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统计学
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基础
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</span>
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</a>
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统计量
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</span>
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</a>
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抽样
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</a>
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假设检验
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</span>
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</a>
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<span class="md-ellipsis">
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推断
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</span>
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</a>
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概率论
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<span class="md-nav__icon md-icon"></span>
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概率论
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</label>
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计数
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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="../02.%20probability%20concepts/" class="md-nav__link">
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概率概念
|
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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="../03.%20distributions/" class="md-nav__link">
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分布
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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="../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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|
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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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信息论
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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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<span class="md-ellipsis">
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信息论
|
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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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<nav class="md-nav md-nav--secondary" aria-label="目录">
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<span class="md-nav__icon md-icon"></span>
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目录
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<li class="md-nav__item">
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<a href="#colab-notebook" class="md-nav__link">
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<span class="md-ellipsis">
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编程练习（使用 CoLab 或 notebook）
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</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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</ul>
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</li>
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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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</span>
|
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<span class="md-nav__icon md-icon"></span>
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</label>
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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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|
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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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梯度机器学习
|
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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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|
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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>
|
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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%2006%3A%20machine%20learning/04.%20reinforcement%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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|
||
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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|
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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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|
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分布式深度学习
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
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</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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<li class="md-nav__item md-nav__item--nested">
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|
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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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|
||
</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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</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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|
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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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<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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|
||
|
||
|
||
</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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|
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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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<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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|
||
|
||
<span class="md-ellipsis">
|
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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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|
||
|
||
<span class="md-ellipsis">
|
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|
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|
||
Transformer 与语言模型
|
||
|
||
|
||
|
||
</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/05.%20advanced%20text%20generation/" class="md-nav__link">
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|
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|
||
|
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<span class="md-ellipsis">
|
||
|
||
|
||
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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|
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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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|
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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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ViT 与生成模型
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视频与 3D 视觉
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音频与语音
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数字信号处理
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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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<span class="md-ellipsis">
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说话人与音频分析
|
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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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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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<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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<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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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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|
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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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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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<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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</a>
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<span class="md-ellipsis">
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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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<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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</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/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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</a>
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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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</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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AI 推理
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量化
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边缘推理
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扩缩与部署
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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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编程练习（使用 CoLab 或 notebook）
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||
<h1 id="_1">信息论<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>信息论量化了信息、惊奇度以及概率分布之间的差异。本文涵盖熵、交叉熵、KL散度、互信息和自信息——这些概念是机器学习中每一个分类损失函数、VAE目标和数据压缩方案背后的理论基础。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>信息论由克劳德·香农于1948年创立，为我们提供了量化信息的数学框架。它回答了诸如此类的问题：一个事件应当让你感到多惊讶？一条消息携带了多少信息？两个概率分布之间有多大的差异？</p>
|
||
</li>
|
||
<li>
|
||
<p>这些问题看似抽象，但它们是机器学习损失函数、数据压缩和通信系统的基础。交叉熵损失——分类中最常见的损失函数——直接源于信息论。</p>
|
||
</li>
|
||
<li>
|
||
<p>从最简单的问题开始：单个事件携带了多少信息？</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>自信息</strong>（surprisal，也称 self-information）衡量一个事件的惊奇程度。如果某件极有可能发生的事情真的发生了，你几乎学不到任何新信息。如果某件罕见的事情发生了，你则会获得大量信息。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果你住在沙漠里，有人告诉你今天是大晴天，这并没有什么信息量。但如果他们告诉你正在下雪，那信息量就极其丰富。自信息将这种直觉形式化：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[I(x) = \log_2 \frac{1}{p(x)} = -\log_2 p(x)\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>使用 <span class="arithmatex">\(\log_2\)</span> 时，单位是<strong>比特</strong>。一枚公平的硬币抛掷的自信息为 <span class="arithmatex">\(-\log_2(0.5) = 1\)</span> 比特。一个概率为 <span class="arithmatex">\(1/8\)</span> 的事件具有 $ \log_2(8) = 3$ 比特的自信息。</p>
|
||
</li>
|
||
<li>
|
||
<p>为什么用对数而不是简单的 <span class="arithmatex">\(1/p\)</span>？三个原因：</p>
|
||
<ul>
|
||
<li>必然事件（<span class="arithmatex">\(p = 1\)</span>）应给出零信息：<span class="arithmatex">\(\log(1) = 0\)</span> 但 <span class="arithmatex">\(1/1 = 1\)</span>。</li>
|
||
<li>独立事件的信息应该是可加的：<span class="arithmatex">\(\log(1/p_1 p_2) = \log(1/p_1) + \log(1/p_2)\)</span>。</li>
|
||
<li>我们需要一个平滑、性质良好的函数。<span class="arithmatex">\(1/p\)</span> 会爆炸；<span class="arithmatex">\(\log(1/p)\)</span> 则平缓增长。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>熵</strong>是自信息的期望值，即从一个分布中每次采样获得的平均信息量。它衡量该分布的不确定性或"不可预测性"：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[H(X) = E[I(X)] = -\sum_{x} p(x) \log_2 p(x)\]</div>
|
||
<p><img alt="柱状图展示高概率事件具有低自信息，反之亦然；熵是加权平均值" src="../../images/surprisal_entropy.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>一枚公平硬币的熵为 <span class="arithmatex">\(H = -0.5\log_2(0.5) - 0.5\log_2(0.5) = 1\)</span> 比特。不确定性最大。</p>
|
||
</li>
|
||
<li>
|
||
<p>一枚偏倚硬币，<span class="arithmatex">\(p = 0.9\)</span>，其熵为 <span class="arithmatex">\(H = -0.9\log_2(0.9) - 0.1\log_2(0.1) \approx 0.469\)</span> 比特。不太确定，因此熵更小。</p>
|
||
</li>
|
||
<li>
|
||
<p>一个确定性事件（<span class="arithmatex">\(p = 1\)</span>）的熵为 <span class="arithmatex">\(H = 0\)</span>。完全没有不确定性。</p>
|
||
</li>
|
||
<li>
|
||
<p>当所有结果等可能时，熵达到最大。对于 <span class="arithmatex">\(n\)</span> 个等可能结果，<span class="arithmatex">\(H = \log_2 n\)</span>。一颗公平骰子的熵为 <span class="arithmatex">\(\log_2 6 \approx 2.585\)</span> 比特。</p>
|
||
</li>
|
||
<li>
|
||
<p>熵的实际意义在于<strong>压缩</strong>。香农的源编码定理指出，如果不丢失信息，你无法将数据压缩到低于其熵率。一幅每个像素都等可能的图像（最大熵）无法压缩。一幅几乎全是白色的图像（低熵）则可以很好地压缩。</p>
|
||
</li>
|
||
<li>
|
||
<p>快速感受一下数量级：一个灰度像素（256 个值）的最大熵为 8 比特。一张 1080p 的灰度图像最多有 <span class="arithmatex">\(1920 \times 1080 \times 8 \approx 1660\)</span> 万比特。真实图像的熵要低得多，因为相邻像素是相关的——这正是 JPEG 压缩能够工作的原因。</p>
|
||
</li>
|
||
<li>
|
||
<p>对于连续随机变量，离散求和变为积分。<strong>微分熵</strong>定义为：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[h(X) = -\int_{-\infty}^{\infty} f(x) \log f(x)\, dx\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>方差为 <span class="arithmatex">\(\sigma^2\)</span> 的高斯分布的微分熵为 <span class="arithmatex">\(h = \frac{1}{2}\log_2(2\pi e \sigma^2)\)</span>。在所有具有相同方差的分布中，高斯分布具有最大熵。这也是高斯分布在建模中如此常见的原因之一：它在指定均值和方差之外做出了最少的假设。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>互信息</strong>衡量知道一个变量能告诉你关于另一个变量的多少信息。它是观察到 <span class="arithmatex">\(Y\)</span> 后 <span class="arithmatex">\(X\)</span> 不确定性的减少量：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[I(X; Y) = H(X) - H(X|Y) = H(Y) - H(Y|X)\]</div>
|
||
<ul>
|
||
<li>等价形式：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[I(X; Y) = \sum_{x,y} p(x,y) \log_2 \frac{p(x,y)}{p(x) p(y)}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>如果 <span class="arithmatex">\(X\)</span> 和 <span class="arithmatex">\(Y\)</span> 独立，则 <span class="arithmatex">\(p(x,y) = p(x)p(y)\)</span>，互信息为零。它们依赖程度越高，互信息就越大。</p>
|
||
</li>
|
||
<li>
|
||
<p>在机器学习中，互信息用于特征选择（挑选与目标具有高 MI 的特征）、信息瓶颈方法以及聚类质量评估。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>交叉熵</strong>衡量使用针对分布 <span class="arithmatex">\(q\)</span> 优化的编码方案来编码来自分布 <span class="arithmatex">\(p\)</span> 的事件所需的平均比特数：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[H(p, q) = -\sum_{x} p(x) \log_2 q(x)\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>如果 <span class="arithmatex">\(q\)</span> 与 <span class="arithmatex">\(p\)</span> 完全匹配，则交叉熵等于熵：<span class="arithmatex">\(H(p, p) = H(p)\)</span>。如果 <span class="arithmatex">\(q\)</span> 是一个糟糕的近似，交叉熵就会更高。"额外"的比特来自这种不匹配。</p>
|
||
</li>
|
||
<li>
|
||
<p>这正是交叉熵成为机器学习中分类标准损失函数的原因。真实标签定义了 <span class="arithmatex">\(p\)</span>（一个 one-hot 分布），模型的预测概率定义了 <span class="arithmatex">\(q\)</span>。最小化交叉熵推动 <span class="arithmatex">\(q\)</span> 趋近于 <span class="arithmatex">\(p\)</span>：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathcal{L} = -\sum_{c} y_c \log \hat{y}_c\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>对于单个样本，若真实类别为 <span class="arithmatex">\(c\)</span>，上式简化为 <span class="arithmatex">\(\mathcal{L} = -\log \hat{y}_c\)</span>。该损失就是模型预测下真实类别的自信息。如果模型对正确类别赋予高概率，则损失较低。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>KL 散度</strong>（Kullback-Leibler 散度，也称相对熵）衡量一个分布与另一个分布的差异程度：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[D_{\text{KL}}(p \| q) = \sum_{x} p(x) \log \frac{p(x)}{q(x)} = H(p, q) - H(p)\]</div>
|
||
<ul>
|
||
<li>KL 散度是"使用分布 <span class="arithmatex">\(q\)</span> 而非真实分布 <span class="arithmatex">\(p\)</span> 的额外代价"。它总是非负的（<span class="arithmatex">\(D_{\text{KL}} \ge 0\)</span>），且仅在 <span class="arithmatex">\(p = q\)</span> 时为零。</li>
|
||
</ul>
|
||
<p><img alt="两个分布 p 和 q，它们之间的间隙表示 KL 散度" src="../../images/kl_divergence.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>KL 散度不是对称的：<span class="arithmatex">\(D_{\text{KL}}(p \| q) \ne D_{\text{KL}}(q \| p)\)</span>。这种不对称性很重要。<span class="arithmatex">\(D_{\text{KL}}(p \| q)\)</span> 惩罚 <span class="arithmatex">\(q\)</span> 在 <span class="arithmatex">\(p\)</span> 具有高概率处放置低概率（因为 <span class="arithmatex">\(\log(p/q)\)</span> 会趋于无穷大）。<span class="arithmatex">\(D_{\text{KL}}(q \| p)\)</span> 则惩罚相反的情况。</p>
|
||
</li>
|
||
<li>
|
||
<p>这种不对称性导致了两种近似风格：</p>
|
||
<ul>
|
||
<li>最小化 <span class="arithmatex">\(D_{\text{KL}}(p \| q)\)</span> 产生<strong>矩匹配</strong>行为：<span class="arithmatex">\(q\)</span> 覆盖 <span class="arithmatex">\(p\)</span> 的所有模态，但可能过于分散。</li>
|
||
<li>最小化 <span class="arithmatex">\(D_{\text{KL}}(q \| p)\)</span> 产生<strong>模式寻找</strong>行为：<span class="arithmatex">\(q\)</span> 集中于 <span class="arithmatex">\(p\)</span> 的某一个模态，但可能错过其他模态。变分推断使用的正是这一种。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>由于 <span class="arithmatex">\(H(p)\)</span> 相对于模型是常数，最小化交叉熵 <span class="arithmatex">\(H(p, q)\)</span> 等价于最小化 <span class="arithmatex">\(D_{\text{KL}}(p \| q)\)</span>。这就是为什么我们可以使用交叉熵损失，同时知道我们也在最小化真实分布与预测分布之间的 KL 散度。</p>
|
||
</li>
|
||
<li>
|
||
<p>KL 散度在<strong>贝叶斯更新</strong>中扮演着核心角色。后验 <span class="arithmatex">\(P(\theta | D)\)</span> 是在 KL 散度意义上与先验 <span class="arithmatex">\(P(\theta)\)</span> 最接近且与观测数据一致的分布。每一次新的观测都会更新后验，减少关于 <span class="arithmatex">\(\theta\)</span> 的不确定性。</p>
|
||
</li>
|
||
<li>
|
||
<p>在变分自编码器（VAE）中，损失函数包含两项：重构损失（交叉熵）和一个 KL 散度项，后者对潜在空间进行正则化，使其保持接近标准正态分布。</p>
|
||
</li>
|
||
<li>
|
||
<p>将所有概念联系起来：熵告诉你一个分布内在的不确定性，交叉熵告诉你的模型对现实的近似程度，而 KL 散度则告诉你两者之间的差距。这三个量构成了现代机器学习优化的基石。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colab-notebook">编程练习（使用 CoLab 或 notebook）<a class="headerlink" href="#colab-notebook" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>计算各种分布的熵，并验证在给定结果数量下，均匀分布的熵最大。
|
||
<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="k">def</span><span class="w"> </span><span class="nf">entropy</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="sd">"""以比特为单位计算熵。过滤掉概率为零的事件。"""</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="n">p</span> <span class="o">></span> <span class="mi">0</span><span class="p">]</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="k">return</span> <span class="o">-</span><span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">p</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">log2</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="c1"># 公平骰子</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">fair</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span> <span class="o">/</span> <span class="mi">6</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"公平骰子熵: </span><span class="si">{</span><span class="n">entropy</span><span class="p">(</span><span class="n">fair</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2"> 比特 (最大 = log2(6) = </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">log2</span><span class="p">(</span><span class="mf">6.</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">)"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="c1"># 灌铅骰子</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="n">loaded</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="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">])</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"灌铅骰子熵: </span><span class="si">{</span><span class="n">entropy</span><span class="p">(</span><span class="n">loaded</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2"> 比特"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="c1"># 确定性</span>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="n">det</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="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">])</span>
|
||
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"确定性: </span><span class="si">{</span><span class="n">entropy</span><span class="p">(</span><span class="n">det</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2"> 比特"</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="c1"># 公平硬币</span>
|
||
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="n">coin</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="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">])</span>
|
||
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"公平硬币熵: </span><span class="si">{</span><span class="n">entropy</span><span class="p">(</span><span class="n">coin</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2"> 比特"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>计算真实分布与多个近似分布之间的交叉熵和 KL 散度。验证 <span class="arithmatex">\(D_{\text{KL}}(p \| q) = H(p, q) - H(p)\)</span>。
|
||
<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.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="k">def</span><span class="w"> </span><span class="nf">cross_entropy</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a> <span class="k">return</span> <span class="o">-</span><span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">p</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">log2</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">clip</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="mf">1e-10</span><span class="p">,</span> <span class="mf">1.0</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="k">def</span><span class="w"> </span><span class="nf">kl_divergence</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a> <span class="n">mask</span> <span class="o">=</span> <span class="n">p</span> <span class="o">></span> <span class="mi">0</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a> <span class="k">return</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">mask</span><span class="p">,</span> <span class="n">p</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">log2</span><span class="p">(</span><span class="n">p</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">clip</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="mf">1e-10</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">)),</span> <span class="mf">0.0</span><span class="p">))</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="k">def</span><span class="w"> </span><span class="nf">entropy</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a> <span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="n">p</span> <span class="o">></span> <span class="mi">0</span><span class="p">]</span>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a> <span class="k">return</span> <span class="o">-</span><span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">p</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">log2</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="n">p</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="mf">0.4</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">])</span> <span class="c1"># 真实分布</span>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">q</span> <span class="ow">in</span> <span class="p">[(</span><span class="s2">"完全匹配"</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a> <span class="p">(</span><span class="s2">"轻微偏差"</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="mf">0.35</span><span class="p">,</span> <span class="mf">0.30</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.10</span><span class="p">])),</span>
|
||
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a> <span class="p">(</span><span class="s2">"严重偏差"</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="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">]))]:</span>
|
||
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a> <span class="n">h_p</span> <span class="o">=</span> <span class="n">entropy</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
|
||
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a> <span class="n">h_pq</span> <span class="o">=</span> <span class="n">cross_entropy</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
|
||
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a> <span class="n">kl</span> <span class="o">=</span> <span class="n">kl_divergence</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
|
||
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></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">name</span><span class="si">:</span><span class="s2">20s</span><span class="si">}</span><span class="s2">: H(p)=</span><span class="si">{</span><span class="n">h_p</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">, H(p,q)=</span><span class="si">{</span><span class="n">h_pq</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">, "</span>
|
||
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a> <span class="sa">f</span><span class="s2">"KL=</span><span class="si">{</span><span class="n">kl</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">, H(p,q)-H(p)=</span><span class="si">{</span><span class="n">h_pq</span><span class="o">-</span><span class="n">h_p</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>通过计算两个不同分布之间的 <span class="arithmatex">\(D_{\text{KL}}(p \| q)\)</span> 和 <span class="arithmatex">\(D_{\text{KL}}(q \| p)\)</span>，证明 KL 散度不是对称的。
|
||
<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.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="k">def</span><span class="w"> </span><span class="nf">kl_div</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="n">mask</span> <span class="o">=</span> <span class="n">p</span> <span class="o">></span> <span class="mi">0</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">mask</span><span class="p">,</span> <span class="n">p</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">log2</span><span class="p">(</span><span class="n">p</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">clip</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="mf">1e-10</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">)),</span> <span class="mf">0.0</span><span class="p">)))</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="n">p</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="mf">0.9</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">])</span>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">q</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="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">])</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="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"D_KL(p || q) = </span><span class="si">{</span><span class="n">kl_div</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="w"> </span><span class="n">q</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"D_KL(q || p) = </span><span class="si">{</span><span class="n">kl_div</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="nb">print</span><span class="p">(</span><span class="s2">"不相同！KL 散度是不对称的。"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>模拟训练过程中交叉熵损失的变化。创建一个"真实"的 one-hot 标签，展示随着模型预测概率的改善，损失如何下降。
|
||
<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="c1"># 真实标签：4 个类别中的第 2 类</span>
|
||
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="n">true_label</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">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</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="c1"># 模拟预测逐步改善</span>
|
||
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="n">steps</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="n">losses</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="k">for</span> <span class="n">confidence</span> <span class="ow">in</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.99</span><span class="p">,</span> <span class="mi">50</span><span class="p">):</span>
|
||
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a> <span class="c1"># 模型对类别 2 的置信度逐渐提高</span>
|
||
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a> <span class="n">remaining</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">confidence</span><span class="p">)</span> <span class="o">/</span> <span class="mi">3</span>
|
||
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a> <span class="n">pred</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">remaining</span><span class="p">,</span> <span class="n">remaining</span><span class="p">,</span> <span class="n">confidence</span><span class="p">,</span> <span class="n">remaining</span><span class="p">])</span>
|
||
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a> <span class="n">loss</span> <span class="o">=</span> <span class="o">-</span><span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">true_label</span> <span class="o">*</span> <span class="n">jnp</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">clip</span><span class="p">(</span><span class="n">pred</span><span class="p">,</span> <span class="mf">1e-10</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">)))</span>
|
||
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a> <span class="n">steps</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">confidence</span><span class="p">))</span>
|
||
<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a> <span class="n">losses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">loss</span><span class="p">))</span>
|
||
<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a>
|
||
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></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-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">steps</span><span class="p">,</span> <span class="n">losses</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">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"模型对真实类别的置信度"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"交叉熵损失"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"交叉熵损失随预测改善而下降"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
|
||
<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></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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