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统计学
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基础
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统计量
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抽样
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假设检验
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推断
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概率论
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概率论
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计数
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</span>
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</a>
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</li>
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概率概念
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</span>
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|
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|
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</a>
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</li>
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<li class="md-nav__item">
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<span class="md-ellipsis">
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分布
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|
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|
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|
||
</span>
|
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|
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|
||
|
||
</a>
|
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</li>
|
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<li class="md-nav__item">
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<a href="../../chapter%2005%3A%20probability/04.%20bayesian/" class="md-nav__link">
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<span class="md-ellipsis">
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|
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贝叶斯
|
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|
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|
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|
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</span>
|
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|
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|
||
|
||
</a>
|
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</li>
|
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<li class="md-nav__item">
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<a href="../../chapter%2005%3A%20probability/05.%20information%20theory/" class="md-nav__link">
|
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<span class="md-ellipsis">
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信息论
|
||
|
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|
||
|
||
</span>
|
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|
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|
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|
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</a>
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</ul>
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</nav>
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</li>
|
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||
|
||
|
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<li class="md-nav__item md-nav__item--nested">
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<span class="md-ellipsis">
|
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|
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机器学习
|
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|
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|
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|
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</span>
|
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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机器学习
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter%2006%3A%20machine%20learning/01.%20classical%20machine%20learning/" class="md-nav__link">
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|
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<span class="md-ellipsis">
|
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|
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经典机器学习
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../../chapter%2006%3A%20machine%20learning/02.%20gradient%20machine%20learning/" class="md-nav__link">
|
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|
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<span class="md-ellipsis">
|
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|
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|
||
梯度机器学习
|
||
|
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|
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|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
|
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<li class="md-nav__item">
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<a href="../../chapter%2006%3A%20machine%20learning/03.%20deep%20learning/" class="md-nav__link">
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|
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<span class="md-ellipsis">
|
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|
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|
||
深度学习
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter%2006%3A%20machine%20learning/04.%20reinforcement%20learning/" class="md-nav__link">
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|
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<span class="md-ellipsis">
|
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|
||
强化学习
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter%2006%3A%20machine%20learning/05.%20distributed%20deep%20learning/" class="md-nav__link">
|
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|
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|
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|
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<span class="md-ellipsis">
|
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|
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|
||
分布式深度学习
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
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</li>
|
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|
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|
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|
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</ul>
|
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</nav>
|
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</li>
|
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|
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|
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|
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|
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|
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|
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|
||
|
||
|
||
|
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|
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<li class="md-nav__item md-nav__item--nested">
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|
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|
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<span class="md-ellipsis">
|
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|
||
|
||
计算语言学
|
||
|
||
|
||
|
||
</span>
|
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|
||
|
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|
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<span class="md-nav__icon md-icon"></span>
|
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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计算语言学
|
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|
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|
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter%2007%3A%20computational%20linguistics/01.%20linguistic%20foundations/" class="md-nav__link">
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|
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|
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|
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<span class="md-ellipsis">
|
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|
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|
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语言学基础
|
||
|
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|
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|
||
</span>
|
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|
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|
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|
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</a>
|
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</li>
|
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|
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<li class="md-nav__item">
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<a href="../../chapter%2007%3A%20computational%20linguistics/02.%20text%20processing%20and%20classic%20NLP/" class="md-nav__link">
|
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|
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|
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|
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<span class="md-ellipsis">
|
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|
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|
||
文本处理与经典 NLP
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter%2007%3A%20computational%20linguistics/03.%20embeddings%20and%20sequence%20models/" class="md-nav__link">
|
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|
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|
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|
||
<span class="md-ellipsis">
|
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|
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|
||
嵌入与序列模型
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
||
</li>
|
||
|
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|
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|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter%2007%3A%20computational%20linguistics/04.%20transformers%20and%20language%20models/" class="md-nav__link">
|
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|
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|
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|
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<span class="md-ellipsis">
|
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|
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|
||
Transformer 与语言模型
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
|
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|
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|
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<li class="md-nav__item">
|
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<a href="../../chapter%2007%3A%20computational%20linguistics/05.%20advanced%20text%20generation/" class="md-nav__link">
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|
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|
||
|
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<span class="md-ellipsis">
|
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|
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|
||
高级文本生成
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
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</a>
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</li>
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|
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</ul>
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</nav>
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</li>
|
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|
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|
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|
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|
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|
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|
||
|
||
|
||
|
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|
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<li class="md-nav__item md-nav__item--nested">
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计算机视觉
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图像基础
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目标检测与分割
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音频与语音
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数字信号处理
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自动语音识别
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语音合成
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</span>
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</a>
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<span class="md-ellipsis">
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说话人与音频分析
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</span>
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|
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|
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</a>
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</li>
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源分离与降噪
|
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</span>
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|
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<span class="md-ellipsis">
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多模态学习
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</span>
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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多模态学习
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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多模态表征
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</span>
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</a>
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视觉语言模型
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</span>
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</a>
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图像与视频 Token 化
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</span>
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|
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</a>
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跨模态生成
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</span>
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|
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|
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|
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</a>
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</li>
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<a href="../../chapter%2010%3A%20multimodal%20learning/05.%20unified%20multimodal%20architectures/" class="md-nav__link">
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<span class="md-ellipsis">
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统一多模态架构
|
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|
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|
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</span>
|
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|
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|
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|
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<li class="md-nav__item md-nav__item--nested">
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自主系统
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</span>
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<span class="md-nav__icon md-icon"></span>
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自主系统
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感知
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机器人学习
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</a>
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视觉-语言-动作模型
|
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</span>
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|
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|
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|
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</a>
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自动驾驶
|
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</span>
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|
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|
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|
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</a>
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<a href="../../chapter%2011%3A%20autonomous%20systems/05.%20space%20and%20extreme%20robotics/" class="md-nav__link">
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<span class="md-ellipsis">
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太空与极端机器人
|
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|
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|
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|
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</span>
|
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|
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|
||
|
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图神经网络
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</span>
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<span class="md-nav__icon md-icon"></span>
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图神经网络
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几何深度学习
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</span>
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ARM 与 NEON
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x86 与 AVX
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边缘推理
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ML 系统设计
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应用 AI
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前沿 AI
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神经形态计算
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太空数据中心
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<h1 id="_1">抽样<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>抽样决定了我们如何收集数据，并直接控制着我们所做每项结论的质量。本文涵盖随机抽样、分层抽样、整群抽样与系统抽样、抽样分布、大数定律以及自助法——这些方法对于机器学习中的训练/测试划分和数据集整理至关重要。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>在理想世界中，你会测量所关心群体中的每一个成员。但在实践中，这几乎永远不可能做到。你无法调查每一位选民，无法测试每一只灯泡，也无法扫描每一位患者。所以你只能抽取一个<strong>样本</strong>，并用它来了解整体。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>总体</strong>是你想研究的个体或项目的完整集合。<strong>样本</strong>是你实际观测到的子集。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>参数</strong>是描述总体的数值（例如，某个国家所有成年人的真实平均身高）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>统计量</strong>是从样本中计算出的数值（例如，你测量的 500 人的平均身高）。统计量用于估计参数。</p>
|
||
</li>
|
||
<li>
|
||
<p>结论的质量完全取决于你如何选择样本。一个有偏的样本会导致有偏的结论，无论你的分析多么复杂。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>抽样框</strong>是你实际从中抽取样本的所有个体的列表。理想情况下，抽样框与总体完全吻合，但在实践中总会存在差距。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，如果你通过电话调查人群，就会漏掉所有没有电话的人。抽样框与总体之间的差异称为<strong>覆盖误差</strong>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>抽样误差</strong>是样本统计量与总体参数之间的自然差异。</p>
|
||
</li>
|
||
<li>
|
||
<p>即使是完全随机的样本也不会与总体完全一致。更大的样本可以减少抽样误差。</p>
|
||
</li>
|
||
<li>
|
||
<p>抽样有两大类：概率抽样和非概率抽样。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>概率抽样</strong>意味着总体中的每一个成员都有已知的、非零的概率被选中。这让你能够量化不确定性并推广结果。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>简单随机抽样</strong>：每个个体被选中的概率相等，且每个大小为 <span class="arithmatex">\(n\)</span> 的可能样本出现的概率相同。就像把每个名字放进一顶帽子里，然后蒙眼抽取。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>分层抽样</strong>：根据某个共同特征（如年龄组、地区）将总体划分为互不重叠的组（层），然后从每一层中随机抽样。这保证了每个群体的代表性，并且当层与层之间存在差异时，可以降低方差。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>整群抽样</strong>：将总体划分为若干组（群），随机选择一些群，然后将所选群中的全部个体都纳入样本。当总体在地理上分散时这种方法很实用，比如在整个学区中抽取整所学校而非单个学生。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>系统抽样</strong>：随机选择一个起点，然后从列表中每隔 <span class="arithmatex">\(k\)</span> 个个体选取一个。例如，从第 7 个人开始，然后每隔 10 个人取一个（7, 17, 27, ...）。这种方法易于实施，但如果列表中存在隐藏模式，则可能引入偏差。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="三种概率抽样方法对比：简单随机、分层和整群" src="../../images/sampling_methods.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>非概率抽样</strong>并不给每个成员已知的入选机会。其结果无法被严格推广，但这些方法通常更快、更便宜。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>便利抽样</strong>：选择最容易接触到的人。在购物中心调查人群很方便，但会遗漏那些不去购物中心的人。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>配额抽样</strong>：与分层抽样类似，但没有随机性。研究者通过从每个群体中选取方便接触的个体来填补配额（例如 50 名男性和 50 名女性）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>雪球抽样</strong>：从少数参与者开始，然后请他们招募其他人。适用于难以接触到的人群（例如研究罕见疾病），但会严重偏向于有社交联系的个体。</p>
|
||
</li>
|
||
<li>
|
||
<p>一旦你有了抽样方法，一个自然的问题就出现了：如果抽取一个不同的样本，会得到不同的统计量吗？几乎肯定会。<strong>抽样分布</strong>是一个统计量（如样本均值）在所有相同大小的可能样本上的分布。</p>
|
||
</li>
|
||
<li>
|
||
<p>想象一下抽取 1000 个不同的 30 人样本，并计算每个样本的平均身高。这 1000 个均值形成了一个分布。有些会略高于真实的总体均值，有些会略低于，而大多数会聚集在真实值周围。</p>
|
||
</li>
|
||
<li>
|
||
<p>这个抽样分布的标准差称为<strong>标准误</strong>：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[SE = \frac{\sigma}{\sqrt{n}}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>注意标准误随着 <span class="arithmatex">\(n\)</span> 的增大而缩小。更大的样本能给出更精确的估计。样本量扩大到四倍，标准误减半。</p>
|
||
</li>
|
||
<li>
|
||
<p>统计学中最重要的结果是<strong>中心极限定理（CLT）</strong>。它指出：无论原始总体的分布形态如何，随着样本量的增大，样本均值的分布都趋近于正态分布。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="CLT：偏态总体产生正态分布的样本均值" src="../../images/central_limit_theorem.svg" /></p>
|
||
<ul>
|
||
<li>更精确地说，如果 <span class="arithmatex">\(X_1, X_2, \ldots, X_n\)</span> 是来自任意分布的独立观测值，该分布具有均值 <span class="arithmatex">\(\mu\)</span> 和有限方差 <span class="arithmatex">\(\sigma^2\)</span>，那么随着 <span class="arithmatex">\(n\)</span> 增大：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\bar{X} \approx \text{Normal}\!\left(\mu, \frac{\sigma^2}{n}\right)\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>CLT 是大部分推断统计得以进行的基础。它让我们能够使用正态分布作为近似，即使底层数据不是正态分布，只要样本量足够大即可。</p>
|
||
</li>
|
||
<li>
|
||
<p>"足够大"是多大？一个常见的经验法则是 <span class="arithmatex">\(n \ge 30\)</span>，但这取决于总体的非正态程度。对于高度偏态的分布，你可能需要更大的样本量。对于大致对称的总体，即使 <span class="arithmatex">\(n = 10\)</span> 也可能足够了。</p>
|
||
</li>
|
||
<li>
|
||
<p>CLT 有三个关键条件：</p>
|
||
<ul>
|
||
<li><strong>独立性</strong>：每个观测值不能影响其他观测值</li>
|
||
<li><strong>有限方差</strong>：总体方差必须存在（排除了某些特殊分布）</li>
|
||
<li><strong>同分布</strong>：所有观测值来自同一分布</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colab-notebook">编程任务（使用 CoLab 或 notebook）<a class="headerlink" href="#colab-notebook" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>可视化演示 CLT：从高度偏态的分布中抽取样本，计算样本均值，观察均值直方图如何变成钟形。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="c1"># 指数分布（高度偏态）</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">population</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">exponential</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">100_000</span><span class="p">,))</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">14</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="n">sample_sizes</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">30</span><span class="p">,</span> <span class="mi">100</span><span class="p">]</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="k">for</span> <span class="n">ax</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">axes</span><span class="p">,</span> <span class="n">sample_sizes</span><span class="p">):</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a> <span class="n">keys</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="mi">2000</span><span class="p">)</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a> <span class="n">means</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">population</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">n</span><span class="p">,))</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">keys</span><span class="p">])</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a> <span class="n">ax</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">means</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">40</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"#3498db"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a> <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="sa">f</span><span class="s2">"n = </span><span class="si">{</span><span class="n">n</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a> <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
|
||
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>
|
||
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="s2">"CLT：随着 n 增大，样本均值趋近正态分布"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">13</span><span class="p">)</span>
|
||
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
|
||
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>比较简单随机抽样与分层抽样。创建一个具有不同分组的总体，展示分层抽样能给出更低的估计方差。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="c1"># 总体：两个不同的组</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">group_a</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">500</span><span class="p">,))</span> <span class="o">+</span> <span class="mi">10</span> <span class="c1"># 均值 ~10</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">key</span><span class="p">,</span> <span class="n">subkey</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">group_b</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">subkey</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">500</span><span class="p">,))</span> <span class="o">+</span> <span class="mi">20</span> <span class="c1"># 均值 ~20</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="n">population</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([</span><span class="n">group_a</span><span class="p">,</span> <span class="n">group_b</span><span class="p">])</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="c1"># 简单随机抽样：1000 次试验，样本量 20</span>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="n">srs_means</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">):</span>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a> <span class="n">key</span><span class="p">,</span> <span class="n">subkey</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a> <span class="n">sample</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">subkey</span><span class="p">,</span> <span class="n">population</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,),</span> <span class="n">replace</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a> <span class="n">srs_means</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sample</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
|
||
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="n">srs_means</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">srs_means</span><span class="p">)</span>
|
||
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a>
|
||
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="c1"># 分层抽样：每组各取 10 个</span>
|
||
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="n">strat_means</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">):</span>
|
||
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a> <span class="n">key</span><span class="p">,</span> <span class="n">k1</span><span class="p">,</span> <span class="n">k2</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
|
||
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a> <span class="n">s_a</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">k1</span><span class="p">,</span> <span class="n">group_a</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,),</span> <span class="n">replace</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a> <span class="n">s_b</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">k2</span><span class="p">,</span> <span class="n">group_b</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,),</span> <span class="n">replace</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||
<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a> <span class="n">strat_means</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([</span><span class="n">s_a</span><span class="p">,</span> <span class="n">s_b</span><span class="p">])</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
|
||
<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="n">strat_means</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">strat_means</span><span class="p">)</span>
|
||
<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a>
|
||
<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"简单随机 - 均值: </span><span class="si">{</span><span class="n">srs_means</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">, 标准差: </span><span class="si">{</span><span class="n">srs_means</span><span class="o">.</span><span class="n">std</span><span class="p">()</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"分层抽样 - 均值: </span><span class="si">{</span><span class="n">strat_means</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">, 标准差: </span><span class="si">{</span><span class="n">strat_means</span><span class="o">.</span><span class="n">std</span><span class="p">()</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"分层抽样降低了方差 </span><span class="si">{</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">strat_means</span><span class="o">.</span><span class="n">var</span><span class="p">()</span><span class="o">/</span><span class="n">srs_means</span><span class="o">.</span><span class="n">var</span><span class="p">())</span><span class="o">*</span><span class="mi">100</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2">%"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>探索样本量如何影响标准误。绘制标准误随样本量变化的曲线，验证 <span class="arithmatex">\(1/\sqrt{n}\)</span> 的关系。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="n">population</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">50_000</span><span class="p">,))</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">+</span> <span class="mi">50</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">sample_sizes</span> <span class="o">=</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">200</span><span class="p">,</span> <span class="mi">500</span><span class="p">,</span> <span class="mi">1000</span><span class="p">]</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">std_errors</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">sample_sizes</span><span class="p">:</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="n">means</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">500</span><span class="p">):</span>
|
||
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a> <span class="n">key</span><span class="p">,</span> <span class="n">subkey</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
|
||
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a> <span class="n">sample</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">subkey</span><span class="p">,</span> <span class="n">population</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">n</span><span class="p">,))</span>
|
||
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a> <span class="n">means</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sample</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
|
||
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a> <span class="n">std_errors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">means</span><span class="p">)</span><span class="o">.</span><span class="n">std</span><span class="p">())</span>
|
||
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a>
|
||
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">sample_sizes</span><span class="p">,</span> <span class="n">std_errors</span><span class="p">,</span> <span class="s2">"o-"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"#e74c3c"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"观测到的 SE"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="n">theoretical</span> <span class="o">=</span> <span class="n">population</span><span class="o">.</span><span class="n">std</span><span class="p">()</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">sample_sizes</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">jnp</span><span class="o">.</span><span class="n">float32</span><span class="p">))</span>
|
||
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">sample_sizes</span><span class="p">,</span> <span class="n">theoretical</span><span class="p">,</span> <span class="s2">"--"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"#3498db"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"σ/√n（理论值）"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"样本量 (n)"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"标准误"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"标准误随样本量增大而缩小"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</code></pre></div></p>
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