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<!DOCTYPE html>
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<title>2 MathBasics | 机器学习白板系列</title>
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<li class="chapter" data-level="1" data-path="introduction.html"><a href="introduction.html"><i class="fa fa-check"></i><b>1</b> Introduction</a>
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<li class="chapter" data-level="1.1" data-path="introduction.html"><a href="introduction.html#频率派的观点"><i class="fa fa-check"></i><b>1.1</b> 频率派的观点</a></li>
<li class="chapter" data-level="1.2" data-path="introduction.html"><a href="introduction.html#贝叶斯派的观点"><i class="fa fa-check"></i><b>1.2</b> 贝叶斯派的观点</a></li>
<li class="chapter" data-level="1.3" data-path="introduction.html"><a href="introduction.html#小结"><i class="fa fa-check"></i><b>1.3</b> 小结</a></li>
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<li class="chapter" data-level="2" data-path="mathbasics.html"><a href="mathbasics.html"><i class="fa fa-check"></i><b>2</b> MathBasics</a>
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<li class="chapter" data-level="2.1" data-path="mathbasics.html"><a href="mathbasics.html#高斯分布"><i class="fa fa-check"></i><b>2.1</b> 高斯分布</a>
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<li class="chapter" data-level="2.1.1" data-path="mathbasics.html"><a href="mathbasics.html#一维情况-mle"><i class="fa fa-check"></i><b>2.1.1</b> 一维情况 MLE</a></li>
<li class="chapter" data-level="2.1.2" data-path="mathbasics.html"><a href="mathbasics.html#多维情况"><i class="fa fa-check"></i><b>2.1.2</b> 多维情况</a></li>
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<li class="chapter" data-level="3" data-path="线性回归.html"><a href="线性回归.html"><i class="fa fa-check"></i><b>3</b> 线性回归</a>
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<li class="chapter" data-level="3.1" data-path="线性回归.html"><a href="线性回归.html#最小二乘法"><i class="fa fa-check"></i><b>3.1</b> 最小二乘法</a></li>
<li class="chapter" data-level="3.2" data-path="线性回归.html"><a href="线性回归.html#噪声为高斯分布的-mle"><i class="fa fa-check"></i><b>3.2</b> 噪声为高斯分布的 MLE</a></li>
<li class="chapter" data-level="3.3" data-path="线性回归.html"><a href="线性回归.html#权重先验也为高斯分布的-map"><i class="fa fa-check"></i><b>3.3</b> 权重先验也为高斯分布的 MAP</a></li>
<li class="chapter" data-level="3.4" data-path="线性回归.html"><a href="线性回归.html#正则化"><i class="fa fa-check"></i><b>3.4</b> 正则化</a>
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<li class="chapter" data-level="3.4.1" data-path="线性回归.html"><a href="线性回归.html#l1-lasso"><i class="fa fa-check"></i><b>3.4.1</b> L1 Lasso</a></li>
<li class="chapter" data-level="3.4.2" data-path="线性回归.html"><a href="线性回归.html#l2-ridge"><i class="fa fa-check"></i><b>3.4.2</b> L2 Ridge</a></li>
</ul></li>
<li class="chapter" data-level="3.5" data-path="线性回归.html"><a href="线性回归.html#小结-1"><i class="fa fa-check"></i><b>3.5</b> 小结</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="线性分类.html"><a href="线性分类.html"><i class="fa fa-check"></i><b>4</b> 线性分类</a>
<ul>
<li class="chapter" data-level="4.1" data-path="线性分类.html"><a href="线性分类.html#两分类-硬分类-感知机算法"><i class="fa fa-check"></i><b>4.1</b> 两分类-硬分类-感知机算法</a></li>
<li class="chapter" data-level="4.2" data-path="线性分类.html"><a href="线性分类.html#两分类-硬分类-线性判别分析-lda"><i class="fa fa-check"></i><b>4.2</b> 两分类-硬分类-线性判别分析 LDA</a></li>
<li class="chapter" data-level="4.3" data-path="线性分类.html"><a href="线性分类.html#两分类-软分类-概率判别模型-logistic-回归"><i class="fa fa-check"></i><b>4.3</b> 两分类-软分类-概率判别模型-Logistic 回归</a></li>
<li class="chapter" data-level="4.4" data-path="线性分类.html"><a href="线性分类.html#两分类-软分类-概率生成模型-高斯判别分析-gda"><i class="fa fa-check"></i><b>4.4</b> 两分类-软分类-概率生成模型-高斯判别分析 GDA</a></li>
<li class="chapter" data-level="4.5" data-path="线性分类.html"><a href="线性分类.html#两分类-软分类-概率生成模型-朴素贝叶斯"><i class="fa fa-check"></i><b>4.5</b> 两分类-软分类-概率生成模型-朴素贝叶斯</a></li>
<li class="chapter" data-level="4.6" data-path="线性分类.html"><a href="线性分类.html#小结-2"><i class="fa fa-check"></i><b>4.6</b> 小结</a></li>
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<li class="chapter" data-level="5" data-path="降维.html"><a href="降维.html"><i class="fa fa-check"></i><b>5</b> 降维</a>
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<li class="chapter" data-level="5.1" data-path="降维.html"><a href="降维.html#线性降维-主成分分析-pca"><i class="fa fa-check"></i><b>5.1</b> 线性降维-主成分分析 PCA</a>
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<li class="chapter" data-level="5.1.1" data-path="降维.html"><a href="降维.html#损失函数"><i class="fa fa-check"></i><b>5.1.1</b> 损失函数</a></li>
<li class="chapter" data-level="5.1.2" data-path="降维.html"><a href="降维.html#svd-与-pcoa"><i class="fa fa-check"></i><b>5.1.2</b> SVD 与 PCoA</a></li>
<li class="chapter" data-level="5.1.3" data-path="降维.html"><a href="降维.html#p-pca"><i class="fa fa-check"></i><b>5.1.3</b> p-PCA</a></li>
</ul></li>
<li class="chapter" data-level="5.2" data-path="降维.html"><a href="降维.html#小结-3"><i class="fa fa-check"></i><b>5.2</b> 小结</a></li>
</ul></li>
<li class="chapter" data-level="6" data-path="支撑向量机.html"><a href="支撑向量机.html"><i class="fa fa-check"></i><b>6</b> 支撑向量机</a>
<ul>
<li class="chapter" data-level="6.1" data-path="支撑向量机.html"><a href="支撑向量机.html#约束优化问题"><i class="fa fa-check"></i><b>6.1</b> 约束优化问题</a></li>
<li class="chapter" data-level="6.2" data-path="支撑向量机.html"><a href="支撑向量机.html#hard-margin-svm"><i class="fa fa-check"></i><b>6.2</b> Hard-margin SVM</a></li>
<li class="chapter" data-level="6.3" data-path="支撑向量机.html"><a href="支撑向量机.html#soft-margin-svm"><i class="fa fa-check"></i><b>6.3</b> Soft-margin SVM</a></li>
<li class="chapter" data-level="6.4" data-path="支撑向量机.html"><a href="支撑向量机.html#kernel-method"><i class="fa fa-check"></i><b>6.4</b> Kernel Method</a></li>
<li class="chapter" data-level="6.5" data-path="支撑向量机.html"><a href="支撑向量机.html#小结-4"><i class="fa fa-check"></i><b>6.5</b> 小结</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="指数族分布.html"><a href="指数族分布.html"><i class="fa fa-check"></i><b>7</b> 指数族分布</a>
<ul>
<li class="chapter" data-level="7.1" data-path="指数族分布.html"><a href="指数族分布.html#一维高斯分布"><i class="fa fa-check"></i><b>7.1</b> 一维高斯分布</a></li>
<li class="chapter" data-level="7.2" data-path="指数族分布.html"><a href="指数族分布.html#充分统计量和对数配分函数的关系"><i class="fa fa-check"></i><b>7.2</b> 充分统计量和对数配分函数的关系</a></li>
<li class="chapter" data-level="7.3" data-path="指数族分布.html"><a href="指数族分布.html#充分统计量和极大似然估计"><i class="fa fa-check"></i><b>7.3</b> 充分统计量和极大似然估计</a></li>
<li class="chapter" data-level="7.4" data-path="指数族分布.html"><a href="指数族分布.html#最大熵"><i class="fa fa-check"></i><b>7.4</b> 最大熵</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="概率图模型.html"><a href="概率图模型.html"><i class="fa fa-check"></i><b>8</b> 概率图模型</a>
<ul>
<li class="chapter" data-level="8.1" data-path="概率图模型.html"><a href="概率图模型.html#有向图-贝叶斯网络"><i class="fa fa-check"></i><b>8.1</b> 有向图-贝叶斯网络</a></li>
<li class="chapter" data-level="8.2" data-path="概率图模型.html"><a href="概率图模型.html#无向图-马尔可夫网络马尔可夫随机场"><i class="fa fa-check"></i><b>8.2</b> 无向图-马尔可夫网络（马尔可夫随机场）</a></li>
<li class="chapter" data-level="8.3" data-path="概率图模型.html"><a href="概率图模型.html#两种图的转换-道德图"><i class="fa fa-check"></i><b>8.3</b> 两种图的转换-道德图</a></li>
<li class="chapter" data-level="8.4" data-path="概率图模型.html"><a href="概率图模型.html#更精细的分解-因子图"><i class="fa fa-check"></i><b>8.4</b> 更精细的分解-因子图</a></li>
<li class="chapter" data-level="8.5" data-path="概率图模型.html"><a href="概率图模型.html#推断"><i class="fa fa-check"></i><b>8.5</b> 推断</a>
<ul>
<li class="chapter" data-level="8.5.1" data-path="概率图模型.html"><a href="概率图模型.html#推断-变量消除ve"><i class="fa fa-check"></i><b>8.5.1</b> 推断-变量消除（VE）</a></li>
<li class="chapter" data-level="8.5.2" data-path="概率图模型.html"><a href="概率图模型.html#推断-信念传播bp"><i class="fa fa-check"></i><b>8.5.2</b> 推断-信念传播（BP）</a></li>
<li class="chapter" data-level="8.5.3" data-path="概率图模型.html"><a href="概率图模型.html#推断-max-product-算法"><i class="fa fa-check"></i><b>8.5.3</b> 推断-Max-Product 算法</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="9" data-path="期望最大.html"><a href="期望最大.html"><i class="fa fa-check"></i><b>9</b> 期望最大</a>
<ul>
<li class="chapter" data-level="9.1" data-path="期望最大.html"><a href="期望最大.html#广义-em"><i class="fa fa-check"></i><b>9.1</b> 广义 EM</a></li>
<li class="chapter" data-level="9.2" data-path="期望最大.html"><a href="期望最大.html#em-的推广"><i class="fa fa-check"></i><b>9.2</b> EM 的推广</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="高斯混合模型.html"><a href="高斯混合模型.html"><i class="fa fa-check"></i><b>10</b> 高斯混合模型</a>
<ul>
<li class="chapter" data-level="10.1" data-path="高斯混合模型.html"><a href="高斯混合模型.html#极大似然估计"><i class="fa fa-check"></i><b>10.1</b> 极大似然估计</a></li>
<li class="chapter" data-level="10.2" data-path="高斯混合模型.html"><a href="高斯混合模型.html#em-求解-gmm"><i class="fa fa-check"></i><b>10.2</b> EM 求解 GMM</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="变分推断.html"><a href="变分推断.html"><i class="fa fa-check"></i><b>11</b> 变分推断</a>
<ul>
<li class="chapter" data-level="11.1" data-path="变分推断.html"><a href="变分推断.html#基于平均场假设的变分推断"><i class="fa fa-check"></i><b>11.1</b> 基于平均场假设的变分推断</a></li>
<li class="chapter" data-level="11.2" data-path="变分推断.html"><a href="变分推断.html#sgvi"><i class="fa fa-check"></i><b>11.2</b> SGVI</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html"><i class="fa fa-check"></i><b>12</b> 马尔可夫链蒙特卡洛</a>
<ul>
<li class="chapter" data-level="12.1" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#蒙特卡洛方法"><i class="fa fa-check"></i><b>12.1</b> 蒙特卡洛方法</a></li>
<li class="chapter" data-level="12.2" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#mcmc"><i class="fa fa-check"></i><b>12.2</b> MCMC</a></li>
<li class="chapter" data-level="12.3" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#平稳分布"><i class="fa fa-check"></i><b>12.3</b> 平稳分布</a></li>
<li class="chapter" data-level="12.4" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#隐马尔可夫模型"><i class="fa fa-check"></i><b>12.4</b> 隐马尔可夫模型</a></li>
<li class="chapter" data-level="12.5" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#hmm"><i class="fa fa-check"></i><b>12.5</b> HMM</a>
<ul>
<li class="chapter" data-level="12.5.1" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#evaluation"><i class="fa fa-check"></i><b>12.5.1</b> Evaluation</a></li>
<li class="chapter" data-level="12.5.2" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#learning"><i class="fa fa-check"></i><b>12.5.2</b> Learning</a></li>
<li class="chapter" data-level="12.5.3" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#decoding"><i class="fa fa-check"></i><b>12.5.3</b> Decoding</a></li>
</ul></li>
<li class="chapter" data-level="12.6" data-path="马尔可夫链蒙特卡洛.html"><a href="马尔可夫链蒙特卡洛.html#小结-5"><i class="fa fa-check"></i><b>12.6</b> 小结</a></li>
</ul></li>
<li class="chapter" data-level="13" data-path="线性动态系统.html"><a href="线性动态系统.html"><i class="fa fa-check"></i><b>13</b> 线性动态系统</a></li>
<li class="chapter" data-level="14" data-path="粒子滤波.html"><a href="粒子滤波.html"><i class="fa fa-check"></i><b>14</b> 粒子滤波</a>
<ul>
<li class="chapter" data-level="14.1" data-path="粒子滤波.html"><a href="粒子滤波.html#sis"><i class="fa fa-check"></i><b>14.1</b> SIS</a></li>
</ul></li>
<li class="chapter" data-level="15" data-path="条件随机场.html"><a href="条件随机场.html"><i class="fa fa-check"></i><b>15</b> 条件随机场</a>
<ul>
<li class="chapter" data-level="15.1" data-path="条件随机场.html"><a href="条件随机场.html#crf-的-pdf"><i class="fa fa-check"></i><b>15.1</b> CRF 的 PDF</a></li>
<li class="chapter" data-level="15.2" data-path="条件随机场.html"><a href="条件随机场.html#边缘概率"><i class="fa fa-check"></i><b>15.2</b> 边缘概率</a></li>
<li class="chapter" data-level="15.3" data-path="条件随机场.html"><a href="条件随机场.html#参数估计"><i class="fa fa-check"></i><b>15.3</b> 参数估计</a></li>
<li class="chapter" data-level="15.4" data-path="条件随机场.html"><a href="条件随机场.html#译码"><i class="fa fa-check"></i><b>15.4</b> 译码</a></li>
</ul></li>
<li class="chapter" data-level="16" data-path="高斯网络.html"><a href="高斯网络.html"><i class="fa fa-check"></i><b>16</b> 高斯网络</a>
<ul>
<li class="chapter" data-level="16.1" data-path="高斯网络.html"><a href="高斯网络.html#高斯贝叶斯网络-gbn"><i class="fa fa-check"></i><b>16.1</b> 高斯贝叶斯网络 GBN</a></li>
<li class="chapter" data-level="16.2" data-path="高斯网络.html"><a href="高斯网络.html#高斯马尔可夫网络-gmn"><i class="fa fa-check"></i><b>16.2</b> 高斯马尔可夫网络 GMN</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="贝叶斯线性回归.html"><a href="贝叶斯线性回归.html"><i class="fa fa-check"></i><b>17</b> 贝叶斯线性回归</a>
<ul>
<li class="chapter" data-level="17.1" data-path="贝叶斯线性回归.html"><a href="贝叶斯线性回归.html#推断-1"><i class="fa fa-check"></i><b>17.1</b> 推断</a></li>
<li class="chapter" data-level="17.2" data-path="贝叶斯线性回归.html"><a href="贝叶斯线性回归.html#预测"><i class="fa fa-check"></i><b>17.2</b> 预测</a></li>
</ul></li>
<li class="chapter" data-level="18" data-path="高斯过程回归.html"><a href="高斯过程回归.html"><i class="fa fa-check"></i><b>18</b> 高斯过程回归</a>
<ul>
<li class="chapter" data-level="18.1" data-path="高斯过程回归.html"><a href="高斯过程回归.html#核贝叶斯线性回归"><i class="fa fa-check"></i><b>18.1</b> 核贝叶斯线性回归</a></li>
<li class="chapter" data-level="18.2" data-path="高斯过程回归.html"><a href="高斯过程回归.html#函数空间的观点"><i class="fa fa-check"></i><b>18.2</b> 函数空间的观点</a></li>
</ul></li>
<li class="chapter" data-level="19" data-path="受限玻尔兹曼机.html"><a href="受限玻尔兹曼机.html"><i class="fa fa-check"></i><b>19</b> 受限玻尔兹曼机</a>
<ul>
<li class="chapter" data-level="19.1" data-path="受限玻尔兹曼机.html"><a href="受限玻尔兹曼机.html#推断-2"><i class="fa fa-check"></i><b>19.1</b> 推断</a>
<ul>
<li class="chapter" data-level="19.1.1" data-path="受限玻尔兹曼机.html"><a href="受限玻尔兹曼机.html#phv"><i class="fa fa-check"></i><b>19.1.1</b> <span class="math inline">\(p(h|v)\)</span></a></li>
<li class="chapter" data-level="19.1.2" data-path="受限玻尔兹曼机.html"><a href="受限玻尔兹曼机.html#pv"><i class="fa fa-check"></i><b>19.1.2</b> <span class="math inline">\(p(v)\)</span></a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="20" data-path="谱聚类.html"><a href="谱聚类.html"><i class="fa fa-check"></i><b>20</b> 谱聚类</a></li>
<li class="chapter" data-level="21" data-path="前馈神经网络.html"><a href="前馈神经网络.html"><i class="fa fa-check"></i><b>21</b> 前馈神经网络</a>
<ul>
<li class="chapter" data-level="21.1" data-path="前馈神经网络.html"><a href="前馈神经网络.html#from-pla-to-dl"><i class="fa fa-check"></i><b>21.1</b> From PLA to DL</a></li>
<li class="chapter" data-level="21.2" data-path="前馈神经网络.html"><a href="前馈神经网络.html#非线性问题"><i class="fa fa-check"></i><b>21.2</b> 非线性问题</a></li>
</ul></li>
<li class="chapter" data-level="22" data-path="配分函数.html"><a href="配分函数.html"><i class="fa fa-check"></i><b>22</b> 配分函数</a>
<ul>
<li class="chapter" data-level="22.1" data-path="配分函数.html"><a href="配分函数.html#包含配分函数的-mle"><i class="fa fa-check"></i><b>22.1</b> 包含配分函数的 MLE</a></li>
<li class="chapter" data-level="22.2" data-path="配分函数.html"><a href="配分函数.html#对比散度-cd-learning"><i class="fa fa-check"></i><b>22.2</b> 对比散度-CD Learning</a></li>
<li class="chapter" data-level="22.3" data-path="配分函数.html"><a href="配分函数.html#rbm-的学习问题"><i class="fa fa-check"></i><b>22.3</b> RBM 的学习问题</a></li>
</ul></li>
<li class="chapter" data-level="23" data-path="近似推断.html"><a href="近似推断.html"><i class="fa fa-check"></i><b>23</b> 近似推断</a></li>
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<div id="mathbasics" class="section level1 hasAnchor" number="2">
<h1><span class="header-section-number">2</span> MathBasics<a href="mathbasics.html#mathbasics" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<div id="高斯分布" class="section level2 hasAnchor" number="2.1">
<h2><span class="header-section-number">2.1</span> 高斯分布<a href="mathbasics.html#高斯分布" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div id="一维情况-mle" class="section level3 hasAnchor" number="2.1.1">
<h3><span class="header-section-number">2.1.1</span> 一维情况 MLE<a href="mathbasics.html#一维情况-mle" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>高斯分布在机器学习中占有举足轻重的作用。在 MLE 方法中：</p>
<p><span class="math display">\[
\theta=(\mu,\Sigma)=(\mu,\sigma^{2}),\theta_{MLE}=\mathop{argmax}\limits _{\theta}\log p(X|\theta)\mathop{=}\limits _{iid}\mathop{argmax}\limits _{\theta}\sum\limits _{i=1}^{N}\log p(x_{i}|\theta)
\]</span>
一般地，高斯分布的概率密度函数PDF写为：</p>
<p><span class="math display">\[
p(x|\mu,\Sigma)=\frac{1}{(2\pi)^{p/2}|\Sigma|^{1/2}}e^{-\frac{1}{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu)}
\]</span>
带入 MLE 中我们考虑一维的情况</p>
<p><span class="math display">\[
\log p(X|\theta)=\sum\limits _{i=1}^{N}\log p(x_{i}|\theta)=\sum\limits _{i=1}^{N}\log\frac{1}{\sqrt{2\pi}\sigma}\exp(-(x_{i}-\mu)^{2}/2\sigma^{2})
\]</span>
首先对 <span class="math inline">\(\mu\)</span> 的极值可以得到 ：
<span class="math display">\[
\mu_{MLE}=\mathop{argmax}\limits _{\mu}\log p(X|\theta)=\mathop{argmax}\limits _{\mu}\sum\limits _{i=1}^{N}(x_{i}-\mu)^{2}
\]</span>
于是：
<span class="math display">\[
\frac{\partial}{\partial\mu}\sum\limits _{i=1}^{N}(x_{i}-\mu)^{2}=0\longrightarrow\mu_{MLE}=\frac{1}{N}\sum\limits _{i=1}^{N}x_{i}
\]</span>
其次对 <span class="math inline">\(\theta\)</span> 中的另一个参数 <span class="math inline">\(\sigma\)</span> ，有：
<span class="math display">\[
\begin{align}
\sigma_{MLE}=\mathop{argmax}\limits _{\sigma}\log p(X|\theta)&=\mathop{argmax}\limits _{\sigma}\sum\limits _{i=1}^{N}[-\log\sigma-\frac{1}{2\sigma^{2}}(x_{i}-\mu)^{2}]\nonumber\\
&=\mathop{argmin}\limits _{\sigma}\sum\limits _{i=1}^{N}[\log\sigma+\frac{1}{2\sigma^{2}}(x_{i}-\mu)^{2}]
\end{align}
\]</span>
于是：
<span class="math display">\[
\frac{\partial}{\partial\sigma}\sum\limits _{i=1}^{N}[\log\sigma+\frac{1}{2\sigma^{2}}(x_{i}-\mu)^{2}]=0\longrightarrow\sigma_{MLE}^{2}=\frac{1}{N}\sum\limits _{i=1}^{N}(x_{i}-\mu)^{2}
\]</span>
值得注意的是，上面的推导中，首先对 <span class="math inline">\(\mu\)</span> 求 MLE， 然后利用这个结果求 <span class="math inline">\(\sigma_{MLE}\)</span> ，因此可以预期的是对数据集求期望时 <span class="math inline">\(\mathbb{E}_{\mathcal{D}}[\mu_{MLE}]\)</span> 是无偏差的：
<span class="math display">\[
\mathbb{E}_{\mathcal{D}}[\mu_{MLE}]=\mathbb{E}_{\mathcal{D}}[\frac{1}{N}\sum\limits _{i=1}^{N}x_{i}]=\frac{1}{N}\sum\limits _{i=1}^{N}\mathbb{E}_{\mathcal{D}}[x_{i}]=\mu
\]</span>
但是当对 <span class="math inline">\(\sigma_{MLE}\)</span> 求 期望的时候由于使用了单个数据集的 <span class="math inline">\(\mu_{MLE}\)</span>，因此对所有数据集求期望的时候我们会发现 <span class="math inline">\(\sigma_{MLE}\)</span> 是 有偏的：</p>
<p><span class="math display">\[
\begin{align}
\mathbb{E}_{\mathcal{D}}[\sigma_{MLE}^{2}]&=\mathbb{E}_{\mathcal{D}}[\frac{1}{N}\sum\limits _{i=1}^{N}(x_{i}-\mu_{MLE})^{2}]=\mathbb{E}_{\mathcal{D}}[\frac{1}{N}\sum\limits _{i=1}^{N}(x_{i}^{2}-2x_{i}\mu_{MLE}+\mu_{MLE}^{2})\nonumber
\\&=\mathbb{E}_{\mathcal{D}}[\frac{1}{N}\sum\limits _{i=1}^{N}x_{i}^{2}-\mu_{MLE}^{2}]=\mathbb{E}_{\mathcal{D}}[\frac{1}{N}\sum\limits _{i=1}^{N}x_{i}^{2}-\mu^{2}+\mu^{2}-\mu_{MLE}^{2}]\nonumber\\
&= \mathbb{E}_{\mathcal{D}}[\frac{1}{N}\sum\limits _{i=1}^{N}x_{i}^{2}-\mu^{2}]-\mathbb{E}_{\mathcal{D}}[\mu_{MLE}^{2}-\mu^{2}]=\sigma^{2}-(\mathbb{E}_{\mathcal{D}}[\mu_{MLE}^{2}]-\mu^{2})\nonumber\\&=\sigma^{2}-(\mathbb{E}_{\mathcal{D}}[\mu_{MLE}^{2}]-\mathbb{E}_{\mathcal{D}}^{2}[\mu_{MLE}])=\sigma^{2}-Var[\mu_{MLE}]\nonumber\\&=\sigma^{2}-Var[\frac{1}{N}\sum\limits _{i=1}^{N}x_{i}]=\sigma^{2}-\frac{1}{N^{2}}\sum\limits _{i=1}^{N}Var[x_{i}]=\frac{N-1}{N}\sigma^{2}
\end{align}
\]</span>
所以：
<span class="math display">\[
\hat{\sigma}^{2}=\frac{1}{N-1}\sum\limits _{i=1}^{N}(x_{i}-\mu)^{2}
\]</span></p>
</div>
<div id="多维情况" class="section level3 hasAnchor" number="2.1.2">
<h3><span class="header-section-number">2.1.2</span> 多维情况<a href="mathbasics.html#多维情况" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>多维高斯分布表达式为：
<span class="math display">\[
p(x|\mu,\Sigma)=\frac{1}{(2\pi)^{p/2}|\Sigma|^{1/2}}e^{-\frac{1}{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu)}
\]</span>
其中 <span class="math inline">\(x,\mu\in\mathbb{R}^{p},\Sigma\in\mathbb{R}^{p\times p}\)</span> ，<span class="math inline">\(\Sigma\)</span> 为协方差矩阵，一般而言也是半正定矩阵。这里我们只考虑正定矩阵。首先我们处理指数上的数字，指数上的数字可以记为 <span class="math inline">\(x\)</span> 和 <span class="math inline">\(\mu\)</span> 之间的马氏距离。对于对称的协方差矩阵可进行特征值分解，<span class="math inline">\(\Sigma=U\Lambda U^{T}=(u_{1},u_{2},\cdots,u_{p})diag(\lambda_{i})(u_{1},u_{2},\cdots,u_{p})^{T}=\sum\limits _{i=1}^{p}u_{i}\lambda_{i}u_{i}^{T}\)</span> ，于是：</p>
<p><span class="math display">\[
\Sigma^{-1}=\sum\limits _{i=1}^{p}u_{i}\frac{1}{\lambda_{i}}u_{i}^{T}
\]</span></p>
<p><span class="math display">\[
\Delta=(x-\mu)^{T}\Sigma^{-1}(x-\mu)=\sum\limits _{i=1}^{p}(x-\mu)^{T}u_{i}\frac{1}{\lambda_{i}}u_{i}^{T}(x-\mu)=\sum\limits _{i=1}^{p}\frac{y_{i}^{2}}{\lambda_{i}}
\]</span></p>
<p>我们注意到 <span class="math inline">\(y_{i}\)</span> 是 <span class="math inline">\(x-\mu\)</span> 在特征向量 <span class="math inline">\(u_{i}\)</span> 上的投影长度，因此上式子就是 <span class="math inline">\(\Delta\)</span> 取不同值时的同心椭圆。</p>
<p>下面我们看多维高斯模型在实际应用时的两个问题</p>
<ol style="list-style-type: decimal">
<li><p>参数 <span class="math inline">\(\Sigma,\mu\)</span> 的自由度为 <span class="math inline">\(O(p^{2})\)</span> 对于维度很高的数据其自由度太高。解决方案：高自由度的来源是 <span class="math inline">\(\Sigma\)</span> 有 <span class="math inline">\(\frac{p(p+1)}{2}\)</span> 个自由参数，可以假设其是对角矩阵，甚至在各向同性假设中假设其对角线上的元素都相同。前一种的算法有 Factor Analysis，后一种有概率 PCA(p-PCA) 。</p></li>
<li><p>第二个问题是单个高斯分布是单峰的，对有多个峰的数据分布不能得到好的结果。解决方案：高斯混合GMM 模型。</p></li>
</ol>
<p>下面对多维高斯分布的常用定理进行介绍。</p>
<p>我们记 <span class="math inline">\(x=(x_1, x_2,\cdots,x_p)^T=(x_{a,m\times 1}, x_{b,n\times1})^T,\mu=(\mu_{a,m\times1}, \mu_{b,n\times1}),\Sigma=\begin{pmatrix}\Sigma_{aa}&\Sigma_{ab}\\\Sigma_{ba}&\Sigma_{bb}\end{pmatrix}\)</span>，已知 <span class="math inline">\(x\sim\mathcal{N}(\mu,\Sigma)\)</span>。</p>
<p>首先是一个高斯分布的定理：</p>
<blockquote>
<p>定理：已知 <span class="math inline">\(x\sim\mathcal{N}(\mu,\Sigma), y\sim Ax+b\)</span>，那么 <span class="math inline">\(y\sim\mathcal{N}(A\mu+b, A\Sigma A^T)\)</span>。</p>
<p>证明：<span class="math inline">\(\mathbb{E}[y]=\mathbb{E}[Ax+b]=A\mathbb{E}[x]+b=A\mu+b\)</span>，<span class="math inline">\(Var[y]=Var[Ax+b]=Var[Ax]=A\cdot Var[x]\cdot A^T\)</span>。</p>
</blockquote>
<p>下面利用这个定理得到 <span class="math inline">\(p(x_a),p(x_b),p(x_a|x_b),p(x_b|x_a)\)</span> 这四个量。</p>
<ol style="list-style-type: decimal">
<li><p><span class="math inline">\(x_a=\begin{pmatrix}\mathbb{I}_{m\times m}&\mathbb{O}_{m\times n})\end{pmatrix}\begin{pmatrix}x_a\\x_b\end{pmatrix}\)</span>，代入定理中得到：
<span class="math display">\[
\mathbb{E}[x_a]=\begin{pmatrix}\mathbb{I}&\mathbb{O}\end{pmatrix}\begin{pmatrix}\mu_a\\\mu_b\end{pmatrix}=\mu_a\\
Var[x_a]=\begin{pmatrix}\mathbb{I}&\mathbb{O}\end{pmatrix}\begin{pmatrix}\Sigma_{aa}&\Sigma_{ab}\\\Sigma_{ba}&\Sigma_{bb}\end{pmatrix}\begin{pmatrix}\mathbb{I}\\\mathbb{O}\end{pmatrix}=\Sigma_{aa}
\]</span>
所以 <span class="math inline">\(x_a\sim\mathcal{N}(\mu_a,\Sigma_{aa})\)</span>。</p></li>
<li><p>同样的，<span class="math inline">\(x_b\sim\mathcal{N}(\mu_b,\Sigma_{bb})\)</span>。</p></li>
<li><p>对于两个条件概率，我们引入三个量：
<span class="math display">\[
x_{b\cdot a}=x_b-\Sigma_{ba}\Sigma_{aa}^{-1}x_a\\
\mu_{b\cdot a}=\mu_b-\Sigma_{ba}\Sigma_{aa}^{-1}\mu_a\\
\Sigma_{bb\cdot a}=\Sigma_{bb}-\Sigma_{ba}\Sigma_{aa}^{-1}\Sigma_{ab}
\]</span>
特别的，最后一个式子叫做 <span class="math inline">\(\Sigma_{bb}\)</span> 的 Schur Complementary。可以看到：
<span class="math display">\[
x_{b\cdot a}=\begin{pmatrix}-\Sigma_{ba}\Sigma_{aa}^{-1}&\mathbb{I}_{n\times n}\end{pmatrix}\begin{pmatrix}x_a\\x_b\end{pmatrix}
\]</span>
所以：
<span class="math display">\[
\mathbb{E}[x_{b\cdot a}]=\begin{pmatrix}-\Sigma_{ba}\Sigma_{aa}^{-1}&\mathbb{I}_{n\times n}\end{pmatrix}\begin{pmatrix}\mu_a\\\mu_b\end{pmatrix}=\mu_{b\cdot a}\\
Var[x_{b\cdot a}]=\begin{pmatrix}-\Sigma_{ba}\Sigma_{aa}^{-1}&\mathbb{I}_{n\times n}\end{pmatrix}\begin{pmatrix}\Sigma_{aa}&\Sigma_{ab}\\\Sigma_{ba}&\Sigma_{bb}\end{pmatrix}\begin{pmatrix}-\Sigma_{aa}^{-1}\Sigma_{ba}^T\\\mathbb{I}_{n\times n}\end{pmatrix}=\Sigma_{bb\cdot a}
\]</span>
利用这三个量可以得到 <span class="math inline">\(x_b=x_{b\cdot a}+\Sigma_{ba}\Sigma_{aa}^{-1}x_a\)</span>。因此：
<span class="math display">\[
\mathbb{E}[x_b|x_a]=\mu_{b\cdot a}+\Sigma_{ba}\Sigma_{aa}^{-1}x_a
\]</span></p>
<p><span class="math display">\[
Var[x_b|x_a]=\Sigma_{bb\cdot a}
\]</span></p>
<p>这里同样用到了定理。</p></li>
<li><p>同样：
<span class="math display">\[
x_{a\cdot b}=x_a-\Sigma_{ab}\Sigma_{bb}^{-1}x_b\\
\mu_{a\cdot b}=\mu_a-\Sigma_{ab}\Sigma_{bb}^{-1}\mu_b\\
\Sigma_{aa\cdot b}=\Sigma_{aa}-\Sigma_{ab}\Sigma_{bb}^{-1}\Sigma_{ba}
\]</span>
所以：
<span class="math display">\[
\mathbb{E}[x_a|x_b]=\mu_{a\cdot b}+\Sigma_{ab}\Sigma_{bb}^{-1}x_b
\]</span></p>
<p><span class="math display">\[
Var[x_a|x_b]=\Sigma_{aa\cdot b}
\]</span></p></li>
</ol>
<p>下面利用上边四个量，求解线性模型：</p>
<blockquote>
<p>已知：<span class="math inline">\(p(x)=\mathcal{N}(\mu,\Lambda^{-1}),p(y|x)=\mathcal{N}(Ax+b,L^{-1})\)</span>，求解：<span class="math inline">\(p(y),p(x|y)\)</span>。</p>
<p>解：令 <span class="math inline">\(y=Ax+b+\epsilon,\epsilon\sim\mathcal{N}(0,L^{-1})\)</span>，所以 <span class="math inline">\(\mathbb{E}[y]=\mathbb{E}[Ax+b+\epsilon]=A\mu+b\)</span>，<span class="math inline">\(Var[y]=A \Lambda^{-1}A^T+L^{-1}\)</span>，因此：
<span class="math display">\[
p(y)=\mathcal{N}(A\mu+b,L^{-1}+A\Lambda^{-1}A^T)
\]</span>
引入 <span class="math inline">\(z=\begin{pmatrix}x\\y\end{pmatrix}\)</span>，我们可以得到 <span class="math inline">\(Cov[x,y]=\mathbb{E}[(x-\mathbb{E}[x])(y-\mathbb{E}[y])^T]\)</span>。对于这个协方差可以直接计算：
<span class="math display">\[
\begin{align}
Cov(x,y)&=\mathbb{E}[(x-\mu)(Ax-A\mu+\epsilon)^T]=\mathbb{E}[(x-\mu)(x-\mu)^TA^T]=Var[x]A^T=\Lambda^{-1}A^T
\end{align}
\]</span>
注意到协方差矩阵的对称性，所以 <span class="math inline">\(p(z)=\mathcal{N}\begin{pmatrix}\mu\\A\mu+b\end{pmatrix},\begin{pmatrix}\Lambda^{-1}&\Lambda^{-1}A^T\\A\Lambda^{-1}&L^{-1}+A\Lambda^{-1}A^T\end{pmatrix})\)</span>。根据之前的公式，我们可以得到：
<span class="math display">\[
\mathbb{E}[x|y]=\mu+\Lambda^{-1}A^T(L^{-1}+A\Lambda^{-1}A^T)^{-1}(y-A\mu-b)
\]</span></p>
<p><span class="math display">\[
Var[x|y]=\Lambda^{-1}-\Lambda^{-1}A^T(L^{-1}+A\Lambda^{-1}A^T)^{-1}A\Lambda^{-1}
\]</span></p>
</blockquote>
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