\lambda}{2\\pi})^{\\frac{1}{2}}exp[-\\frac{\\lambda}{2}(x_n-\\mu)^2]\\\\\n", "\\propto exp(-\\frac{N\\lambda\\mu^2}{2}+\\lambda\\mu\\sum_{n=1}^Nx_n)[\\lambda^{\\frac{N}{2}}exp(-\\frac{\\lambda}{2}\\sum_{n=1}^Nx_n^2)]\n", "$$ \n", "\n", "可以发现，右侧$\\lambda^{\\frac{N}{2}}exp(-\\frac{\\lambda}{2}\\sum_{n=1}^Nx_n^2)$是关于$\\lambda$的Gamma分布的形式，左侧$exp(-\\frac{N\\lambda\\mu^2}{2}+\\lambda\\mu\\sum_{n=1}^Nx_n)$可以看做$\\lambda$已知的关于$\\mu$的高斯分布的形式，所以，我们的先验分布可以写作如下形式： \n", "\n", "$$\n", "p(\\mu,\\lambda)=p(\\mu\\mid\\lambda)p(\\lambda)\n", "$$ \n", "\n", "其中$p(\\mu\\mid\\lambda)$是一个高斯分布，$p(\\lambda)$是一个Gamma分布，经过归一化整理，我们的先验分布可以写作如下形式： \n", "\n", "$$\n", "p(\\mu,\\lambda)=N(\\mu\\mid\\mu_0,(\\beta\\lambda)^{-1})Gam(\\lambda\\mid a,b)\n", "$$ \n", "\n", "这里，$\\beta$是一个超参数，类比上面的样本量$N$，该分布也被称为**高斯-Gamma分布**" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.5" } }, 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