PRML/Chap2-Probability-Distributions/2.1-binary-variables.ipynb

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    "# 2.1 二元变量\n",
    "本书的第2章主要介绍了常见的概率分布以及它们的性质，这些分布之所以重要是因为它们是复杂模型的主要构件，同时也为后续学习统计学中的一些关键概念（如贝叶斯推断、变分推断、概率图模型等）作铺垫。我们研究概率分布，其中有一个重要的应用就是根据一组数据$\\mathcal{D}=\\{\\mathsf{x}_1,\\mathsf{x}_2,...,\\mathsf{x}_N\\}$来推测随机变量$\\mathsf{x}$的真实分布，这个过程称为密度估计（density estimation）。密度估计是一个ill-posed的问题，因为任何满足在$\\{\\mathsf{x}_1,\\mathsf{x}_2,...,\\mathsf{x}_N\\}$上概率大于0的分布都是一个可能的候选，而从中选择一个合适的分布有助于我们进行模型选择。\n",
    "\n",
    "密度估计方法一般可以分为两种：  \n",
    "1.参数化方法  \n",
    "这类方法的特点是分布的形式由一个或多个参数决定，比如高斯分布的形状取决于均值和标准差。参数化方法分为两大流派：\n",
    "* 频率学派：  \n",
    "频率学派认为参数是已知的定值，通过求解一个目标函数（如最大似然）来估计参数值；  \n",
    "* 贝叶斯学派：  \n",
    "贝叶斯学派认为参数是随机变量，首先假设参数服从一个先验分布，接着通过收集到的数据来修正先验分布，得到后验分布  \n",
    "\n",
    "2.非参数化方法  \n",
    "参数化方法的缺点是当假设的分布形式不符合实际数据的分布时，效果会很差。因此有学者提出了非参数方法，这类方法的特点是分布的形式不依赖于特定的参数，而是依赖于数据集的大小。但需要注意的是，“非参”并不意味着这类模型没有参数，它与参数化方法的区别在于，它的参数控制的不是分布的形式，而是模型的复杂度。  \n",
    "非参数方法主要代表有k近邻、非参数贝叶斯等。\n",
    "\n",
    "## 伯努利分布\n",
    "我们首先考虑一个二元随机变量$x\\in\\{0,1\\}$，可以将其理解为投掷一次硬币的结果，如果出现正面则$x=1$，出现反面则$x=0$。我们进一步假设这枚硬币是不均匀的，那么$x=1$的概率可以由参数$\\mu$来表示\n",
    "$$p(x=1|\\mu)=\\mu$$\n",
    "其中$0\\leq \\mu \\leq 1$，从中可以得出$p(x=0|\\mu)=1-\\mu$。那么关于$x$的分布可以写为如下形式：\n",
    "$$ p(x|\\mu)=\\mu^x(1-\\mu)^{1-x}$$\n",
    "这个分布称为伯努利分布。可以验证其均值和方差分别为\n",
    "$$\\mathbb{E}[x]=\\mu*1+(1-\\mu)*0=\\mu\\\\ var[x]=\\mathbb{E}[x^2]-\\mathbb{E}^2[x]=\\mu(1-\\mu)$$\n"
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    "## 二项式分布\n",
    "\n",
    "伯努利分布只考虑了一个二元变量的情况（即$N$只能等于1），然而只进行一次实验通常是无法准确估计参数的。如果我们放宽对于$N$的限制，进行$N$次独立重复实验，通过这$N$次实验的结果来更精确地估计$\\mu$的值。记$\\{x_1,x_2,...,x_N\\}$为$N$次实验的结果对应的随机变量集合，$\\mathcal{D}=\\{\\mathsf{x}_1,\\mathsf{x}_2,...,\\mathsf{x}_N\\}$为实验获得的观测值，我们希望从中估计出参数$\\mu$的值从而确定$x$的分布。在此之前，我们应假定这组样本是独立同分布产生，在此假设下$x=1$的总次数$m=\\sum_{i=1}^N x_i$服从二项式分布（binomial distribution）：\n",
    "$$ p(m|N,\\mu)=\\binom N m \\mu^m(1-\\mu)^{N-m}$$\n",
    "其中$\\binom N m=\\frac{N!}{(N-m)!m!}$  \n",
    "其均值，方差分别为\n",
    "$$ \\mathbb{E}[m]=\\sum_{m=0}^N m p(m|N,\\mu) = N\\mu\\\\ var[m]=\\sum_{m=0}^N (m-\\mathbb{E}[m])^2 p(m|N,\\mu)=N\\mu(1-\\mu)$$\n",
    "简单证明一下这两个结论：  \n",
    "1）$\\mathbb{E}[m]= N\\mu$  \n",
    "证：  \n",
    "$$\\begin{aligned}\\mathbb{E}[m]&=\\sum_{m=0}^N m p(m|N,\\mu) \\\\&=\\sum_{m=0}^N m \\frac{N!}{(N-m)!m!}\\mu^m(1-\\mu)^{N-m}\\\\&=0+\\sum_{m=1}^N m \\frac{N!}{(N-m)!m!}\\mu^m(1-\\mu)^{N-m}\\\\&=N\\mu\\sum_{m=1}^N\\frac{(N-1)!}{(N-m)!(m-1)!}\\mu^{m-1}(1-\\mu)^{N-m}\\\\&=N\\mu\\sum_{k=0}^{N-1}\\frac{(N-1)!}{(N-k-1)!k!}\\mu^k(1-\\mu)^{N-k-1}\\\\&=N\\mu(\\mu+1-\\mu)^{N-1}=N\\mu\\end{aligned}$$\n",
    "2）$var[m]=N\\mu(1-\\mu)$  \n",
    "证：  \n",
    "$$\\begin{aligned}var[m]&=\\mathbb{E}[m^2]-\\mathbb{E}^2[m]\\\\&=\\sum_{m=0}^N m^2 \\frac{N!}{(N-m)!m!}\\mu^m(1-\\mu)^{N-m}-N^2\\mu^2\\\\&=\\sum_{m=1}^N m \\frac{N!}{(N-m)!(m-1)!}\\mu^m(1-\\mu)^{N-m}-N^2\\mu^2\\\\&=\\sum_{m=1}^N [(m-1)+1] \\frac{N!}{(N-m)!(m-1)!}\\mu^m(1-\\mu)^{N-m}-N^2\\mu^2\\\\&=N(N-1)\\mu^2\\sum_{m=2}^N\\frac{(N-2)!}{(N-m)!(m-2)!}\\mu^{m-2}(1-\\mu)^{N-m}+N\\mu\\sum_{m=1}^N \\frac{(N-1)!}{(N-m)!(m-1)!}\\mu^{m-1}(1-\\mu)^{N-m}-N^2\\mu^2\\\\&=N(N-1)\\mu^2[\\mu+(1-\\mu)]^{N-2}+N\\mu[\\mu+(1-\\mu)]^{N-1}-N^2\\mu^2\\\\&=N^2\\mu^2-N\\mu^2+N\\mu-N^2\\mu^2=N\\mu-N\\mu^2=N\\mu(1-\\mu)\\end{aligned}$$\n",
    "\n",
    "实际上，练习1.10的结论告诉我们：对于一组独立随机变量之和$z=x_1+x_2+...+x_N$，$z$的均值等于这组随机变量的均值之和$\\sum_{i=1}^N mean(x_i)$，$z$的方差等于$\\sum_{i=1}^N var(x_i)$。利用这个结论，我们可以很轻易地得出上面的结论。  \n",
    "\n",
    "按照频率学派的观点，我们可以通过最大似然法来求解$\\mu$的值。由独立同分布假设，我们可以将这组数据关于$\\mu$的条件概率（也称为关于$\\mu$的似然函数）写为\n",
    "$$L(\\mu)=p(\\mathcal{D}|\\mu)=\\prod_{i=1}^N p(x_i|\\mu)$$\n",
    "为了方便后续的求导，我们对似然函数取对数，得到对数似然函数：\n",
    "$$\\mu_{ML}=\\arg\\max_{\\mu}=\\mathcal{L}(\\mu)=\\log L(\\mu)=\\sum_{i=1}^N \\log p(x_i|\\mu)=\\sum_{i=1}^N [x_i\\log (\\mu)+(1-x_i)\\log(1-\\mu)]$$\n",
    "最大化对数似然等价于最小化一个负对数似然损失函数（negative log-likelihood，nll）\n",
    "$$\\mu_{ML}=\\arg\\min_{\\mu}=-\\sum_{i=1}^N \\log p(x_i|\\mu)=-\\sum_{i=1}^N [x_i\\log (\\mu)+(1-x_i)\\log(1-\\mu)]$$\n",
    "\n",
    "注意到方括号里的项有些类似二元交叉熵（实际上交叉熵就是从最大似然目标函数推出来的），我们可以将其理解为$\\mu$在每个点$x_i$上的损失，如果$x_i=1$，则损失取决于前一项，当$\\mu$越接近于1，这种损失就越小，而当$\\mu$越接近与0，损失就越大；而如果$x_i=0$，情况则相反。负对数似然损失函数反映了$\\mu$的某一个特定取值在样本集上的平均损失，最小化nll损失函数等价于寻找使得整体损失最小的参数值。  \n",
    "我们将nll函数的导数设为0，可以得到参数的极大似然估计：\n",
    "$$\\mu_{ML}=\\frac{1}{N}\\sum_{i=1}^N x_i$$\n",
    "也就是样本均值（sample mean）。如果我们把$x_i=1$的次数记为$m$，那么$\\mu_{ML}=\\frac{m}{N}$  \n"
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    "## beta分布\n",
    "最大似然方法在统计推断中应用广泛，然而它也存在一个致命的缺陷。举个例子来说，假设我们连续地投掷一枚硬币3次，结果都是正面朝上，那么根据MLE，这枚硬币出现正面的概率就一定是$\\mu=1$，这显然是违背直觉的，出现这种情况的原因在于MLE在小数据集上容易发生过拟合。  \n",
    "为了解决MLE容易过拟合的问题，有统计学家提出了贝叶斯方法。贝叶斯学派认为参数不是定值，而是具有不确定性的随机数，比如投掷10次硬币，其中正面出现了6次，那么贝叶斯派认为$\\mu$的值不一定是0.6，而是有可能大于0.6，也可能小于0.6，只不过概率稍低一些。为此他们还引入了先验分布来刻画参数的不确定性，先验在一定程度上反映了观测者对于参数的最初认知，如何选取合适的先验分布是一门学问，通常的做法是结合领域知识来进行初始化；在先验分布的基础上，通过不断收集数据，参数的不确定性发生了改变，在此过程中我们更正对于参数的置信度，由此得到了后验分布。这个过程可以用贝叶斯公式来表示：  \n",
    "$$p(\\mathbf{w}|\\mathcal{D})=\\frac{p(\\mathcal{D}|\\mathbf{w})p(\\mathbf{w})}{p(\\mathcal{D})}$$\n",
    "其中$\\mathbf{w}$表示分布的参数，$\\mathcal{D}$表示我们收集到的数据。    \n",
    "回到文章主题，在伯努利分布和二项式分布中，似然函数表现为$\\mu^x(1-\\mu)^{1-x}$乘以某个常数的形式，这启发我们，如果我们在为$\\mu$选择先验时，使其与$\\mu$的幂以及$(1-\\mu)$的幂成正比，那么后验分布将具有与先验分布相同的函数形式。这个性质称为共轭（conjugacy），而满足这个性质的先验被称为共轭先验（conjugate prior）。特别地，我们选择beta分布作为二项式分布的共轭先验，其表达式如下：\n",
    "$$ Beta(\\mu|a,b)=\\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)}\\mu^{a-1}(1-\\mu)^{b-1}=\\frac{1}{Beta(a,b)}\\mu^{a-1}(1-\\mu)^{b-1}$$\n",
    "其中$\\Gamma(x)$是伽马函数，$Beta(a,b)=\\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)}$，上式的系数保证了beta分布是归一化的，即\n",
    "$$\\int_0^1 Beta(\\mu|a,b)d\\mu=1$$\n",
    "注意到beta分布是关于二项分布参数$p(x_i=1|\\mu)=\\mu$的参数，因此定义域为$[0,1]$。  \n",
    "Beta分布的均值、方差分别为：\n",
    "$$\\mathbb{E}[\\mu]=\\frac{a}{a+b}\\\\ var[\\mu]=\\frac{ab}{(a+b)^2(a+b+1)}$$\n",
    "简单证明一下。  \n",
    "1）$\\mathbb{E}[\\mu]=\\frac{a}{a+b}$  \n",
    "证：  \n",
    "$$\\begin{aligned}\\mathbb{E}[\\mu]&=\\int_0^1 \\mu \\frac{1}{Beta(a,b)}\\mu^{a-1}(1-\\mu)^{b-1}\\\\&=\\frac{1}{Beta(a,b)}\\int_0^1 \\mu^{a}(1-\\mu)^{b-1}\\\\&=\\frac{Beta(a+1,b)}{Beta(a,b)}\\\\&=\\frac{\\frac{\\Gamma(a+1)\\Gamma(b)}{\\Gamma(a+b+1)}}{\\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)}}=\\frac{\\Gamma(a+1)\\Gamma(a+b)}{\\Gamma(a)\\Gamma(a+b+1)}\\\\&=\\frac{a\\Gamma(a)\\Gamma(a+b)}{(a+b)\\Gamma(a)\\Gamma(a+b)}=\\frac{a}{a+b}\\end{aligned}$$\n",
    "2）$var[\\mu]=\\frac{ab}{(a+b)^2(a+b+1)}$  \n",
    "证：  \n",
    "$$\\begin{aligned}var[\\mu]&=\\mathbb{E}[m^2]-\\mathbb{E}^2[m]\\\\&=\\frac{1}{Beta(a,b)}\\int_0^1\\mu^{a+1}(1-\\mu)^{b-1}d\\mu-\\big(\\frac{a}{a+b}\\big)^2\\\\&=\\frac{Beta(a+2,b)}{Beta(a,b)}-\\frac{a^2}{(a+b)^2}\\\\&=\\frac{\\frac{\\Gamma(a+2)\\Gamma(b)}{\\Gamma(a+b+2)}}{\\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)}}-\\frac{a^2}{(a+b)^2}\\\\&=\\frac{a(a+1)\\Gamma(a)\\Gamma(a+b)}{(a+b)(a+b+1)\\Gamma(a)\\Gamma(a+b)}-\\frac{a^2}{(a+b)^2}\\\\&=\\frac{(a^2+a)(a+b)-a^2(a+b+1)}{(a+b)^2(a+b+1)}\\\\&=\\frac{a^3+a^2b+a^2+ab-a^3-a^2b-a^2}{(a+b)^2(a+b+1)}\\\\&=\\frac{ab}{(a+b)^2(a+b+1)}\\end{aligned}$$\n",
    "我们把$a,b$叫作$\\mu$的超参数（hyperparameter），因为它们决定了$\\mu$的分布。有时我们也把它们称为形状参数（shape parameter），下图展示了它们是如何控制beta分布的形状的"
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SlbPN1X9Lsv4JZuOAac65i2Kf/wvgnHP3N/I1HwCnO+f21Gt3TV2vxfbs8Qpk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\nkRQlxyIiIiIiKf8PGKs+1wGPdoYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x40eba90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "from scipy.stats import beta\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "params = [(.1,.1), (1,1), (2,3), (8,4)]\n",
    "\n",
    "x = np.linspace(0, 1, 100)\n",
    "fig, axes = plt.subplots(2, 2, figsize=(12,8))\n",
    "axes = np.ravel(axes)\n",
    "for i, ax in enumerate(axes):\n",
    "    a, b = params[i][0], params[i][1]\n",
    "    ax.plot(x, beta.pdf(x, a, b), 'r-', lw=5, alpha=0.6, label='beta pdf')\n",
    "    ax.set_ylim(0,3)\n",
    "    ax.set_xlabel('$\\mu$', fontsize=20)\n",
    "    ax.text(0.1, 2.7, '$a={}$'.format(a), size=20)\n",
    "    ax.text(0.1, 2.4, '$b={}$'.format(b), size=20)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "下面我来说明为什么beta分布是二项分布参数的共轭先验。  \n",
    "假设二项分布的$N$次独立实验中，事件发生（$x=1$）的次数为$m_1$，未发生（$x=0$）的次数为$m_2$，且$m_1+m_2=N$。那么似然函数可以表示为：\n",
    "$$count(m_1,m_2)=\\binom {m_1+m_2} {m_2}\\mu^{m_1}(1-\\mu)^{m_2}$$\n",
    "依据贝叶斯公式，后验概率分布可以写成\n",
    "$$\\begin{aligned}p(\\mu|m_1,m_2)&=\\frac{\\frac{1}{Beta(a,b)}\\mu^{a-1}(1-\\mu)^{b-1}\\cdot \\binom {m_1+m_2} {m_2}\\mu^{m_1}(1-\\mu)^{m_2}}{\\int_0^1 \\frac{1}{Beta(a,b)}\\mu^{a-1}(1-\\mu)^{b-1}\\cdot \\binom {m_1+m_2} {m_2}\\mu^{m_1}(1-\\mu)^{m_2}d\\mu} \\\\&=\\frac{\\frac{1}{Beta(a,b)}\\mu^{a+m_1-1}(1-\\mu)^{b+m_2-1}}{\\frac{Beta(a+m_1,b+m_2)}{Beta(a,b)}\\int_0^1 \\frac{1}{Beta(a+m_1,b+m_2)}\\mu^{a+m_1-1}(1-\\mu)^{b+m_2-1}d\\mu}\\\\&=\\frac{1}{Beta(a+m_1,b+m_2)}\\mu^{a+m_1-1}(1-\\mu)^{b+m_2-1}\\\\&=Beta(\\mu|a+m_1,b+m_2) \\end{aligned}$$\n",
    "至此，证明了后验分布也是一个beta分布。也许推导的公式有点难懂，我们可以从另一个更直观的角度来理解。  \n",
    "$$Beta(\\mu|a,b)\\times count(m_1,m_2)=Beta(\\mu|a+m_1,b+m_2)$$\n",
    "我们发现，从先验转变到后验的过程中，观测数据的作用相当于将$a$增加到$a+m_1$，将$b$增加到$b+m_2$。那么，我们可以认为beta分布的两个超参数$a$和$b$分别代表$x=1$和$x=0$的伪计数（因为它们可以是实数）。我们可以将这个过程简单地表达为：  \n",
    "><center>伪计数+真实计数=总计数<center/>  \n",
    "\n",
    "共轭性质使得贝叶斯框架天然具有学习序列数据的能力。举个例子来说，假设先验分布为$Beta(a_0,b_0)$，在$t=1$时刻观测到的数据集记为$\\mathcal{D}_1$，那么我们可以根据$\\mathcal{D}_1$修正先验为后验$Beta(a_1,b_1)$。在$t=2$时，我们可以利用上一轮的后验$Beta(a_1,b_1)$作为本轮的先验，通过观测集$\\mathcal{D}_2$得到后验$Beta(a_2,b_2)$，以此类推。这么做的好处如下：  \n",
    "1）可以在无法获取完整数据的情况下进行学习，在线更新参数  \n",
    "2）具有良好的扩展性，很容易处理较大规模的数据集  "
   ]
  },
  {
   "cell_type": "markdown",
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    "如果我们的目标是给定观测数据集，预测下一次实验的观测值，根据求和公式和乘法公式，有\n",
    "$$ p(x=1|\\mathcal{D})=\\int_0^1 p(x=1|\\mu)p(\\mu|\\mathcal{D})d\\mu = \\int_0^1 \\mu p(\\mu|\\mathcal{D})d\\mu =\\mathbb{E}[\\mu|\\mathcal{D}]$$\n",
    "我们来求解上式：\n",
    "$$\\begin{aligned}\\mathbb{E}[\\mu|\\mathcal{D}]&=\\int_0^1 \\mu p(\\mu|\\mathcal{D})d\\mu\\\\&=\\frac{1}{Beta(a+m_1,b+m_2)}\\int_0^1 \\mu^{a+m_1}(1-\\mu)^{b+m_2-1}\\\\&=\\frac{Beta(a+m_1+1,b+m_2)}{Beta(a+m_1,b+m_2)}\\\\&=\\frac{\\Gamma(a+m_1+1)\\Gamma(b+m_2)}{\\Gamma(a+b+m_1+m_2+1)}\\cdot\\frac{\\Gamma(a+b+m_1+m_2)}{\\Gamma(a+m_1)\\Gamma(b+m_2)}\\\\&=\\frac{(a+m_1)\\Gamma(a+m_1)\\Gamma(b+m_2)}{(a+b+m_1+m_2)\\Gamma(a+b+m_1+m_2)}\\cdot\\frac{\\Gamma(a+b+m_1+m_2)}{\\Gamma(a+m_1)\\Gamma(b+m_2)}\\\\&=\\frac{a+m_1}{a+b+m_1+m_2}\\end{aligned}$$"
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   "source": [
    "上式可以简单解释为$x=1$的样本数和占总样本数的比例（先验的有效样本数与观测集的真实样本数之和）。我们可以证明$\\mu$的后验均值一定处于先验均值和极大似然估计之间，且二者的次序取决于$am_1-bm_2$的符号。  \n",
    "证明：  \n",
    "首先考察先验均值\n",
    "$$\\begin{aligned}\\frac{a+m_1}{a+b+m_1+m_2}-\\frac{a}{a+b}&=\\frac{(a+m_1)(a+b)-a(a+b+m_1+m_2)}{(a+b+m_1+m_2)(a+b)}\\\\&=\\frac{bm_1-am_2}{(a+b+m_1+m_2)(a+b)}\\end{aligned}$$\n",
    "接着考察极大似然估计\n",
    "$$\\begin{aligned}\\frac{a+m_1}{a+b+m_1+m_2}-\\frac{m_1}{m_1+m_2}&=\\frac{(a+m_1)(m_1+m_2)-m_1(a+b+m_1+m_2)}{(a+b+m_1+m_2)(m_1+m_2)}\\\\&=\\frac{am_2-bm_1}{(a+b+m_1+m_2)(m_1+m_2)}\\end{aligned}$$\n",
    "1）如果$am_2-bm_1>0$，则\n",
    "$$\\mu_{ML}<\\mathbb{E}[\\mu|\\mathcal{D}]<\\mathbb{E}[\\mu]$$\n",
    "2）如果$am_2-bm_1<0$，则\n",
    "$$\\mathbb{E}[\\mu]<\\mathbb{E}[\\mu|\\mathcal{D}]<\\mu_{ML}$$\n",
    "3）如果$am_2-bm_1=0$，则\n",
    "$$\\mu_{ML}=\\mathbb{E}[\\mu|\\mathcal{D}]=\\mathbb{E}[\\mu]$$\n",
    "当$m_1,m_2\\to\\infty$时，$a,b$可以忽略不计，那么后验均值的极限将趋近于\n",
    "$$\\frac{m_1}{m_1+m_2}$$\n",
    "也就是$\\mu$的极大似然估计。这说明了贝叶斯后验估计最终将收敛于极大似然估计。"
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