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统计学
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基础
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统计量
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假设检验
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</a>
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推断
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推断
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</a>
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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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概率概念
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</span>
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</a>
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分布
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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%2005%3A%20probability/04.%20bayesian/" class="md-nav__link">
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<span class="md-ellipsis">
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贝叶斯
|
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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%2005%3A%20probability/05.%20information%20theory/" class="md-nav__link">
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信息论
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</span>
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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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|
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|
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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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<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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<span class="md-ellipsis">
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经典机器学习
|
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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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梯度机器学习
|
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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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<span class="md-ellipsis">
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|
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深度学习
|
||
|
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|
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|
||
</span>
|
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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/04.%20reinforcement%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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强化学习
|
||
|
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|
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|
||
</span>
|
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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/05.%20distributed%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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分布式深度学习
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
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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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|
||
|
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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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计算语言学
|
||
|
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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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<span class="md-nav__icon md-icon"></span>
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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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</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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<span class="md-ellipsis">
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|
||
语言学基础
|
||
|
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|
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|
||
</span>
|
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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%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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|
||
<span class="md-ellipsis">
|
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|
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|
||
文本处理与经典 NLP
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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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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<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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|
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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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<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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<span class="md-ellipsis">
|
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|
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|
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Transformer 与语言模型
|
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|
||
|
||
|
||
</span>
|
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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%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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</ul>
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</nav>
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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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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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ViT 与生成模型
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视频与 3D 视觉
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<span class="md-ellipsis">
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音频与语音
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<span class="md-ellipsis">
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数字信号处理
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</span>
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</a>
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<span class="md-ellipsis">
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自动语音识别
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</span>
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</a>
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</li>
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<a href="../../chapter%2009%3A%20audio%20and%20speech/03.%20text%20to%20speech%20and%20voice/" class="md-nav__link">
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<span class="md-ellipsis">
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语音合成
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</span>
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</a>
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</li>
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<a href="../../chapter%2009%3A%20audio%20and%20speech/04.%20speaker%20and%20audio%20analysis/" class="md-nav__link">
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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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|
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</a>
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</li>
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<span class="md-ellipsis">
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源分离与降噪
|
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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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</ul>
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</nav>
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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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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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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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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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<span class="md-ellipsis">
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图像与视频 Token 化
|
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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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<span class="md-ellipsis">
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跨模态生成
|
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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%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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|
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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 class="md-nav__item md-nav__item--nested">
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<span class="md-ellipsis">
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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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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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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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</span>
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</a>
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机器人学习
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</span>
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|
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</a>
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</li>
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<span class="md-ellipsis">
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视觉-语言-动作模型
|
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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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<a href="../../chapter%2011%3A%20autonomous%20systems/04.%20self-driving/" class="md-nav__link">
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自动驾驶
|
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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%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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</span>
|
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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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<span class="md-ellipsis">
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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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</label>
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图神经网络
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</label>
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几何深度学习
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<span class="md-ellipsis">
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|
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硬件基础
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</a>
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ARM 与 NEON
|
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x86 与 AVX
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Triton、TPU 与 Pallas
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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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量子机器学习
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神经形态计算
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编程练习（在 CoLab 或 notebook 中完成）
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<h1 id="_1">统计推断<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>统计推断超越了简单的"是/否"决策，以量化的不确定性来估计总体参数。本节涵盖置信区间、点估计与区间估计、极大似然估计、矩法以及回归分析——这是连接原始数据与机器学习预测模型的桥梁。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>假设检验给出一个"是/否"的结论：拒绝或不拒绝原假设。但通常你希望得到更有信息量的结果——你正在估计的参数的一个合理取值区间。这正是<strong>置信区间</strong>所提供的。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>点估计</strong>是从样本中计算出的单一数值，比如样本均值 <span class="arithmatex">\(\bar{x}\)</span>。它是你对总体参数的最佳猜测，但仅凭它本身无法反映估计的精确程度。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>置信区间</strong>在点估计周围包裹一个反映不确定性的范围。其形式为：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\text{CI} = \bar{x} \pm \text{ME}\]</div>
|
||
<ul>
|
||
<li><strong>误差范围</strong>取决于三个因素：你希望多高的置信度、数据的变异程度有多大、以及样本量有多大：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\text{ME} = z^\ast \cdot \frac{\sigma}{\sqrt{n}}\]</div>
|
||
<ul>
|
||
<li>其中 <span class="arithmatex">\(z^\ast\)</span> 是从正态分布中查得的临界值，与你期望的置信水平对应。对于 95% 置信度，<span class="arithmatex">\(z^\ast = 1.96\)</span>；对于 99% 置信度，<span class="arithmatex">\(z^\ast = 2.576\)</span>。</li>
|
||
</ul>
|
||
<p><img alt="置信区间：点估计及其两侧的误差范围" src="../../images/confidence_interval.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>95% 置信区间</strong>的含义是：如果你重复进行多次实验，每次构建一个区间，那么大约 95% 的区间会包含真实的总体参数。这并不意味着该参数有 95% 的概率落在这个特定的区间内。参数是一个固定值；变化的是区间本身。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>示例</strong>：你测量了 50 人的身高，得到 <span class="arithmatex">\(\bar{x} = 170\)</span> cm，<span class="arithmatex">\(\sigma = 8\)</span> cm。构建一个 95% 置信区间。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\text{ME} = 1.96 \cdot \frac{8}{\sqrt{50}} = 1.96 \cdot 1.131 = 2.22 \text{ cm}\]</div>
|
||
<div class="arithmatex">\[\text{CI} = [170 - 2.22, \; 170 + 2.22] = [167.78, \; 172.22]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>你可以说，有 95% 的把握认为真正的平均身高介于 167.78 cm 和 172.22 cm 之间。</p>
|
||
</li>
|
||
<li>
|
||
<p>当 <span class="arithmatex">\(\sigma\)</span> 未知时（这是常见情况），改用样本标准差 <span class="arithmatex">\(s\)</span> 和 t 分布：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\text{CI} = \bar{x} \pm t^\ast_{n-1} \cdot \frac{s}{\sqrt{n}}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>越宽的区间置信度越高，但精度越低；越窄的区间精度越高，但置信度越低。在不降低置信度的前提下，增大样本量可以缩小区间。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>功效分析</strong>帮助你在实验开始前进行规划。要回答的问题是：为了检测到某个给定大小的效应并达到指定的检验功效，我需要多大的样本量？</p>
|
||
</li>
|
||
<li>
|
||
<p>回顾上一节的内容，功效 = <span class="arithmatex">\(1 - \beta\)</span>，即正确拒绝错误原假设 <span class="arithmatex">\(H_0\)</span> 的概率。常见的功效目标是 80%。</p>
|
||
</li>
|
||
<li>
|
||
<p>对于 z 检验，检测差异 <span class="arithmatex">\(\delta\)</span> 所需样本量（给定显著性水平 <span class="arithmatex">\(\alpha\)</span> 和功效 <span class="arithmatex">\(1-\beta\)</span>）为：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[n = \left(\frac{(z_{\alpha/2} + z_{\beta}) \cdot \sigma}{\delta}\right)^2\]</div>
|
||
<ul>
|
||
<li>例如，要检测平均身高 2 cm 的差异（<span class="arithmatex">\(\sigma = 8\)</span>），取 <span class="arithmatex">\(\alpha = 0.05\)</span>、功效 80%（<span class="arithmatex">\(z_{0.025} = 1.96\)</span>，<span class="arithmatex">\(z_{0.20} = 0.84\)</span>）：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[n = \left(\frac{(1.96 + 0.84) \cdot 8}{2}\right)^2 = \left(\frac{22.4}{2}\right)^2 = 11.2^2 \approx 126\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>你大约需要每组 126 人。</p>
|
||
</li>
|
||
<li>
|
||
<p>功效分析可以防止两种常见错误：实验规模太小，无法检测到真实的效应（功效不足）；或者浪费资源做远超必要规模的实验（功效过剩）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>蒙特卡洛方法</strong>利用随机抽样来求解难以或无法解析求解的问题。其核心思想是：如果你无法精确计算某个量，那就多次模拟并用结果作为近似值。</p>
|
||
</li>
|
||
<li>
|
||
<p>名称来源于蒙特卡洛赌场，寓意随机性的角色。这些方法是机器学习中的重要工具，用于估计积分、评估模型不确定性以及近似复杂分布等任务。</p>
|
||
</li>
|
||
<li>
|
||
<p>蒙特卡洛的一般步骤：</p>
|
||
<ul>
|
||
<li>定义可能输入的取值范围</li>
|
||
<li>从该范围中随机生成输入</li>
|
||
<li>对每个输入评估某个函数</li>
|
||
<li>汇总结果（平均值、计数等）</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>一个经典例子是估算 <span class="arithmatex">\(\pi\)</span>。想象一个边长为 2 的正方形，中心在原点，内切一个半径为 1 的圆。正方形的面积为 4，圆的面积为 <span class="arithmatex">\(\pi\)</span>。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="正方形及其内切圆，随机点按圆内/圆外着色" src="../../images/monte_carlo_pi.svg" /></p>
|
||
<ul>
|
||
<li>在正方形内均匀地随机投点。落在圆内的点的比例近似 <span class="arithmatex">\(\pi/4\)</span>：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\pi \approx 4 \times \frac{\text{圆内点数}}{\text{总点数}}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>点 <span class="arithmatex">\((x, y)\)</span> 在圆内的条件是 <span class="arithmatex">\(x^2 + y^2 \le 1\)</span>。投的点越多，估算值就越接近 <span class="arithmatex">\(\pi\)</span> 的真实值。</p>
|
||
</li>
|
||
<li>
|
||
<p>在机器学习中，蒙特卡洛方法出现在：</p>
|
||
<ul>
|
||
<li><strong>蒙特卡洛 Dropout</strong>：多次执行推理（启用 dropout）来估计预测不确定性</li>
|
||
<li><strong>MCMC（马尔可夫链蒙特卡洛）</strong>：在贝叶斯模型中从复杂的后验分布中抽样</li>
|
||
<li><strong>策略梯度方法</strong>：通过采样轨迹来估计强化学习中的梯度</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>因子分析</strong>是一种发现隐藏（潜在）变量的技术，这些变量解释了观测变量之间的相关性。如果 10 个个性调查问题可以由 3 个潜在特质（外向性、宜人性、责任心）解释，因子分析就能找出这些特质。</p>
|
||
</li>
|
||
<li>
|
||
<p>该模型假设每个观测变量 <span class="arithmatex">\(x_i\)</span> 是少数潜在因子 <span class="arithmatex">\(f_j\)</span> 的线性组合加上噪声：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[x_i = \lambda_{i1} f_1 + \lambda_{i2} f_2 + \ldots + \lambda_{ik} f_k + \epsilon_i\]</div>
|
||
<ul>
|
||
<li>
|
||
<p><span class="arithmatex">\(\lambda\)</span> 值称为<strong>因子载荷</strong>，表示每个观测变量与各因子的关联强度。这与第 2 章的矩阵分解直接相关；因子分析与特征值分解和 SVD 密切相关。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>实验设计</strong>是安排实验结构的艺术，使你能够得出有效的结论。糟糕的设计甚至会使大量数据变得毫无价值。</p>
|
||
</li>
|
||
<li>
|
||
<p>良好实验设计的关键要素：</p>
|
||
<ul>
|
||
<li><strong>自变量</strong>：你操控的变量（例如药物剂量、模型架构）</li>
|
||
<li><strong>因变量</strong>：你测量的变量（例如恢复时间、准确率）</li>
|
||
<li><strong>对照组</strong>：不接受处理（或接受安慰剂），提供比较的基线</li>
|
||
<li><strong>随机分配</strong>：参与者被随机分配到各组，从而平衡掉未测量的混杂变量</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>常见的实验设计</strong>：</p>
|
||
<ul>
|
||
<li><strong>完全随机设计</strong>：受试者被随机分配到处理组。在各组可比的情况下，简单有效。</li>
|
||
<li><strong>随机区组设计</strong>：受试者先按区组分组（例如按年龄），然后在每个区组内随机分配到处理组。这降低了区组因素带来的变异，类似于分层抽样的思路。</li>
|
||
<li><strong>析因设计</strong>：同时测试多个自变量。一个 <span class="arithmatex">\(2 \times 3\)</span> 的析因设计包含一个变量的 2 个水平和另一个变量的 3 个水平，共 6 种处理组合。这使你能够检测到<strong>交互作用</strong>——即一个变量的效应取决于另一个变量的水平。</li>
|
||
<li><strong>交叉设计</strong>：每个受试者按顺序接受所有处理（其间有洗脱期）。每个受试者作为自身的对照，减少了个体差异的影响。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>在机器学习实验中，这些原则至关重要。比较模型时，应控制随机种子、数据集划分和硬件环境。交叉验证是一种交叉设计形式。逐次移除一个组件的消融研究则遵循析因设计的逻辑。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colab-notebook">编程练习（在 CoLab 或 notebook 中完成）<a class="headerlink" href="#colab-notebook" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>为身高示例构建一个 95% 置信区间，然后尝试不同的置信水平和样本量。
|
||
<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.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">x_bar</span> <span class="o">=</span> <span class="mf">170.0</span> <span class="c1"># 样本均值</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="n">sigma</span> <span class="o">=</span> <span class="mf">8.0</span> <span class="c1"># 总体标准差（已知）</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">n</span> <span class="o">=</span> <span class="mi">50</span> <span class="c1"># 样本量</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"># 常用置信水平的临界值</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">z_stars</span> <span class="o">=</span> <span class="p">{</span><span class="mf">0.90</span><span class="p">:</span> <span class="mf">1.645</span><span class="p">,</span> <span class="mf">0.95</span><span class="p">:</span> <span class="mf">1.960</span><span class="p">,</span> <span class="mf">0.99</span><span class="p">:</span> <span class="mf">2.576</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="k">for</span> <span class="n">conf</span><span class="p">,</span> <span class="n">z_star</span> <span class="ow">in</span> <span class="n">z_stars</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="n">me</span> <span class="o">=</span> <span class="n">z_star</span> <span class="o">*</span> <span class="p">(</span><span class="n">sigma</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">n</span><span class="p">))</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a> <span class="n">lower</span><span class="p">,</span> <span class="n">upper</span> <span class="o">=</span> <span class="n">x_bar</span> <span class="o">-</span> <span class="n">me</span><span class="p">,</span> <span class="n">x_bar</span> <span class="o">+</span> <span class="n">me</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">conf</span><span class="o">*</span><span class="mi">100</span><span class="si">:</span><span class="s2">.0f</span><span class="si">}</span><span class="s2">% CI: [</span><span class="si">{</span><span class="n">lower</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">, </span><span class="si">{</span><span class="n">upper</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">] (ME = </span><span class="si">{</span><span class="n">me</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">)"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>使用蒙特卡洛模拟估算 <span class="arithmatex">\(\pi\)</span>。绘制随着点数增加估算值收敛的曲线。
|
||
<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><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-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></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">42</span><span class="p">)</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="c1"># 在 [-1, 1] x [-1, 1] 内生成随机点</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">n_points</span> <span class="o">=</span> <span class="mi">100_000</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><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>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="n">x</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">uniform</span><span class="p">(</span><span class="n">k1</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">n_points</span><span class="p">,),</span> <span class="n">minval</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">maxval</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="n">y</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">uniform</span><span class="p">(</span><span class="n">k2</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">n_points</span><span class="p">,),</span> <span class="n">minval</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">maxval</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="c1"># 检查哪些点在单位圆内</span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="n">inside</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o"><=</span> <span class="mf">1.0</span>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="n">cumulative_inside</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">inside</span><span class="p">)</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="n">counts</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_points</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="n">pi_estimates</span> <span class="o">=</span> <span class="mf">4.0</span> <span class="o">*</span> <span class="n">cumulative_inside</span> <span class="o">/</span> <span class="n">counts</span>
|
||
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a>
|
||
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></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">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">pi_estimates</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">linewidth</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
|
||
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="n">jnp</span><span class="o">.</span><span class="n">pi</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">linestyle</span><span class="o">=</span><span class="s2">"--"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">"π = </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">pi</span><span class="si">:</span><span class="s2">.6f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"点数"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"π 的估算值"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"蒙特卡洛估算 π"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="mf">2.8</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">)</span>
|
||
<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</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">"最终估算值: </span><span class="si">{</span><span class="n">pi_estimates</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="si">:</span><span class="s2">.6f</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">"真实值: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">pi</span><span class="si">:</span><span class="s2">.6f</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">"误差: </span><span class="si">{</span><span class="nb">abs</span><span class="p">(</span><span class="n">pi_estimates</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">jnp</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span><span class="si">:</span><span class="s2">.6f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>执行一个简单的功效分析：给定效应大小和标准差，计算所需样本量并通过模拟验证。
|
||
<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>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="c1"># 参数</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">delta</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="c1"># 效应大小（均值差）</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="n">sigma</span> <span class="o">=</span> <span class="mf">8.0</span> <span class="c1"># 总体标准差</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.05</span>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">power_target</span> <span class="o">=</span> <span class="mf">0.80</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="c1"># 解析计算的样本量</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="n">z_alpha</span> <span class="o">=</span> <span class="mf">1.96</span> <span class="c1"># 双尾，alpha=0.05</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="n">z_beta</span> <span class="o">=</span> <span class="mf">0.84</span> <span class="c1"># power=0.80</span>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="n">n_required</span> <span class="o">=</span> <span class="p">((</span><span class="n">z_alpha</span> <span class="o">+</span> <span class="n">z_beta</span><span class="p">)</span> <span class="o">*</span> <span class="n">sigma</span> <span class="o">/</span> <span class="n">delta</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span>
|
||
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"每组所需样本量: </span><span class="si">{</span><span class="n">n_required</span><span class="si">:</span><span class="s2">.0f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a>
|
||
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="c1"># 通过模拟验证</span>
|
||
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></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-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">n_required</span><span class="p">))</span>
|
||
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="n">n_sims</span> <span class="o">=</span> <span class="mi">5000</span>
|
||
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="n">rejections</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a>
|
||
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></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="n">n_sims</span><span class="p">):</span>
|
||
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></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-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></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">k1</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">sigma</span> <span class="o">+</span> <span class="mi">50</span>
|
||
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></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">k2</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">sigma</span> <span class="o">+</span> <span class="mi">50</span> <span class="o">+</span> <span class="n">delta</span>
|
||
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a> <span class="n">pooled_se</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="mi">2</span> <span class="o">*</span> <span class="n">sigma</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="n">n</span><span class="p">)</span>
|
||
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a> <span class="n">z</span> <span class="o">=</span> <span class="p">(</span><span class="n">group_b</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="o">-</span> <span class="n">group_a</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span> <span class="o">/</span> <span class="n">pooled_se</span>
|
||
<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a> <span class="n">p</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="nb">__import__</span><span class="p">(</span><span class="s2">"jax"</span><span class="p">)</span><span class="o">.</span><span class="n">scipy</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
|
||
<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a> <span class="k">if</span> <span class="n">p</span> <span class="o"><=</span> <span class="n">alpha</span><span class="p">:</span>
|
||
<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a> <span class="n">rejections</span> <span class="o">+=</span> <span class="mi">1</span>
|
||
<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a>
|
||
<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"模拟功效: </span><span class="si">{</span><span class="n">rejections</span><span class="o">/</span><span class="n">n_sims</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-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"目标功效: </span><span class="si">{</span><span class="n">power_target</span><span class="si">:</span><span class="s2">.3f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>可视化置信区间宽度随样本量的变化。这展示了为什么收集更多数据可以得到更精确的估计。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></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-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></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-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a>
|
||
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="n">sigma</span> <span class="o">=</span> <span class="mf">8.0</span>
|
||
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="n">z_star</span> <span class="o">=</span> <span class="mf">1.96</span> <span class="c1"># 95% 置信度</span>
|
||
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a>
|
||
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="n">sample_sizes</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="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">30</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> <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-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="n">margins</span> <span class="o">=</span> <span class="n">z_star</span> <span class="o">*</span> <span class="n">sigma</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">sample_sizes</span><span class="p">)</span>
|
||
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a>
|
||
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></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-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">([</span><span class="nb">str</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">))</span> <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> <span class="n">margins</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>
|
||
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></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"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"误差范围 (cm)"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"95% CI 误差范围随样本量增大而缩小"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></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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