{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "概率论在解决模式识别问题时具有重要作用,它是构成更复杂模型的基石。\n", "\n", "概率分布的一个作用是在给定有限次观测x1, . . . , xN的前提下,对随机变量x的概率分布p(x)建模。这个问题被称为密度估计(density estimation)。选择一个合适的分布与模型选择的问题相关,这是模式识别领域的一个中心问题。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 二元变量" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "import numpy as np" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 1. 伯努利分布" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "考虑一个二元随机变量$x\\in\\{0,1\\}$,$x=1$的概率被记作参数$\\mu$,因此$p(x=1|\\mu)=\\mu$,$p(x=0|\\mu)=1-\\mu$,其中$\\mu\\leq1$。\n", "\n", "x的概率分布可以写成$Bern(x|\\mu)=\\mu^x(1-\\mu)^{1-x}$,称为伯努利分布($Bernoulli$ $distribution$)。\n", "\n", "伯努利分布的均值和方差分别为$\\mathbb{E}[x]=\\mu$,$var[x]=\\mu(1-\\mu)$。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "假设有x的观测值数据集$\\mathbb{D}={x_1,...,x_N}$。假设每次观测都是独立地从$p(x|\\mu)$中抽取的,因此关于$\\mu$的似然函数是$p(\\mathbb{D}|\\mu)=\\prod\\limits_{n=1}^N p(x_n|\\mu)=\\prod\\limits_{n=1}^N \\mu^{x_n}(1-\\mu)^{1-x_n}$。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "其最大似然的估计值是$\\mu_{ML}=\\frac{1}{N}\\sum\\limits_{n=1}^N x_n=\\frac{m}{N}$,如果把数据集里$x=1$的观测的数量记为m。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**因此,在最大似然的框架中,x=1的概率是数据集中x=1的观测所占的比例。**假设扔一枚硬币5次,碰巧5次硬币都正面朝上,那么$\\mu_{ML}=1$。最大似然的结果会预测所有未来的观测值都是正面向上。这是最大似然中过拟合现象的一个极端例子。我们通过引入$\\mu$的先验分布,来得到更加合理的解释。" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from scipy.stats import bernoulli" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**伯努利分布直方图,u=0.6**" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "u = 0.6\n", "x = np.arange(2)\n", "bern = bernoulli.pmf(x, u)\n", "\n", "plt.bar(x, bern, facecolor='green', alpha=0.5)\n", "plt.xlabel('x')\n", "plt.xlim(-0.2, 2)\n", "plt.title('Histogram of the nernoulli distribution')\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2. 二项分布" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "给定数据集规模N的条件下,x = 1的观测出现的数量m的概率分布,称为二项分布(binomial distribution)。\n", "\n", "$Bin(m|N, \\mu)=\\binom{N}{m}\\mu^m(1-\\mu)^{N-m}$,其中$\\binom{N}{m}=\\frac{N!}{(N-m)!m!}$。\n", "\n", "其均值和方差分别是$\\mathbb{E}[m]=\\sum\\limits_{m=0}^{N}mBin(m|N,\\mu)=N\\mu$,$var[m]=\\sum\\limits_{m=0}^{N}(m-\\mathbb{E}[x])^2Bin(m|N,\\mu)=N\\mu(1-\\mu)$。" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from scipy.stats import binom" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 5.63135147e-02, 1.87711716e-01, 2.81567574e-01,\n", " 2.50282288e-01, 1.45998001e-01, 5.83992004e-02,\n", " 1.62220001e-02, 3.08990479e-03, 3.86238098e-04,\n", " 2.86102295e-05, 9.53674316e-07])" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 10\n", "u = 0.25\n", "m = np.arange(N+1)\n", "binData = binom.pmf(m, N, u)\n", "binData" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**关于m的函数的二项分布直方图,其中N=10且u=0.25**" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.bar(m, binData, facecolor='green', alpha=0.5)\n", "plt.xlabel('m')\n", "plt.xlim(-0.2, 10.5)\n", "plt.title('Histogram of the binomial distribution')\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 3. Beta分布" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "伯努利分布的似然函数是某个因子与$\\mu^x(1-\\mu)^{1-x}$的乘积的形式,如果我们选择一个正比于$\\mu$和$(1-\\mu)$的幂指数的先验概率分布,那么后验概率分布(正比于先验和似然函数的乘积)就会有着与先验分布相同的函数形式。这种性质叫做共轭性(conjugacy)。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Beta分布定义如下:\n", "\n", "$$Beta(\\mu|a, b) = \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)}\\mu^{a-1}(1-\\mu)^{b-1}$$\n", "\n", "$\\Gamma(x)$是Gamma函数,保证Beta分布归一化。\n", "\n", "Beta分布的均值和方差为$$\\mathbb{E}[\\mu]=\\frac{a}{a+b},var[\\mu]=\\frac{ab}{(a+b)^2(a+b+1)}$$\n", "\n", "参数a和b被称为超参数(hyperparameter),因为它们控制了参数$\\mu$的概率分布。" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from scipy.stats import beta" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "D:\\Python27_32\\lib\\site-packages\\matplotlib\\transforms.py:656: RuntimeWarning: invalid value encountered in absolute\n", " inside = ((abs(dx0 + dx1) + abs(dy0 + dy1)) == 0)\n" ] }, { "data": { "image/png": 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i/Hw48UTvRLt58/wejYiEwWuvwYEDMH58OO9kJyIiHZOWPcUAt98Od9wB06fD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\npIn5Y/Gii+DAAaujEkKECO3WbLpnEwCp96XSpFMTiyMSwa75iOYA7Pt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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = plt.figure(figsize=(12,8))\n", "ax = fig.add_subplot(221)\n", "x = np.linspace(0, 1, 100)\n", "pbeta = beta.pdf(x, 0.1, 0.1)\n", "ax.plot(x, pbeta, label='a=0.1\\nb=0.1',lw=2)\n", "plt.ylim(0,3)\n", "plt.xlabel('$\\mu$')\n", "plt.grid(True)\n", "plt.legend(loc=\"best\")\n", "\n", "ax = fig.add_subplot(222)\n", "x = np.linspace(0, 1, 100)\n", "pbeta = beta.pdf(x, 1, 1)\n", "ax.plot(x, pbeta, 'g', label='a=1\\nb=1',lw=2)\n", "plt.xlabel('$\\mu$')\n", "plt.ylim(0,3)\n", "plt.grid(True)\n", "plt.legend(loc=\"best\")\n", "\n", "ax = fig.add_subplot(223)\n", "x = np.linspace(0, 1, 100)\n", "pbeta = beta.pdf(x, 2, 3)\n", "ax.plot(x, pbeta, 'r', label='a=2\\nb=3',lw=2)\n", "plt.xlabel('$\\mu$')\n", "plt.ylim(0,3)\n", "plt.grid(True)\n", "plt.legend(loc=\"best\")\n", "\n", "ax = fig.add_subplot(224)\n", "x = np.linspace(0, 1, 100)\n", "pbeta = beta.pdf(x, 8, 4)\n", "ax.plot(x, pbeta, 'm', label='a=8\\nb=4',lw=2)\n", "plt.xlabel('$\\mu$')\n", "plt.ylim(0,3)\n", "plt.grid(True)\n", "plt.legend(loc=\"best\")\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4. 后验概率" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "将Beta先验和二项似然函数相乘,得到后验概率分布的形式为:\n", "\n", "$$p(\\mu|m,l,a,b)\\propto\\mu^{m+a-1}(1-\\mu)^{l+b-1}$$\n", "其中$l=N-m$对应硬币反面朝上的样本数量。\n", "\n", "具体得到归一化系数,公式如下:\n", "\n", "$$p(\\mu|m,l,a,b)=\\frac{\\Gamma(m+a+l+b)}{\\Gamma(m+a)\\Gamma(l+b)}\\mu^{m+a-1}(1-\\mu)^{l+b-1}$$\n", "\n", "我们可以发现,**如果数据集里有m次观测为x=1,有l次观测为x=0,那么从先验概率到后验概率,a的值变大了m,b的值变大了l。**" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from scipy.misc import comb\n", "\n", "pbinom = lambda m, N, p: comb(N, m) * p**m * (1-p)**(N-m)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "上面代码是二项式分布的分布函数$Bin(m|N, p)=\\binom{N}{m}p^m(1-p)^{N-m}$,其似然函数和该分布函数具有相同的形式,只是其参数p是未知的。" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "image/png": 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FEvaXnEflo/v7yaVke+lhhXOiWt1qvL/5fb7840tOX9P/fwYGjUs0pn3Z9rR4rAWl85R+\n5CjAri6ueLp5ktc7L2XylqEZd2/WXQi/wMpjK1l0aBGr/17NwoMLWXhwIbUL12Zsg7G0Lt062RGG\nU8IK30VaOXSlLSwsLMn3KlWqlLCckgOgYMGCXLhwIdF7sbGxXLp0iTx58iR63xYHlBBCxJs2bRof\nf/wxzZs3Z8OGDQk5J/7flStXJnnBUaZMGZvGkZI8WLBgQQ4ePPjA7164cCFhnQIFCnDjxg2io6MT\nVdzCwsLSlD/jLx6jo6Px9vZOeP/KlSup3lbevHkT3dRLq4IFCwIP/g2K//+7/++GEI4kqWunCxcu\nUKlSJQoWLEh4ePgDo2NfuHABb2/vhH6pderUYdWqVURFRfHrr78ybNgwevTowe+///7Q/ebJkwc3\nNze2bt2Ki8uDDbnurUymJFcUKlSIAwcOJFmWvHnzJnpPrt0e7vLNy3zy+ydM2zGNa1G6X+JjeR5j\nYI2B9KjUg4LZC9pkP77ZfOlXrR/9qvXj7PWzfLvnW6btmMaOcztoM78NTxZ9ks9bfU71gtVtsr/M\nyKGbR4aFhSVKEKdPn2bPnj3JDs18/8lbp04dlixZkqhz6uLFi4mJieGpp55KcTxWmBvCCmUAa5TD\nCmUQycuRIwdr1qwBoEWLFty4cQOAJ554Ai8vL86dO0f16tUfeGXLls2mcaQkD9atW5ddu3Zx8uTJ\nhHXOnTvH77//nrBO/ETfS5cuTVgnPDz8gdHnUqpIkSIAHDx4MOGc2L59O9evX0+0Xkouypo2bcrl\ny5cTBiZJiru7Ozdv3kw2pkKFCvHTTz8lev+nn34iZ86ciW4c3s8K57UVypCZPeraqVatWhiGwcKF\nCxM+V0oRFBRE/fr1H9iWh4cHbdq0oW/fvolu6CR1HjVu3JjY2FiuXr2aKJddv36d6tWrJ1QIU1rB\nqlOnTrL5KKM44zlxK+YWk7dMpuTUkryz6R2uHb5GI79GrOq5iiOvHmH4k8NtVmG7X5EcRRjbcCwn\nhp7gk+afkD9rfrae2UrNr2oyaMUgrkddT34jD+GM34WtOPSTtnz58vHcc88lGgHJ19eXPn36PPL3\n1N0OvACMGTOGatWq0b59ewYOHMjZs2cZOXIkLVu2pE6dOnYuhRAis8uTJw+//vor9evXp02bNqxe\nvZpcuXIREBDA0KFDOXXqFPXr1ycuLo6jR48SEhLC4sWLbRpDSvJgnz59+OCDD2jVqhUTJkzAxcWF\n8ePH4+Pjw4ABAwDdnLNt27YMGjSI69evU6BAASZPnkzWrFnT1DyyTp06FC5cmCFDhtC5c2fOnj3L\n5MmTyZEjR6Lt3Z/Xk9KsWTNatGhBjx49GDt2LNWqVSM0NJRNmzYxY8YMAMqVK8fKlStp2bIlWbNm\npVy5cg9UkF1cXAgICGDAgAHkzZuXpk2bsmHDBmbMmMF77733QNNQIRzJo66d3N3d6d69O6+++io3\nbtygZMmSzJo1i6NHjzJz5kxAP/3/9ttv6dChA0WLFuXcuXPMnDmTJk2aJOwjqfOobNmyDBw4kG7d\nuvHmm29So0YNbt26xfLly5k7dy6zZs0CUnYuQ8rykUjassPLGLp6KKeunQKgacmmtCvfjle7vJrM\nb9pWVvesvP7E6/Sr1o/xG8bz+Y7PmbFrBqv/Wc3sdrNp6NcwQ+NxemkZvSQtL9IwAlLNmjXVkiVL\nVJkyZZSHh4d66qmn1IEDBxLWuXe0r3v5+/snGj1SKaXWrVun6tSpozw9PVX+/PnVK6+8kmgEs+Dg\nYOXi4pJo+0KIR8MCI7SlNjfF69OnT6IR0b777jvl4uKSaFRHFxcXtXLlyoR1Tp06pYoVK6ZatWql\noqOjlVJK/fjjj6pGjRrKy8tL5c6dW9WtW1dNmTIlXftJ6veUSj4PKqXU8ePHVfv27VX27NlVtmzZ\n1DPPPKP+/vvvROtcuXJFdevWTWXNmlUVKFBATZw4UY0YMSJFo0feG3u8nTt3qlq1ailvb29VvXp1\ntWXLlgdGj0wqryeVt2/evKlGjBihihQpojw8PFSJEiXUmDFjEj7ftWuXqlu3rsqaNatycXFRGzZs\nUEol/ffk888/V4899phyd3dXpUqVUp9++mmizwMCApSPj88D5XzY3yaRcayQm5Sdrp0iIyPV4MGD\nla+vr/Lw8FC1atVSa9euTfj8yJEjqlOnTqpo0aLKw8NDFSlSRA0aNEhduXIlYZ2HnUdK6VFdK1as\nqDw8PJSPj4/y9/dPNKJkUueyUkmfTynJR3K+3XXyykn1zLxnEkZ0rPRFJbX62Gqzw0qw/8J+VWNm\nDUUAyggw1Fu/vaViYmPMDivDpTU/Gfp37c8wDJWaffXp04cDBw4kjFQmhHA8hmGglHLqzgSpzU1C\nCMdnhdwEcu0kUkYpxde7v2bY2mGER4eTwyMHkxpNYlCtQbi5OFajutuxt5m0cRKTNk0iTsXRvFRz\n5nWcR17vvMn/skWkNT85dJ82R2OFdrRWKANYoxxWKINwDFY5lqxQDimDEIlZ4Xhy5DKE3gil1dxW\n9F/Rn/DocDqW78jhVw4zuM7gRBU2RylDFtcsjG80nt+e/4183vlY+89aas2qxZGLR1L0+45SDjM4\nbKXNMAwZDUgIIYQQIoXk2ilzWXVsFVVmVGHNP2vI65WX+c/OJ6hzkN0GGLGlRiUasav/LmoWqsmJ\nqyeo9209tp3dZnZYDs1hm0cKIRyfFZogSW4SwnqskJtA8pNIWkxcDKPXjebDrR8C0KREE+Z0mOMU\nlbX7RURH0DWoKyuPrcTLzYtFXRbRqnQrs8Oyq7TmJ6m0CSHSzAoXRpKbhLAeK+QmkPwkHhQWEUa3\noG4EnwzG1XBlUuNJvFnvTVwMh208l6yYuBgG/DyAb/d+i7urO0u6LqF16dZmh2U30qctA1ihHa0V\nygDWKIcVyiAcg1WOJSuUQ8ogRGJWOJ4cpQx7QvdQ46saBJ8MxjerL+t7r2fUU6NSVGFzlDIkxc3F\nja/bfs2Q2kOIjo2mQ2AHVh1bleS6jlwOe5NKmxBCCCGEEA5s8aHFPPXdU5y9fpYnijzB7gG7aVC8\ngdlh2YxhGHza8lMG1x6cUHHbcHKD2WE5FGkeKYRIMys0QZLcJIT1WCE3geQnoYfzn7x1MiN/GwlA\nryq9+KrNV3i4eZgcmX0opXh55cvM2DWDnB452dR3E5V8K5kdlk1JnzYhRIazwoWR5CYhrMcKuQkk\nP2V2sXGxDF09lOk7pwPwfpP3ebPem5YfITQ2LpauQV1ZdGgRhbIXYmu/rRTPVdzssGxG+rRlACu0\no7VCGcAa5bBCGYRjsMqxZIVySBmESMwKx5MZZbgVc4vOCzszfed03F3dCewUyMinRqa5wuZM34Or\niys/dvyRhsUbcv7GedouaEt4dDjgXOWwNam0CSGEEEII4SBuRN2g9dzWLDm8hFyeufj1+V/pUrGL\n2WFlKE83T5Z2W0rZvGX568Jf9FnahzgVZ3ZYppLmkUKINLNCEyTJTUJYjxVyE0h+yowuRV6i1dxW\n7Dy/k4LZCrL2+bU8nv9xs8MyzZGLR6j9dW2uR11nvP94xjYca3ZI6Wa35pGGYXxrGMYFwzD2PeRz\nf8MwrhmGsefOa0xqgxBCiNSS3CSEcFSSn0RahEWE0ej7Ruw8v5MSuUqwqe+mTF1hAyibryzzn52P\ngcG4kHGs/nu12SGZJiXNI78DWiazzgalVLU7r0k2iMshWaEdrRXKANYohxXKYDLJTXdY5ViyQjmk\nDOIOyU93WOF4yogyhN4IxX+2P/vC9lEuXzk299tMqTylbLZ9Z/4eWpduzcRGEwHo+lFXzt84b3JE\n5ki20qaU2gRcSWY1p2+CIIRwLpKbhBCOSvKTSI3QG6E0+r4Rhy4eoqJPRUJ6h1AoeyGzw3Io/6v/\nP5qVbMb1W9fpsagHMXExZoeU4VLUp80wDD/gZ6XUAxMlGIbREFgMnAXOASOUUgeTWE/aZTu5mzfh\n+HE4eRJOnYILF+C//+DKFYiIgMhIiP+KXV3B2xuyZYM8ecDHBwoUAD+/u68sWcwri7ANs/uNSG4S\nQiTF7Nx0JwY/JD+JZFwIv5BQYavsW5nfnv8Nn6w+ZoflkC6EX6DqzKr8G/4vAQ0DGOc/zuyQ0iSt\n+cnNBvveDRRVSkUahtEKWAqUSWrFPn364OfnB0CuXLmoWrUq/v7+wN3HtrLsGMsrVoRw8CBER/uz\naxfs2BFCaCgopT+HkDv/pm3Z1TWEokWhbl1/atYEN7cQypWDFi0co/yynPRy/M8nT57ECUhukmVZ\nziTL8T87SW4CyU+yDCxbvYzX1rzGyVwnqehTkfHFx3Ng5wGHic/Rlg/9cYg3Cr3B8KPDmbhxIgUu\nFqBsvrIOE5+981O6n7Qlse4JoIZS6vJ97zv93aKQkJCEL8JZPawMUVGwcSOsXg3BwbB3792nZvFc\nXaFkSf2UrHhxKFhQP0HLk0c/UfPy0usAxMToJ2/h4XDpEoSFwfnz+gnd8eNw+vSDsbm7Q+3a0Lgx\ntGoFtWrd3V5Ky+FMrFAGs+9mS27SrHAsgTXKIWVwDGbnpjsx+CH5yRLHkz3KcO3WNRr/0Jjdobsp\nn688wb2D8c3ma9N93MsK3wPociyPWs6UbVMol68cu/vvxiuLl9lhpYppT9oMw/AFwpRSyjCM2uiK\n4OXkfk+YLzwcVqyAoCBdWYuIuPtZlixQs6auRNWqBVWqQJkyumJlCxERcOiQrhzu3Anbt8Nff8Hm\nzfo1YQLkzQtt20LnztCkie32LTIHyU1CCEcl+Slzi7wdSZv5bdgduptSuUuxrtc6u1bYrOadxu+w\n+u/VHLp4iLfWvcWUllPMDilDJPukzTCM+UBDIB9wARgHZAFQSs00DOMVYBAQA0QCw5RS25LYjtPf\nLbKC2FhYtw6+/x6WLNH91OJVqaKfcDVtCk88ofukZaQrV2DTJli7Fn75BU6cuPtZnjzQrRv07q0r\nkYZ033YIZt7NltwkhHgYs5+0SX4SDxMdG027Be1Y/fdqiuQowqa+m/DL5Wd2WE7nj/N/UPfrusSq\nWDb33Uy9YvXMDinF0pqfZHLtTCIsDL75BmbO1E0U49Wrp59kdewIRYuaF9/9lNJP4oKCYOFC2L//\n7meVK8PLL0PPnrpZpjCP2RdGtiC5SQjrsUJuAslPVhOn4nhu8XPM3z8fH28fNvXdRNl8Zc0Oy2mN\nXjeadze/SwWfCuwZsAd3V+dokmW3ybXFXfd2KHQWBw/CCy/oCtlbb8GpUyGUKAHjx+snWZs3w9Ch\njlVhA/0krUIFGDsW9u3TzShffx3y5dPNKAcODKFIERg1Cs6dMzvatHHG40k4JqscS1Yoh5RBiMSs\ncDzZogxKKV5f/Trz988nm3s2VvVclaEVNit8D5C4HGMajOGxPI9x8L+DfLjlQ/OCyiBSabOoXbug\nfXuoWBG+/RZu34ZnnoEPPoC//9aVoTuDUTmFKlXgk0/g7FmYO1eX69o1XZ4SJeCllxI3pxRCCCGE\ncBQfbvmQz3Z8hrurO0u7LqVGoRpmh+T0vLJ48VWbrwCYuHEiRy4eMTki+5LmkRbz118wZgz8/LNe\n9vSEvn31U6rSpc2Nzda2b4ePP4ZFiyAuDtzcoE8fGDcOihQxO7rMwQpNkCQ3CWE9VshNIPnJKub+\nNZfnljyHgcGCTgvoUrGL2SFZSr9l/fhu73c0LdmUtc+txXDwgQ+kT1smd/q0rqz9+KPuD+btrft9\njRgBvhYfkOjIEXj3XV32uDhdUR08WDcHzZXL7OiszQoXRpKbhLAeK+QmkPxkBetPrKfljy25HXeb\nKS2m8Frd18wOyXIuRl6kzOdluHLrCou7LKZD+Q5mh/RI0qctAzhie+CICN3UsWxZmDNHP20aMkQ3\nFZw8+cEKmyOWIS3uLUfZsno0zMOHoUsXuHVLl710aT3wSmyseXE+ilW+C2E+qxxLViiHlEGIxKxw\nPKW1DAfCDtAhsAO3427zet3XTa2wWeF7gKTLkc87HxMbTQRg2Nph3Lx984F1rEAqbU5KKT1kf7ly\nMHGirqh066afOk2dCvnzmx1hxitdGgID9bxvDRrAxYswcKCeb277drOjE0IIIURmcSH8Ak/Pe5rr\nUdfpVKHqVAxTAAAgAElEQVQTHzX/yOyQLG1AzQFUyl+Jk1dPMnnrZLPDsQtpHumETp+GV17RE2MD\nVK+uK2pPPWVuXI5EKd3XbcQIPcWBYcCAAfDee9Jk0pas0ARJcpMQ1mOF3ASSn5xV5O1IGn3fiB3n\ndlCncB2CewfjlcXL7LAsb8PJDfh/74+nmyfHBh+jSA7HHOBAmkdmAnFx8OWXeuTEFSsgRw6YNg12\n7JAK2/0MAzp10lMejBoFrq4wYwY8/jisXGl2dEIIIYSwojgVR++lvdlxbgd+ufxY3n25VNgySEO/\nhnSq0IlbMbcYGzzW7HBsTiptqWBme+BTp6BpUz24SHg4PPus7sP1yiu6QpJSVm7TnBRvb/10be9e\nqFtXz+nWpg307q2nDDCTVb4LYT6rHEtWKIeUQYjErHA8paYMASEBBB0MIodHDlb2WEn+rI7RX8UK\n3wMkX453G7+Lm4sbs/fOZt+FfRkTVAaRSpsTmDsXKleG4GDw8YGFCyEoCAoWNDsy51Gxop5I/KOP\n9OiSP/yg537btMnsyIQQQghhBfP3zWfixom4GC4Edgqkgk8Fs0PKdErnLc2gmoNQKEb+NtLscGxK\n+rQ5sBs3YNAgXWkDaNcOZs3SFTeRdkeOQM+eegJyw9BTJYwdq0feFKljhX4jkpuEsB4r5CaQ/ORM\ndp7bSf3v6hMVG8XUllMZUmeI2SFlWv9F/Eepz0pxI/oGvz3/G01KNjE7pESkT5vF7N0LNWroCpu3\nN3z9tR4tUips6Ve2LPz+O4werZcnToTGjeHsWXPjEkIIIYTzCb0RSvvA9kTFRvFS9ZcYXHuw2SFl\naj5ZfRj11CgARq8fjVVufEilLRUyqj3wt9/q/lfHjulmkbt2wQsv6KdC6ZVZ2jQnJ0sWmDQJ1q3T\nzUw3bYJq1WD9etvElxJW+S6E+axyLFmhHFIGIRKzwvH0qDLcirlFh8AOnL9xnvrF6jOt9TQMW1yw\n2ZgVvgdIeTmG1BmCj7cP289t55djv9g3qAwilTYHEhWlh6V/4QX980svwbZtei42YR+NGumnms2a\n6XndmjWDDz/UUwYIIYQQQjyMUoqXV77M9nPbKZazGEFdgnB3dTc7LAFkc8+W8LRtbMhYSzxtkz5t\nDuLff6FDB11J8/DQQ/v37Wt2VJlHbCyMGwfvvKOXu3bVTzy9vc2Ny9FZod+I5CYhrMcKuQkkPzm6\n6Tum8+qqV/Fy82JLvy1UK1jN7JDEPW7evkmpz0oRGh7K4i6L6VC+g9khAdKnzant3g21aukKW7Fi\nsHWrVNgymqurbi65bBlkzw6BgdCggfRzE0IIIcSDNp7ayGtrXgPgm7bfSIXNAXll8WJ0fT2AwbiQ\nccSpOJMjSh+ptKWCPdoDL1umJ8Y+exbq1YOdO6F6dZvvJkFma9OcWm3b6kFKSpbUfQlr19aVanuw\nynchzGeVY8kK5ZAyCJGYFY6n+8tw9vpZOi/sTExcDCOeGEH3St3NCSwVrPA9QOrL8WL1Fymaoyj7\nwvax/Mhy+wSVQaTSZqKpU3WTyJs3oU8fPShGfseYgzFTq1gRduyAhg0hNFQ/cVu50uyohBBCCGG2\nqJgoOv3UibCIMJqUaMJ7Td8zOyTxCB5uHrzx5BsAvLPpHafu2yZ92kwQFwcjRsCUKXp50iR46y3b\njA4pbCcqCl58EX78EVxcdD/D/v3NjsqxWKHfiOQmIazHCrkJJD85ooErBjJz10yK5SzGrv67yOed\nz+yQRDJu3r6J31Q/wiLCWPPcGpqXam5qPNKnzUlER8Pzz+sKW5YsukIwerRU2ByRhwf88IOeeDsu\nTo/sOXGijCwphBBCZEbf7fmOmbtm4uHqwaIui6TC5iS8sngxrO4wACZtnGRyNGknlbZUSG974IgI\n3Wdq3jzIlg1++QV69rRNbCmVWds0p5VhwPjxMGOGfto2diwMHqwrcellle9CmM8qx5IVyiFlECIx\nKxxPISEh7Andw8u/vAzAF09/Qc1CNU2OKnWs8D1A2ssxqNYgcnnmYtPpTWw6tcm2QWUQqbRlkGvX\noEULWLMGfHwgJASaNjU7KpFSAwZAUJB++jZ9uh7dMybG7KiEEEIIYW83om7w7E/PcivmFi9We5F+\n1fqZHZJIpRweORhSewgAH2790ORo0kb6tGWAixd1hW33bihaFH77DcqUMTsqkRbr1+unpRER0LEj\nzJ8P7pl4Hk0r9BvJzLlJCKuyQm4CyU+OIE7F0W5BO1YcXUH1gtXZ0m8Lnm6eZocl0uC/iP8o9mkx\nbsXc4tArhyiXr5wpcUifNgf133/QuLGusJUqBZs2SYXNmTVuDL/+CjlzwuLF8OyzesASIYQQQljP\nh1s+ZMXRFeT2zE1Q5yCpsDkxn6w+9K7SG4BPfv/E5GhSTyptqZDadrRhYfoif98+KFdOV9iKF7dP\nbCmV2ds028ITT0BwMOTJAytW6Cdut26lfjtW+S6E+axyLFmhHFIGIRJz5uMp5GQIo9ePhhMwp8Mc\nSuQuYXZIaebM38O90luO1+u+joHBD3/+QFhEmG2CyiBSabOT+Cds+/dDhQr6Ir9gQbOjErZSrZpu\nKpk3rx5QpmNHeeImhBBCWMW/4f/SLagbcSqOnpV68nSZp80OSdhA2XxleabsM0TFRjF9x3Szw0kV\n6dNmB5cv6wrbn3/qCtv69eDra3ZUwh727dPf9cWL0K4dLFyop3LILKzQbyQz5SYhMgsr5CaQ/GSW\n2LhYms5pSsjJEBr5NWLt82txc3EzOyxhI5tObaLB7Abk9crLmdfP4JXFK0P3L33aHMS1a9C8ua6w\nlSkD69ZJhc3KKlXSA8vkzg3LlkGPHjKqpBBCCOHMAkICCDkZgm9WX+Y9O08qbBbzVLGnqFGwBpdu\nXmLB/gVmh5NiUmlLheTa0UZGQps2sGsXlCypn7AVKJAxsaWUtGm2vSpVYO1ayJFDTwvw0kspm8fN\nkcognJtVjiUrlEPKIERiznY8rf57NZM2TcLFcGH+s/MpkK2A05UhKVYoA9imHIZhMLj2YAA+3/E5\nzvI0WyptNhIdrUcS3LwZihTRFbbChc2OSmSUmjVh1Srw9obZs2H4cHCSHCCEEEII4Oz1szy/5HkA\nxvuPp1GJRiZHJOyl6+Ndyeedjz3/7mHrma1mh5Mi0qfNBuLidLO4wEDIl0+PElnOnKkfhMnWrtVP\nW2/fhgkT4O23zY7IvqzQb8TKuUmIzMoKuQkkP2WkmLgYGn3fiM2nN9O8VHNW9VyFiyHPNqxs9LrR\nvLv5XbpW7MqCThnXTFL6tJlEKXjtNV1hy54d1qyRCltm1rw5zJsHLi4wdizMmmV2REIIIYRIzrjg\ncWw+vZmC2Qoyp8McqbBlAgNrDsTVcGXRoUWcv3He7HCSJUdkKiTVjvbDD+Hzz8HdXQ9EUb16xseV\nGtKm2f46dYIvvtA/DxwIy5cnvZ4jl0E4F6scS1Yoh5RBiMSc4Xha8/ca3t38bkI/tvxZ8yf63BnK\nkBwrlAFsW46iOYvSvlx7YuJi+GrXVzbbrr0kW2kzDONbwzAuGIax7xHrfGYYxjHDMP40DKOabUN0\nXHPmwKhRYBjw44/QSJo+izsGDIBx43TT2a5d4fffzY7IeiQ3CSEcleQn53H+xvmEfmwBDQNo6NfQ\n5IhERnq51ssAfL37a2LiHHv472T7tBmGUR8IB35QSlVK4vPWwKtKqdaGYdQBpiql6iaxnqXaZQcH\nQ4sWuu/SZ5/B4MFmRyQcjVK68jZrlu7ruG0blCpldlS2ZWa/EclNQoiHMbtPm+Qn53DvfGxNSjRh\nzXNrcHVxNTsskYGUUpSdVpZjl4+xrNsy2pZta/d92q1Pm1JqE3DlEau0Bb6/s+52IJdhGJaemezg\nQejQQVfYhg2TCptImmHoZpItW+rJt1u31hOvC9uQ3CSEcFSSn5zDxI0TE+Zjm9txrlTYMiHDMOhf\noz8AM3fNNDmaR7NFn7bCwJl7ls8CRWywXYcTEhJCWBg8/bSeRLtDB5g82eyoUkfaNGcsNzc9SE3l\nynD0qD5moqP1Z85SBieWqXKTFVihHFIG80RF6b7lPXqYHUmKSH4yWfCJYCZsmICBwdyOc/HN9vA6\ns6OWITWsUAawTzn6VO2Du6s7q46t4tTVUzbfvq3Yaor3+x/xJfksv0+fPvj5+QGQK1cuqlatir+/\nP3D3S3Dk5Z079zJmjD8nT0LZsiH07w8uLo4TX0qW4zlKPGld3rt3r0PF86jlHDlgzJgQBg2CjRv9\nGTQInnsuhD//3OsQ8aVmOf7nkydP4iQyRW7au9f5jiWrLjtTbnrYsjMdT7/+GsKuXbrLQkhICNHR\nJ3Eikp9MWq5QqwI9F/dEnVA8X+V5mpRs8sj14zlK/Jl52V7H07Pln2X+z/MZ8+0Y5gybY9Ptx/+c\n3munFM3TZhiGH/DzQ9plzwBClFIL7iwfBhoqpS7ct55Tt8tWCvr0gR9+0JNn79gBBQuaHZVwJn/8\nAQ0awM2b8NFHegJuZ+cA/Ub8yOS5SYjMJiYGQkJgwQJYvBiu3NMIsVo1PfjTqFHmz9Mm+ckxxak4\nnp73NKv/Xk39YvVZ33s9bi62eoYhnNWGkxvw/96fgtkKcvr103Y9Jsycp2050OtOEHWBq/cnHSv4\n+GNdYfP21kO4S4VNpFbNmvD99/rnN96AlSvNjScTyBS5SYjMIDZWV9QGDYJChaBZM/jmG11he/xx\nmDgRjhyB3bth5Eizo00RyU8m+Xjrx6z+ezV5vfIy79l5UmETADQo3oAyecsQGh7Kmr/XmB1OkpKt\ntBmGMR/YCpQ1DOOMYRj9DMMYYBjGAACl1C/AccMw/gZmAi/bNWITrFkT/0cghDlz9J08Z3X/o35n\n5azl6NwZxo/XT267dAnhyBGzI3Jekpvuctbz4X5WKIeUwXbi4mDLFhgyRLdwadQIZsyA//6DsmVh\n7Fg4cAD27YMxY6BMGbMjvkvy012OcjwBbDu7jbfWvwXA7PazKZIjZd0IHakMaWWFMoD9ymEYBn2r\n9gXgu73f2WUf6ZXs7QWlVPcUrPOqbcJxPH//Dd266T8evXtDx45mRySc3Zgx8OefullPu3awfTvk\nzGl2VM4ns+cmIaxIKdi5Uw/gtHAhnLlnqI4SJXTTx65doUoVPUKvo5L85Hiu3rpK90XdiYmL4fW6\nr9OmTBuzQxIOpleVXoxeP5rlR5ZzMfIi+bzzmR1SIinq02aTHTlhu+zwcKhbV9/Ja9dOX2S72KJB\nqcj0wsPhiSdg/3545hlYutQ5jy2z+7TZgjPmJiGsRCl9IyswUL9OnLj7WdGi0KWLrqjVrJnyipoV\nchNIfrIVpRRdgroQdDCIGgVrsKXfFjzcPMwOSzigp+c9zS/HfuHTFp8ytO5Qu+wjrflJKm0PoZR+\nwvbTT1CunH4akiOH2VEJK/nnH6hVS/fJmDhRP4FzNla4MHK23CSEVRw4cLeidvTo3fcLFtRNybt0\n0Te30nJDywq5CSQ/2crMP2YycOVAsrtnZ/eA3TyW5zGzQxIOKuhgEJ0Xdqayb2X2DtiLYYdH+mYO\nRGJJn36qK2zZs8OSJbrCZoX2wFYoA1ijHGfOhDB3rr5zPHas7jspRFpY4XwAa5RDyvBoR4/qm1SP\nP353AJGjR8HHRw8yEhKim0ROnQr16jlnCwSRmNnnxL4L+3htzWsAzGwzM00VNrPLYAtWKAPYvxzP\nlHmGPF55+OvCX+wO3W3XfaWWDJmThE2b9Oh+ALNn6ydtQthDq1YwbhwEBOgJYXfvhuLFzY5KCCFs\n5/hxfRM0MBDuTGMHQJ48up94167g7w9uckUibCwiOoKuQV25FXOLF6q9QPdKyXY1zLSuX4dTp+D8\neQgNhcuX4do1uHEDoqPh9m19E8XdHTw9IVcu/SpQAAoXhmLFwNfXsfuapoSHmwfPVXqOz3Z8xg9/\n/kCNQjXMDimBNI+8T1gYVK2qD9g334QPPjA7ImF1cXG6X9svv0Dt2vqmgbu72VGljBWaIDlLbhLC\nmZw5oytqCxboOSrj5cgBHTroilrTppAli332b4XcBJKf0uvF5S/yzZ5vKJ+vPDtf2klW96xmh2S6\niAh9g3jPHt2vfv9+OHYMLl5M/7azZ9cju1auDDVq6C4gVava7zy3lz/O/0GtWbXw8fbh3LBzZHG1\nbQGkT5sNxMZCy5bw229Qvz6sXy93/kTGuHQJqleH06dh6FDdPNcZWOHCyBlykxDOIDRUj/gYGAhb\nt959P1s2aNtW91Fr2RI8MmD8ByvkJpD8lB7z982nx+IeeLp5suPFHVTyfWCO80zh4kUIDoaNG/VN\n4X379M3i+3l5gZ+fngexYEHIl08/ScuWTd9IzpJF/97t23DzJly9qvvkh4bCuXN6AKF7J7qPlzWr\nburcvDm0aaOn5nD0p3FKKSp8UYHDFw+zssdKWpdubdPtpzk/KaUy5KV35dgCApQCpXx8lDp37sHP\ng4ODMzwmW7NCGZSyRjnuL8O2bUplyaKPwYULzYkpte6c1xmWR+zxcobclBwrnA9KWaMcma0MYWFK\nffGFUg0bKmUYOn+BUl5eSnXqpHNZZKTdQn0oK+QmJfkpzY5dOqayv5tdEYCasXNGurfnTOd1bKy+\nnnjrLaVq1Lj3vAxWoJSrq1JVqyr14otKffaZUuvW6Wve2Nj07TcuTueDjRuVmjpVqV69lCpT5m5O\niH+VLq3UmDFK7d+ftv1k1HcxacMkRQCqe1B3m287rflJniPdERysJz02DJg3T99pECIj1akDkyfD\na6/Biy/qpgUlSpgdlRBCJHb5sp4CJzBQ/+2MjdXvu7vrfrpdu+om39mymRunyJyiYqLoFtSNG9E3\n6FShE/1r9Dc7JLuLjdVP0n76SQ+ed+HC3c88POCpp3R/+V69dJNFb2/bx2AYekAhHx/dWi1eaKge\nYOiXX/Tr2DGYNEm/qleH/v2he3fHG6G9Z+WejAkew9LDS7kRdYPsHtnNDkmaRwL895+eqDM0FN5+\nGyZMMDsikVkppTvmL12qK3GbNjl2W3ArNEFy5NwkhKO4dg2WLdMVtbVrISZGv+/mBs2a6Sly2rWD\nnDnNjTOeFXITSH5Ki2FrhjFl2xSK5yzO3oF7yeWZy+yQ7ELdmd9wzhz9sOHff+9+VqyYPh9bt4YG\nDexTSUuLmBhduVywAIKC7janzJYNXnhBdw9xpJvVDb5rwKbTm5jdbja9q/a22XalT1saxcXpNrar\nVkk/NuEYLl/WHXfPnIGRI+H9982O6OGscGHkqLlJCLOFh8PPP+sLrNWr9QhyAK6u0LixfqLWoYMe\nBdLRWCE3geSn1FpxdAXPzH8GNxc3NvXdRN0idc0OyeauXoW5c+HrrxOPxlqqlO432rmzvoZw9H5j\nt27pJ/ZffQUbNuj3XFz0DaC333aMkdtn7ZpF/xX9aVKiCb/1+s1m25V52tJo6lRdYcuTR58Ej6qw\nWWGOCyuUAaxRjoeVIU8emD9fXxh98AH8+mvGxiWcjxXOB7BGOZy9DJGRMH58CJ07Q/78eiqS5cv1\n4AP+/vDll7pVytq1+s64I1bYhGPJqHPi3PVz9FnaB4B3Gr9j0wqbI5zXe/borhOFCsGrr+oKW548\n8MorsG2bbnb47rtQrVrSFTZHKMO9PD11fgkJ0WXp1Utf98ybBxUrwvPP6ykI7peR5ehUoRPuru6s\nP7Ge0BuhGbbfh8nUlbY//4RRo/TP33wDRYuaG48Q8erV03O3AfTubZuheIUQIim3bummjz166Ipa\nQIBuunTzps5Fn32mR4cLDoaBA3WfFSEcSUxcDN0XdefSzUu0KNWCEU+OMDskm4iN1X3UGjTQ/b++\n+Uafl02a6Cfg58/DtGm6O4WjP1l7lCpV4PvvdcVzwABdefvxRz19wKhReg45M+T2yk3Lx1qiUAQd\nDDIniHtk2uaRN29CzZpw8KA+QGbMMDsiIRKLjYVGjXS/tnbtdOJ2tKRshSZIjpabhMgI0dF6epvA\nQN2H9t6Lolq1dBOlzp2d92amFXITSH5KqbHBY5m4cSIFsxVk78C95M+a3+yQ0uXWLZg9Gz76CP75\nR7+XIwf07QuDBunKjJWdPAmjR+unbqCnIJgyRTf/zOjroPipI54s+iRb+m2xyTalT1sqvfoqTJ+u\n28zu2uU4nTSFuNepU/oO1LVr+sbCgAFmR5SYFS6MHC03CWEvMTH6aVlgoO5Lcu+cStWq6T5qnTtD\nyZLmxWgrVshNIPkpJdafWE/TH5oCsK7XOhqVaGRyRGl344ZugvzJJ3dHgCxRQo8q3bevnrw6M9m5\nE4YM0c0/Qc/19tVXeiTMjBIeHU7+yfm5GXOTk0NPUjxX+ncufdpSYfVqXWHLkkXX4lNaYXO09sBp\nYYUygDXKkZIyFC9+9ynwsGG66YAQ97PC+QDWKIejlSE2VvcZefll3RemeXPdxOrKFd1vZMIEOHIE\ndu/WAx+VLOl4ZRDOzZ7H04XwC/Rc3BOF4u0Gb9utwmbvc+LGDXjnHT259ciRusJWrZq+wXLsmK64\npLfC5oznda1asGWLrqjlzq370pYvH8KsWXr0zIyQzT0bz5R9BoCfDvyUMTt9iExXabt0Cfr10z9P\nnKhPCiEcWbduuq9JZKTumBs/1LYQQiQlLk5f6AwZops3Nmqk797/9x+UKaNHZtu/X7/eflu/J4Sz\niVNxPL/kef4N/5cGxRvwdsO3zQ4p1SIi9IBjJUrAmDF69Ognn9QD5O3apZsDurqaHaW5XFzgpZfg\n0CE9JdLNm3put2ee0TktI3St2BWAwAOBGbPDh8hUzSOV0s0/Fi7UEw2GhMjJIJzDlStQuTKcPatv\nNowZY3ZEmhWaIDlCbhIivZTSTYl++km/zpy5+1nJkvpvX9euOo84Wt9Ye7BCbgLJT4/y7qZ3Gb1+\nNPm887F3wF4K5yhsdkgpFh2th+yfMOFuM8h69fRyo0aZ4xxNC6X008eXX9bXRQUL6gFLGje2735v\n3r6J70e+3Ii+wdFXj1I6b+l0bU/6tKXAvHnQs6eexO/PP63Rbl9kHuvWQdOmelqK7dv1SFJms8KF\nkSPkJiHSQik9VHZgoK6onThx97OiRfVd+q5d9aBbme0i0Aq5CSQ/PcymU5to9H0jYlUsq3quouVj\nLc0OKUWU0g8O/vc/OH5cv1erlm4a2bRp5jtP0+r0aX09v3mzfhL3zju6Wak9//96LenFnL/mMKnR\nJEY3GJ2ubUmftmSEhurBR0B38ExLhc0Z2wPfzwplAGuUI7VlaNJEN3eKidHTAERF2Scu4XyscD6A\nNcqREWU4cADGjtUjyFWvrptXnTih7zoPGaKbRp48qUeeq1Ur9RcyVvgehOOw9fH0X8R/dF/UnVgV\ny5tPvpkhFTZblGHLFnjiCX0j5fhxPRDeokX6JmyzZvavsFnlvA4JCaFYMT2o0pgxujn4//4HnTrp\nvoH20rlCZwAWHVpkv50kI1NU2pTS7V+vXIFWrfTkhEI4o/feg9KldV+U8ePNjkYIkVGOHNFNpx5/\nXL8mTtQDFPj46CHAQ0J0k8ipU3WfGJdM8dddZDZxKo5eS3tx7sY56hWtx6TGk8wOKVknTuin3k89\npStovr56gLF9+3QfLXm6ljZubjoPLl+up0NYvFg3MT192j77a1aqGdnds7Pn3z38c/kf++wkGZmi\neeT330OfPpAzp75DWdh5mj0L8YCtW3XyNwz9c5065sVihSZI0vxIOKrjx3Wzx8BA3QwyXp480KGD\nHqTI319fvIjErJCbQPLT/d7f/D7/W/c/8nrlZc+APRTN6bgTCUZEwPvvw+TJumWMlxeMGAFvvqm7\n6QjbOXZMD0xy5IiuFC9fDrVr234/PRb1YP7++XzQ9APerPdmmrcjfdoe4vx5qFBBz3P1/ffQq1eG\nhyCEzb3xhm76VL487NkDHh7mxGGFCyO5KBKO5MyZuxW1nTvvvp8jh66ode2q+75kyWJejM7ACrkJ\nJD/da+OpjTT+vjGxKpaVPVbSunRrs0NKklL6HB4xQg8eBrr/1XvvOe9k9c7gyhXdRHL9el1BDgqC\n1jY+RBYfWsyzPz1L7cK12f7i9jRvR/q0JUEpGDhQV9jatNHDpaeHFdoDW6EMYI1ypKcMEyfq/iyH\nDumfReZmhfMBrFGOtJQhNBQ++0w37SlWTF/s7dwJWbNC9+6wdKkeYW72bN3E394VNit8D8Jx2OJ4\nCosIo1tQN2JVLKPqjcrwCltKy3DwoL6p0q2brrDVqKH7sv34o/kVNquc1w8rR+7ceh7mfv30tABt\n28KcObbdd8vHWuKdxZsd53Zw6uop2248BSxdaVuwAH7+Wd+hnDFD2g0L6/D01BPkGoZufrFnj9kR\nCSFSIyxMz53m76+b7A8dqps7e3lB5856hLmwMD3qcbt2+pwXIjOKjYvlucXPERoeSv1i9ZnY2PHu\nVEZE6NELq1TRT3ry5tUTQm/frvuYioyRJYueSuF//4PYWN267vPPbbd97yzeCTcMFh9abLsNp5Bl\nm0eGhelmkZcu6S/whRcybNdCZJjXXtMDD1Spou/MZ3STKSs0QZLmRyKjXL6sO8sHBuoLu7g4/b67\nu36C1q2bbhUi/V3Szwq5CSQ/AYwPGU/AhgB8vH3YM2CPw83H9vPPenTy06f1jdT+/fUQ9Hnzmh1Z\n5jZlCgwbpn+ePFm3YLCFwP2BdFvUjSeLPsmWflvStA3p03afHj1g/nz9mHrtWnnKJqwpIgIqVdKj\nU733HowalbH7t8KFkVwUCXu6dk03bwwMhF9/1VN2gB48pHlz3UetXTs9UJawHSvkJpD8tPaftbT8\nUQ/pv+a5NTQr1czkiO46e1ZPsbFkiV6uXl0/PbfHABgibb76CgYM0D+/8w689Vb6txkeHU6+D/MR\nHRvN+eHnKZCtQKq3IX3a7vHLL7rC5u2tvzBbVdis0B7YCmUAa5TDFmXImhVmztQ/BwToEZRE5mOF\n83CE7RcAACAASURBVAGsUY6QkBDCw+82a8yfX49evGqV7mfdrJlu/XHhAqxcqZvvOFqFzQrfg3Ac\naT2ezlw7Q49FPVAoAvwDTK2w3VuG2FiYNk235lqyRD8ZnzoVduxw7AqbVc7r1JSjf3/49ltdDxg9\nWg/gll7Z3LPRtGRTFIqfj/yc/g2mguUqbeHhes4a0HPalChhbjxC2FuzZncn2+7fX18YCiEyVmSk\nHq1s3DhdUevZUw87ffu27rf25Zd6wJG1a3Vz/Tx5zI5YCMcVHRtNl6AuXLp5iRalWjCmwRizQwL0\ntFFPPQWDB+uJnNu31wOCDRkCrq5mRyeS0revrriBHnn7iy/Sv8325doDsOzIsvRvLBUs1zwyvo9P\njRqwbZvMXyMyh0uX9PD///2XsX04rdAEKbM3PxJpFxWlRysLDNQVtIiIu5/Vq6ebPj77LBQqZF6M\nmZUVchNk3vw0+JfBTNs5jaI5irJ7wG7yeeczNZ6oKHj3Xd0N4fZtfU5Pm6an4RDO4csv4eWX9c/p\nnQLsQvgFCn5cEHdXd/574z+ye2RP1e9Lnzbgjz/0RMOGoQdlqFbNrrsTwqHMm6fv7ufODYcP67v9\n9maFC6PMelEk0iY6Gn77TVfUli6F69fvfla7tq6ode5s/vDemZ0VchNkzvw0b988ei7uiburO5v6\nbqJ2YXPbHG7froeRP3hQLw8YAB984HjNmkXyPv5YD0ji6qoHkGnVKu3bqvdtPbae2crCzgvpVKFT\nqn430/dpi4nRJ1JcnH7aZo8KmxXaA1uhDGCNcti6DN2766aSV67A8OE23bRwcFY4H8BxyxETowcR\nefFFKFAAnn4afvhBV9iqVtXTbhw/ri/uqlcPcfoKm6N+D8I5peZ42h+2n5d+fgmAqS2nmlphi4zU\nF/hPPgkHD4ZQujSEhOgppJyxwmaV8zo95Rg+XE/NEBurJ+Lenvb5sWlfNuObSFqm0jZ9OuzerScm\nDQgwOxohMp5h6Mf/np56Is9168yOSAjnFRurL9AGDdJNoZo313MjXrkCjz+uJ7U/ckTPkThypPSf\nFiK9rt66SofADkTejuT5ys8zoMYA02LZtElPpfPxx3q5Wzf4809o2NC0kISNvPeeHgcgMlLfgPvn\nn7Rtp125dgCsOLqC27G3bRjhw1mieeTZs7o/T3i47lfwzDN22Y0QTuGdd2DMGChdGv76y76T8lqh\nCVJmbH4kkhYXB7//rps+BgXpgUPilSmjmz527QoVK5oXo0gZK+QmyDz5KU7F0W5BO1YcXUHVAlXZ\n0m8L3lm8MzyOiAg9LPznn+tBvR5/XA9iUatWhoci7Oj2bWjbVvdJLlcOtm7VXUtSq8L0Chy6eIjf\nnv+NJiWbpPj37NY80jCMloZhHDYM45hhGCOT+NzfMIxrhmHsufPK8CF+hg3TFbYOHaTCJsQbb+ib\nGMeO6QklrcwZ8pNwbErpPtDDh4Ofnx4Z7vPPdYWtRAk99+GePbqf6IQJUmETKSO5KXUmbZzEiqMr\nyO2Zm8VdFptSYdu4UT9d++wz3efp7bf1WAlSYbOeLFn0zblKlXRu79RJV+RSq11Z/bTt56MZNPS/\nUuqhL8AV+BvwA7IAe4Hy963jDyx/1HburKfsYc0apUApb2+lTp+2yy4SBAcH23cHGcAKZVDKGuWw\nZxmCg/V54emp1D//2G036s55/chz314vW+Une+WmjGSF80GpjCtHXJxSu3crNXKkUiVK6HMl/lW0\nqFLDhyu1Y4deL7Ws8F1YoQxWyE0qk+Sn5YeXKwJQRoChVh1blTFB3SMiQqmhQ5UyDJ0DKldWateu\nxOtY4ZywQhmUsm05Tp1SytdXf++DBqX+9zef2qwIQJWcWlLFpeIPRlrzU3JP2moDfyulTiqlbgML\ngHZJrGdKE4SoKHj1Vf3zuHEyWpcQ8fz99UiSt27p+WMs2rrGofOTcDwHDui752XLQvXqegS4Eyeg\nYEF9nmzZAidP6glYa9XS/USFSAPJTSl0+OJhnlvyHADvNnmXlo+1zND9b9min65NnQouLjo/7Nyp\n84OwvmLFdLcqDw89JsDXX6fu9+sWqUter7wcv3KcI5eO2CfIezyyT5thGJ2AFkqpl+4sPwfUUUoN\nvmedhsBi4CxwDhihlDqYxLbUo/aVFpMm6ROsfHnYuxfc3W26eSGc2r//6ovT69f10OTtkrpkSCcz\n+43YKj9llj4jmdXRo7oZTGCgrrTF8/HRTWK6dtVNImViXGuxQm66s55l89O1W9eo83Udjlw6QucK\nnQnsFIiRQXdKbt3S148ff3y379r330tlLbOaPVtPwu3urgegeuKJlP9uryW9mPPXHCY3m8yIJ0ek\n6HfSmp+Sm3o6JZliN1BUKRVpGEYrYClQJqkV+/Tpg5+fHwC5cuWiatWq+Pv7A3eH8Ezp8oIFIUyY\nAODP9OmwdWvqfl+WZTkzLE+a5M+QITBgQAientCiRfq2F//zyZMncQA2y0+2zE2ybP7y+fNw+rQ/\ngYGwd6/+HPzJnRueeCKExo1h6FB/3Nz0+ps2OVb8siy56V5WzE/1G9Snx+IeHPnjCH65/fi23bcY\nhpEh+z98GD77zJ9Dh8AwQujRA7791h8PD8f5/5HljF3u08ef3bvh889DaNMGDh70x9c3Zb9f4qoe\nOnjF0RXUjK6Z5PrxP6c7Pz2q7SRQF1h9z/L/gJHJ/M4JIE8S76e4rWdKPPusboParZtNN/tIVmgP\nbIUyKGWNcmREGW7f1u3zQamAANtvH3P7jdgkP9k6N5nBCueDUukrx+nTSn30kVK1aqlEfdRy5FCq\nVy+lVq5UKirKdrE+jBW+CyuUwQq5SVk4P438daQiAJXngzzqn8t27Hh9j6gopcaMUcrVVeeGsmWV\n2rYtZb9rhXPCCmVQyn7liI5WqkEDfWw0bqxUTEzKfu/KzSvKbYKbch3vqi5HXk7R76Q1P7kkU6f7\nAyhtGIafYRjuQFdg+b0rGIbha9x5nm0YRm10k8vLqa49psJvv8GiReDtbf3R8YRIDzc3PRIe6AmA\nHeMmtM04ZH4SGSc0VI/0Vq+e7pswYoTuj5I1q55sftkyCAvTzZ5at5Ym9CLDSG56hHn75vHBlg9w\nNVwJ6hxEydwl7b7PffugTh3drSYuTo86vmePfk8I0CNKLlgA+fPD+vUwfnzKfi+XZy7qF6tPrIpl\n9d+r7RpjsvO03Xls/yl6NKRvlFLvGYYxAEApNdMwjFeAQUAMEAkMU0ptS2I7Krl9pcTt27rT6KFD\neoK8UaPSvUkhLK9nT5g3D9q3hyVLbLdds+dCskV+snKfESsKC9M37QID9RDd8V+dl5eeKLVrV11B\n8874EcOFA7FCbrqzHUvlp21nt+E/25+o2CimtZrGK7Vfsev+YmL0zf1x4/T1Y4kSuv9SgwZ23a1w\nYuvXQ9Om+ufVq6F58+R/55PfP2H42uH0qNSDuR3nJrt+WvOT002uPWWKvkNSurS+c+LhYYPghLC4\nc+f0oCQREbB2LTRrZpvtmn1hZAtWuyiyosuX9c2GwEBYt07fKQf95Kx1a11Ra9MGsmUzN07hOKyQ\nm8Ba+en0tdPUnlWbCxEXGFRzENNbT7frwCNHjkDv3rB9u14eNAg+/FDyhPh/e3caHkWZ/X38e7MJ\nEWQXBVQ2UQQXXEABJYjIoqKiAooiooz7yigu44YoMuOoqA84IKLOAEbQGVARwQWGYR8RUFYBQWT4\nh03DkkAgqefFSUwMIekk3anq4ve5rr7oTror9yFVJ3Wq7qVwzz0HTz0FderAsmV2960ga3as4ZQ3\nTqF6xepse3gbZcsUPLNVzBbXDpKtW3NuV778cukXbLkHFMarMMQA4YijNGOoVw/+lLV06wMP2NVH\nCY8wHA/w+zhSUnK6NdapA7fdBjNm2LTc3brZ97ZutWKud+/gnIiF4XcRhhgkOGbOnMnu/bvpPqE7\nyXuT6diwI8O7DI9ZwZaZad2mW7a0gq1+ffj8cxgxovh5IgzHRBhigNKJ4/HHITERkpOhX7/Cl01q\nWrMpjas35pd9v7Bw88KYtSuuirYnn7Q/5F26WDcYEYncAw9Ao0awYgW8+abfrRE5VFoaTJhg3XiP\nPdb+WH72mf3B7NTJ1tBJToZPP4W+faFqVb9bLCKFycjMoPeHvVmavJSmNZsy8bqJlC9bPiY/68cf\noWNHuP9+yyc332y9siLp4iaSrWxZ+PvfoUYN+xv02muFfyZ7jcHP1n4Ws3bFTffIJUts/YyyZe1W\nZbNmUWycyBHiX/+Cq6+G6tXhhx+gZs2SbS8MXZDC1P0oHqWmWhGWlGT/7ttnX3cO2re3ro89ehTe\nPUUktzDkJghHfrp36r28segNalaqyfzb5tOkRpOo/wzPg1GjbDKiPXssX4waFZv1SeXIkX3OdNRR\n8M030Lz54d/76ZpPuXzC5ZxX9zwWDij4bluox7R5HnToALNm2d2CV16JcuNEjhDZdyy+/BLuuSdn\nZsniCsOJURhOiuLNvn3WXSkpCaZMsbGW2dq0se6O114Lxx/vXxslvoUhN0H856fh84fzwOcPUKFs\nBb7s+yXtTmwX9Z+xaZN1n54+3V5fd511haxVK+o/So5AAwZYL4+WLWH+/MPPQpx6IJUaw2qQnpFO\n8h+TqX107cNuM9Rj2iZPtoKtZk0bGOiXMPQHDkMMEI44/IjBObvoUaYMjBwJq1aVehMkBuLheEhP\nh6lTrbtSnTrWBXLCBCvYWrWCv/4VkpJmMmcO3Htv/BZs8fC7KEwYYhD/TVoxiQc/fxB+hLFXjo16\nweZ5NhNkixZWsNWoYVO2f/BB9Au2MBwTYYgBSj+Ol1+GBg1siYghQw7/voTyCbRv0B4Pj8/XfR6T\ntgS+aEtPh4cftufPPGPdukSk+E4/3a5KZmTkHFsisXDwoE0ectttcNxxNhb5vfdg1y67ajl0KKxb\nZ5MFPPSQukCKhMXsjbO58aMb8fAYcM4Abjj9hqhu/3//gyuugFtusXxyxRWwfLl1pxaJpipVbOIr\n5+CFF2BhAT0fuzS2cW2xWq8t8N0js6f4P/VUG8tWPjZjV0WOKMnJtmzG7t12Up29JklRhaELUrx3\nPwqajAyYPdu6Pn74IWzblvO9Fi3spKpnT2ja1L82SviFITdBfOan5VuXc+HYC/ll3y9Rn9rf82yC\niPvvh19/hWrVbJKIG2+0k2qRWBk40O66NW9u49vym8F+1fZVNPt/zaiVUIvkPyZTxuV/byyUY9p2\n7IAmTezA/OQTzRgpEk0vvgiPPQZnnAGLF9skP0UVhhOjeDwpCprMTJg3z7olTZwIW7bkfO+UU3IK\ntYIGcYtEUxhyE8Rffvop5SfajGnD5t2b6X5Kdz7s+SHlypSLyrY3b4Y77rDzQbClP0aNsiVtRGIt\nNRXOPBPWrrWhWtlLkOXmeR6NXmvEhl83sOC2BbSq1yrfbYVyTNtzz1nBdskldnD6LQz9gcMQA4Qj\nDr9jeOABOOkku4P97ru+NkVKyI99yfOsm8jAgbYftWtnV7y3bIGGDeHRR20MwMqV9sctkoLN72Mi\nGhSDHKm2p26n8z86s3n3Ztqd2I73r3mfcmXKlXh/8jwYO9ZyyCef2FIfY8fa89Iq2MJwTIQhBvAv\njoQEGDPGnr/wAixdeuh7nHN0bdIVgOnrpke9DYEt2tats9l/nIOXXtJtb5Foq1jREg/YGoi5Z/AT\nyY/n2fIrjz0GjRtD69bWXeTnn+GEE6yAW7jQ8vfQoXDWWcrdIkeCXft30XVcV1ZtX8Xpx57Ox9d/\nTKXylUq83Y0bbW3e/v1tnd7LL7exa/36KbdI6bvoIrjrLhuvfeut9m9elza2RQFjUbQFtntkz57W\nzaZfP7uiIiLRl5lpM/d9843d2f7Tn4r2+TB0QYq37kd+WL7cxqglJcGaNTlfP/54m167Vy84/3yb\nlVQkCMKQmyA+8lPqgVS6juvKvzf+m0bVGzH7ltnUrVK3RNvMzIQ334RBg2zdtRo1YPhw6NNHxZr4\na/duu+u7aRO8+qqNr8wtZV8KNf9cE+ccOx/ZSZWjqhyyjVCNaZs3z9bqqVTJThDq149x40SOYDNn\n2jqIlStbX+06dSL/bBhOjOLhpMgPq1fbGLWkJCvastWuDddcY2uptWtXvLGQIrEWhtwEwc9P+w/u\n56qkq5i2dhr1qtRj9i2zaVi9YYm2uXKlzTg7d669vvZaeOONov1tEomlKVNs4fbKlW1/zVuntH27\nLXM3zWVK7ylcccoVh3w+NGPaPA8eecSeP/hgsAq2MPQHDkMMEI44ghJDYqJNl7xnT/4DayX4orUv\nrV9v3RpbtrQZe596ygq2GjXsJGrGDJtqe+RIaN8++gVbUI6JklAMcqRIz0in56SeTFs7jVoJtZhx\n04x8C7ZI96f0dBg82LpVz51ry4R8+KH1uvK7YAvDMRGGGCAYcXTvDldfbedNee+0AVzaKDZdJANX\ntH3yCfznP7Yw4qBBfrdG5MgwbJh1bRs16vfd3yT8Nm2yha1btbJxao8/buPWqlaFvn1tQez/+z8Y\nPdomhSoXnYngRCSOHcg4wPUfXs+U1VOoXrE6M26aQbPazYq9vTlz7GLR009b8XbrrXYHo0ePKDZa\nJIpee83utH30Uc6Mptk6Ne4EwPT10S3aAtU9MiPDph9fscL6Lt93X6k0TUSwOyljxtgYpQ8+iOwz\nYeiCFPTuR7GwZQtMmgTvv5/TBQng6KOty0evXtC5c/7r0IjEgzDkJghmfjqQcYAbPrqBSSsmUfWo\nqnzZ90vOqXtOsba1c6fNNDt6tL0++WS7eJiYGL32isTKq69ar8BGjaxXSsWK9vWDmQep+eea7Nq/\niw33b+Ckaif97nOh6B753ntWsDVsCLff7ndrRI4szzxjCWfiRJsBUMJl61br1piYaNNk33efFWyV\nKtmYkYkT7T3jxlnXDxVsIpLX/oP7uW7idb8VbNNvml6sgi17kexmzaxgK18ennjClqBRwSbx4p57\noEULG1rw0ks5Xy9XphwdG3YEYMb6GVH7eYEp2tLSbPwEwJAhwTxhCEI/2pIKQwwQjjiCFkP9+jl9\nswcNsj+qEh8Oty/t3AlvvQWdOtlMj3fdBbNmQYUKcNVVMGGCFWoTJ1rhlpBQuu3OK2jHRHEoBgmr\ntANp9PigB5NXT6Z6xep80feLwy4enFve/Wn5cpv8qm9fyz8XXmhrXg0ZknOnImjCcEyEIQYIVhzl\nysHrr9vzF16wJSqydWqU1UUyiuPaAlO0jRhha/2cdZbNSiY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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig = plt.figure(figsize=(15,3))\n", "ax = fig.add_subplot(131)\n", "x = np.linspace(0, 1, 100)\n", "pbeta = beta.pdf(x, 2, 2)\n", "ax.plot(x, pbeta,lw=2)\n", "plt.xlim(0,1)\n", "plt.ylim(0,2)\n", "plt.xlabel('$\\mu$')\n", "plt.grid(True)\n", "plt.text(0.05, 1.7, \"prior\", fontsize=15)\n", "\n", "ax = fig.add_subplot(132)\n", "x = np.linspace(0, 1, 100)\n", "pbin = pbinom(1, 1, x)\n", "ax.plot(x, pbin,lw=2)\n", "plt.xlim(0,1)\n", "plt.ylim(0,2)\n", "plt.xlabel('$\\mu$')\n", "plt.grid(True)\n", "plt.text(0.05, 1.7, \"likelihood function\", fontsize=15)\n", "\n", "ax = fig.add_subplot(133)\n", "x = np.linspace(0, 1, 100)\n", "post = pbeta * x\n", "ax.plot(x, post,lw=2)\n", "ax.plot(x, beta.pdf(x, 3, 2), lw=2)\n", "plt.xlim(0,1)\n", "plt.ylim(0,2)\n", "plt.xlabel('$\\mu$')\n", "plt.grid(True)\n", "plt.text(0.05, 1.7, \"posterior\", fontsize=15)\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "上面第三幅图中,绿色的线是实际的后验概率的曲线,蓝色曲线是直接将先验概率和似然函数相乘后,没有进行归一化的结果。\n", "\n", "图中,beta先验分布的参数是a=2,b=2,似然函数N=m=1,对应获得一个x=1的样本数据,修正的beta后验分布参数是a=3,b=2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 补充: 贝叶斯推断的三个步骤\n", "- **步骤1:指定概率模型。**,贝叶斯统计学通过使用概率模型来解决问题,所以第一步是要指定具体的概率分布,这一步要做的考虑的问题是,样本的分布模型是什么,什么样的形式可以最好的描述未知参数的不确定性。\n", "- **步骤二:计算后验概率。**后验概率告诉了我们经过样本数据的观测后未知参数是什么样子,其中有多中贝叶斯计算方法来计算后验概率。一旦后验概率计算得到,再得到点估计、区间估计、分位数和预测就相对容易了。\n", "- **步骤三:验证模型。**在将模型进行统计推断之前,我们需要对模型和输出进行评估。我们需要问自己,模型是不是对数据拟合,推论的结果是不是合理,结果是不是对于模型结构的变化很敏感?" ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.5" } }, "nbformat": 4, "nbformat_minor": 0 }