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&figure&&img src=&https://pic4.zhimg.com/v2-010ae91f347c4d05c5ea3_b.jpg& data-rawwidth=&591& data-rawheight=&762& class=&origin_image zh-lightbox-thumb& width=&591& data-original=&https://pic4.zhimg.com/v2-010ae91f347c4d05c5ea3_r.jpg&&&/figure&&p&首先感谢大家还关注这个专栏，前段时间一直在写一本书，题为《21个项目玩转深度学习——基于TensorFlow的实践详解》。&/p&&p&简单地介绍一下，这本书的定位是深度学习+TensorFlow的入门实验书，&b&希望提供一些在TensorFlow下常用的、并且真正能work的实践项目，方便初学者通过动手实验来快速入门。&/b&&/p&&p&目前该书已全面上架京东（&a href=&http://link.zhihu.com/?target=https%3A//item.jd.com/.html& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&https://&/span&&span class=&visible&&item.jd.com/.ht&/span&&span class=&invisible&&ml&/span&&span class=&ellipsis&&&/span&&/a&），天猫（&a href=&http://link.zhihu.com/?target=https%3A//detail.tmall.com/item.htm%3Fid%3D& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&https://&/span&&span class=&visible&&detail.tmall.com/item.h&/span&&span class=&invisible&&tm?id=&/span&&span class=&ellipsis&&&/span&&/a&）、当当（&a href=&http://link.zhihu.com/?target=http%3A//product.dangdang.com/.html& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&product.dangdang.com/25&/span&&span class=&invisible&&245282.html&/span&&span class=&ellipsis&&&/span&&/a&）价格非常便宜，只需要60块钱。如果不想买书的话也欢迎来star这本书的Github：&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&hzy46/Deep-Learning-21-Examples&/a&，全部项目的代码和运行方式已经整理上传了~~&/p&&h2&&b&有哪些实践案例&/b&&/h2&&p&从难易度、实用性、趣味性考量，这本书选择了21个入门的项目，主要有三个大类：&/p&&ul&&li&CNN、图像相关：包含&b&图像分类&/b&、&b&目标检测&/b&、&b&人脸识别&/b&、&b&风格迁移&/b&，同时包含&b&GAN&/b&、&b&cGAN&/b&、&b&CycleGAN&/b&等和GAN相关的内容&/li&&li&RNN、序列相关：&b&文本生成&/b&、&b&序列分类&/b&、&b&训练词嵌入&/b&、&b&时间序列预测&/b&、&b&机器翻译&/b&等等。&/li&&li&强化学习：主要复现一些基础的算法，如&b&Q Learning&/b&、&b&SARSA&/b&、&b&Deep Q Learning&/b&等。&/li&&/ul&&p&所有项目都基于TensorFlow大于等于1.4.0的版本，具体案例如下：&/p&&p&&b&一、CNN与计算机视觉：&/b&&/p&&p&&b&1、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_1/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&MNIST机器学习入门&/a&&/b&&/p&&p&基础的MNIST手写数字识别，你懂的。&/p&&figure&&img src=&https://pic3.zhimg.com/v2-cd3a645fd6351a1fadea_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1840& data-rawheight=&876& class=&origin_image zh-lightbox-thumb& width=&1840& data-original=&https://pic3.zhimg.com/v2-cd3a645fd6351a1fadea_r.jpg&&&/figure&&p&&b&2、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_2/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&CIFAR-10与ImageNet图像识别&/a&&/b&&/p&&p&在CIFAR-10（同样也是一个常用数据集，不过和MNIST不同的是它是三通道彩色图像）上训练一个图像识别模型。&/p&&figure&&img src=&https://pic4.zhimg.com/v2-3d4dc157dd06b3a73c6df971d6de8703_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&487& data-rawheight=&365& class=&origin_image zh-lightbox-thumb& width=&487& data-original=&https://pic4.zhimg.com/v2-3d4dc157dd06b3a73c6df971d6de8703_r.jpg&&&/figure&&p&&b&3、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_3/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&打造自己的图像识别模型&/a&&/b&&/p&&p&从零开始，如何在自己的数据集上训练ImageNet模型进行图像识别。&/p&&p&书中提供了一个网上爬取的卫星图像数据集，共有森林、水域、岩石、湿地、冰川、城市六类共6000章图像。你将体验数据转tfrecord =& 训练模型 =& 调整参数 =& 验证正确率 =& 导出模型 =& 使用导出模型测试单张图片的全过程。&/p&&p&&b&4、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_4/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&Deep Dream&/a&&/b&&/p&&p&一个比较有趣的图像生成模型，最初由Google公布，可以帮助我们理解卷积神经网络的特征背后的含义。&/p&&p&简单来说，极大化卷积网络某个通道的输出，就可以得到如下所示的展示该通道特征的图片：&/p&&figure&&img src=&https://pic1.zhimg.com/v2-3b6aa022e7d54a25b982ca8c8ae2b748_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&2042& data-rawheight=&864& class=&origin_image zh-lightbox-thumb& width=&2042& data-original=&https://pic1.zhimg.com/v2-3b6aa022e7d54a25b982ca8c8ae2b748_r.jpg&&&/figure&&p&&b&5、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_5/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&深度学习中的目标检测&/a&&/b&&/p&&p&目标检测，计算机视觉经典问题之一。这里以TensorFlow Object Detection API为例，在VOC2009数据集上，训练了一个Faster R-CNN模型，并且导出模型对单张图片进行检测，测试效果如下：&/p&&figure&&img src=&https://pic3.zhimg.com/v2-54d56ac961c58f34647c41e_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1264& data-rawheight=&546& class=&origin_image zh-lightbox-thumb& width=&1264& data-original=&https://pic3.zhimg.com/v2-54d56ac961c58f34647c41e_r.jpg&&&/figure&&p&&b&6、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_6/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&人脸检测和人脸识别&/a&&/b&&/p&&p&人脸检测和识别也是常见的项目。检测和对齐人脸用的是MTCNN，训练人脸模型用的是带有中心损失的CNN。&/p&&p&这章包含的内容有：验证已经训练好的模型的准确率、使用已经训练好的模型进行人脸识别、训练自己的模型。&/p&&p&&b&7、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_7/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&图像风格迁移&/a&&/b&&/p&&p&神经网络的风格迁移，可以生成比较酷炫的图片，例如：&/p&&figure&&img src=&https://pic1.zhimg.com/v2-c38c119251dbc5fe0f08_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1918& data-rawheight=&924& class=&origin_image zh-lightbox-thumb& width=&1918& data-original=&https://pic1.zhimg.com/v2-c38c119251dbc5fe0f08_r.jpg&&&/figure&&p&&b&二、GAN以及它的几个重要变体&/b&&/p&&p&&b&8、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_8/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&GAN与DCGAN入门&/a&&/b&&/p&&p&这章介绍了用GAN和DCGAN的基本原理，并利用DCGAN做了一个动漫人物生成Demo，生成的图片如下所示：&/p&&figure&&img src=&https://pic1.zhimg.com/v2-c010248adb1f0c902e03a_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1092& data-rawheight=&620& class=&origin_image zh-lightbox-thumb& width=&1092& data-original=&https://pic1.zhimg.com/v2-c010248adb1f0c902e03a_r.jpg&&&/figure&&p&&b&9、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_9/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&pix2pix模型与自动上色技术&/a&&/b&&/p&&p&使用pix2pix模型将黑白图像转换成彩色图像，这一章有两个实验，分别是将黑白的食物图像和动漫图像转换为彩色图像。&/p&&p&上色效果如下（从左到右依次是，黑白图片、自动上色的图片、真实图片）：&/p&&figure&&img src=&https://pic2.zhimg.com/v2-81d11c28c5eeedc33cfd815_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&818& data-rawheight=&268& class=&origin_image zh-lightbox-thumb& width=&818& data-original=&https://pic2.zhimg.com/v2-81d11c28c5eeedc33cfd815_r.jpg&&&/figure&&p&&b&10、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_10/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&超分辨率：让图像变得更清晰&/a&&/b&&/p&&p&使用pix2pix模型，将模糊的图片还原为清晰的图片。&/p&&p&效果如下（从左到右依次为，模糊处理后的图片、清晰化后的图片、真实图片）：&/p&&figure&&img src=&https://pic4.zhimg.com/v2-bbc9e22a3d6dfdf8ce0e290e24ea1ff3_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1052& data-rawheight=&342& class=&origin_image zh-lightbox-thumb& width=&1052& data-original=&https://pic4.zhimg.com/v2-bbc9e22a3d6dfdf8ce0e290e24ea1ff3_r.jpg&&&/figure&&p&&b&11、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_11/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&CycleGAN与非配对图像转换&/a&&/b&&/p&&p&使用CycleGAN进行图片转换，做了两个实验：苹果变成橘子，男人变成女人。&/p&&p&&b&三、RNN与自然语言处理&/b&&/p&&p&&b&12、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_12/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&RNN基本结构与Char RNN文本生成&/a&&/b&&/p&&p&主要内容为：RNN、LSTM基础结构。介绍了一个比较简单的Char RNN的例子，它文本生成、机器写诗。&/p&&p&写诗示例：&/p&&div class=&highlight&&&pre&&code class=&language-text&&&span&&/span&何人无不见，此地自何如。
一夜山边去，江山一夜归。
山风春草色，秋水夜声深。
何事同相见，应知旧子人。
何当不相见，何处见江边。
一叶生云里，春风出竹堂。
何时有相访，不得在君心。
&/code&&/pre&&/div&&p&&b&13、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_13/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&序列分类问题详解&/a&&/b&&/p&&p&一类数列为等差数列，一类数列为随机数列，使用RNN，达到99%的分类准确率。&/p&&p&&b&14、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_14/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&词的向量表示：word2vec与词嵌入&/a&&/b&&/p&&p&词嵌入在各种自然语言处理问题时经常用到，这章使用Skip-gram方法训练一个词嵌入，并进行简单的可视化。&/p&&figure&&img src=&https://pic1.zhimg.com/v2-a0d297de65c2df4b1a20_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1736& data-rawheight=&608& class=&origin_image zh-lightbox-thumb& width=&1736& data-original=&https://pic1.zhimg.com/v2-a0d297de65c2df4b1a20_r.jpg&&&/figure&&p&&b&15、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_15/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&在TensorFlow中进行时间序列预测&/a&&/b&&/p&&p&用TensorFlow来解决时间序列预测问题，预测方法包括传统的AR模型和LSTM模型。&/p&&p&预测效果如下（红色部分为预测值）：&/p&&figure&&img src=&https://pic3.zhimg.com/v2-061f118aeee53ff8209c74a_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1500& data-rawheight=&500& class=&origin_image zh-lightbox-thumb& width=&1500& data-original=&https://pic3.zhimg.com/v2-061f118aeee53ff8209c74a_r.jpg&&&/figure&&p&&b&16、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_16/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&神经网络机器翻译技术&/a&&/b&&/p&&p&这一章会训练一个把英文翻译成中文的模型。&/p&&p&例如，输入英文：It is learned that china has on many occasions previously made known its stand on its relations with the g-8.&/p&&p&可以翻译为中文：据悉，中国曾多次表明同八国集团之间的关系。&/p&&p&使用的代码为TensorFlow NMT。&/p&&p&&b&17、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_17/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&看图说话：将图像转换为文字&/a&&/b&&/p&&p&给定一张图片：&/p&&figure&&img src=&https://pic2.zhimg.com/v2-3adff6ec008e302bc100f122d2431965_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1648& data-rawheight=&712& class=&origin_image zh-lightbox-thumb& width=&1648& data-original=&https://pic2.zhimg.com/v2-3adff6ec008e302bc100f122d2431965_r.jpg&&&/figure&&p&输出一句描述它的话，如&a man riding a wave on top of a surfboard .&&/p&&p&&b&四、强化学习基础与深度强化学习&/b&&/p&&p&&b&18、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_18/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&强化学习入门之Q Learning&/a&&/b&&/p&&p&借助一个走迷宫问题，来讲解强化学习基础算法 —— Q Learning。&/p&&p&&b&19、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_19/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&强化学习入门之SARSA算法&/a&&/b&&/p&&p&解决的问题和上一章类似，不过使用的算法是另一种强化学习基础算法SARSA。&/p&&p&&b&20、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_20/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&深度强化学习：Deep Q Learning&/a&&/b&&/p&&p&训练一个DQN，来玩打砖块游戏。&/p&&figure&&img src=&https://pic4.zhimg.com/v2-4a02b2504aeaebc_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&2144& data-rawheight=&854& class=&origin_image zh-lightbox-thumb& width=&2144& data-original=&https://pic4.zhimg.com/v2-4a02b2504aeaebc_r.jpg&&&/figure&&p&&b&21、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_21/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&策略梯度（Policy Gradient）算法&/a&&/b&&/p&&p&训练一个策略梯度网络，来玩Cartpole。&/p&&figure&&img src=&https://pic2.zhimg.com/v2-5fb3abbbf27d4f58af41_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1728& data-rawheight=&676& class=&origin_image zh-lightbox-thumb& width=&1728& data-original=&https://pic2.zhimg.com/v2-5fb3abbbf27d4f58af41_r.jpg&&&/figure&&h2&&b&为什么写这本书&/b&&/h2&&p&去年6月份，电子工业出版社的一位编辑朋友找到我，希望能够帮忙编写一本以项目合集为形式的深度学习入门书。当时没有怎么犹豫就答应下来了，一方面，市面上确实缺少一本这样以实践案例组成的入门书，另一方面，我个人也觉得，&b&实践是入门深度学习的的极佳方式，原因如下：&/b&&/p&&ul&&li&在学习的初期，这种方式具有趣味感、成就感，比较容易坚持&/li&&li&深度学习需要深厚的数理基础，对于初学者来说不是很友好，另外TensorFlow这个框架也不是很容易学习。使用具体例子来理解更加直观有效。&/li&&li&深度学习更偏向是一门“实验科学”。很多深度学习算法都存在一些无法用理论解释的trick和训练技巧，这些东西如果不是自己去经历一次是很难有直观感受的。&/li&&/ul&&p&另外，在这个专栏中，我曾经写过很多比较有意思的深度学习实践项目。这本书是在专栏的基础上，添加了更多的项目，整理成册，希望提供一个比较完整的深度学习入门案例合集。&/p&&h2&&b&最后&/b&&/h2&&p&最后大家如果对这本书有兴趣的话，可以猛戳下面的购买链接购买：&/p&&ul&&li&京东（&a href=&http://link.zhihu.com/?target=https%3A//item.jd.com/.html& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&https://&/span&&span class=&visible&&item.jd.com/.ht&/span&&span class=&invisible&&ml&/span&&span class=&ellipsis&&&/span&&/a&）&/li&&li&天猫（&a href=&http://link.zhihu.com/?target=https%3A//detail.tmall.com/item.htm%3Fid%3D& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&https://&/span&&span class=&visible&&detail.tmall.com/item.h&/span&&span class=&invisible&&tm?id=&/span&&span class=&ellipsis&&&/span&&/a&）&/li&&li&当当（&a href=&http://link.zhihu.com/?target=http%3A//product.dangdang.com/.html& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&product.dangdang.com/25&/span&&span class=&invisible&&245282.html&/span&&span class=&ellipsis&&&/span&&/a&）&/li&&li&亚马逊（&a href=&http://link.zhihu.com/?target=https%3A//www.amazon.cn/dp/B07BF39RHQ& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&https://www.&/span&&span class=&visible&&amazon.cn/dp/B07BF39RHQ&/span&&span class=&invisible&&&/span&&/a&）&/li&&/ul&&p&也欢迎随时访问本书的Github：&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&hzy46/Deep-Learning-21-Examples&/a& 吐槽、提issue，在这里谢谢各位啦，比心！&/p&
首先感谢大家还关注这个专栏，前段时间一直在写一本书，题为《21个项目玩转深度学习——基于TensorFlow的实践详解》。简单地介绍一下，这本书的定位是深度学习+TensorFlow的入门实验书，希望提供一些在TensorFlow下常用的、并且真正能work的实践项目，方便…
&figure&&img src=&https://pic2.zhimg.com/v2-678fe1ce18bdfd_b.jpg& data-rawwidth=&1482& data-rawheight=&1080& class=&origin_image zh-lightbox-thumb& width=&1482& data-original=&https://pic2.zhimg.com/v2-678fe1ce18bdfd_r.jpg&&&/figure&此文的灵感来自360云盘关闭。&p&&figure&&img src=&https://pic1.zhimg.com/v2-ec1e44faf51_b.jpg& data-rawwidth=&800& data-rawheight=&450& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic1.zhimg.com/v2-ec1e44faf51_r.jpg&&&/figure&NAS（网络附加存储），其实就是存储服务器。&br&在这个网盘纷纷关闭的萧瑟的深秋，我觉得有必要聊聊这个话题。&br&这个东西，对我来说是一个偶然，但是用了之后，成了必然。&br&最初，我的MacBook air是128G的，我还喜欢拍照用RAW格式，你说怎么就这么命苦？&br&移动硬盘倒是可以，但是我有三台笔记本、两个平板、三个手机，这个可怎么办？&br&于是，遇到NAS这个名词，脑海中有个声音说，来一台吧，然后来了一台。&br&&figure&&img src=&https://pic3.zhimg.com/v2-c66f1c2cd748a977adc25bd3_b.jpg& data-rawwidth=&2011& data-rawheight=&1718& class=&origin_image zh-lightbox-thumb& width=&2011& data-original=&https://pic3.zhimg.com/v2-c66f1c2cd748a977adc25bd3_r.jpg&&&/figure&最初使用它的作用主要是扩展MBA的存储，那时候我都是用iPhoto修图，有了NAS之后，再也不担心MBA的存储了，同时，NAS可以作为苹果TimeMachine的备份盘，让苹果时光机发挥作用；而且可以多个设备共用，就像网盘客户端。&br& 不乱说了，系统一点讲吧。本篇是入门篇，主要讲一些简单的应用，高级应用将在下一篇里讲解。&br&哦，对了，还有一句要说的就是：这玩意不贵，门槛很低。&br&&br&&br&&b&1 NAS能做什么&/b&&br&NAS作为一台存储服务器，其实就是一台配置不高但是存储空间巨大，功耗很低可以长时间运行的电脑。它的主要功能就是存储，形成家庭或者办公室的数据中心，所以最主要的功能就是存储空间的共享，然后就是在此基础上附加的一些功能：数据同步（个人云）、照片共享、音乐库、影音库、下载、数据备份等功能，一项一项说说吧。&/p&&p&&figure&&img src=&https://pic4.zhimg.com/v2-bf23cdf0a37ea139a84a6_b.jpg& data-rawwidth=&1024& data-rawheight=&966& class=&origin_image zh-lightbox-thumb& width=&1024& data-original=&https://pic4.zhimg.com/v2-bf23cdf0a37ea139a84a6_r.jpg&&&/figure&&br&&b&共享存储：&/b&你所有的设备，只要可以和NAS连接，就可以访问NAS中存储的数据，所以你可以在手机、笔记本、平板、智能电视等等各种设备上直接访问NAS，就像使用本地数据一样使用NAS中存储的数据，所以这是一件相当方便的事。&br&比如你把拍摄的照片存在NAS里，然后你用你的MBA修图，修好后的图片可以直接在平板和智能电视观看，是不是很方便？而且，假如你的NAS总共有4T空间，则相当于你的所有设备都有了这4T的空间。&br&&b&多媒体共享&/b&：这一部分包括照片、音乐和视频，存在NAS中的多媒体文件可以直接以你想要的方式呈现到你希望呈现的设备。比如你在看NAS上的电影、你女朋友可以查看NAS上存储的照片，你出差在外，可以直接网页登录到家中的NAS上，欣赏你存储在NAS里的音乐。&br&&b&挂机下载：&/b&这个很简单，因为NAS就是一台电脑，不过就是功耗低，一直开着而已，所以你想下载的东西可以直接丢给它，让它给你下载就是了，反正它有的是时间。&br&&b&数据备份：&/b&NAS都具备强大的数据备份功能，确保数据安全，当然，人家NAS就是干这个的。你可以对你的数据进行RAID的保护，也可以设置备份任务直接备份到网盘，也可以将重要数据自动备份到移动硬盘中，反正就是可以保证你的数据非常安全，这一部分内容很多，到下一篇里面会详细讲解，你所要知道的就是，这玩意存储可靠性非常高，有三四层楼那么高。&br&除了这些功能，很多NAS还有一些特色功能，比如说直接解码高清电影，相当于备具了硬盘播放机的功能；还有可以直接外接HIFI声卡作为音源；因为一直开着，所以可以作为自己的网站服务器，把自己的个人主页挂在上面；还可以作为监控摄像头的存储盘，直接存储摄像头录制的视频；有的还带有Android系统，兼容各种安卓APP，还可以打植物大战僵尸～～哎，太多了，一时半会说不完。&br&这么多功能，是不是用起来很麻烦？我当初也是这么想，后来了解了一下就不这么想了，下面就说说NAS的用法。&br&&br&&b&2 NAS怎么用&/b&&br&话不多说，直接上图，这就是一台NAS的管理界面。&br&这个界面怎么来的？网页登录上去就是了，既可以在家中登录，也可以在任何一个网络能够联通的地方登录。&br&&br&这是QNAP（著名NAS厂家）的NAS管理页面，系统名称叫QTS，一看就明白了，用起来非常简单。&br&&figure&&img src=&https://pic3.zhimg.com/v2-5ad6c3eef7b75aeaeaebbba_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic3.zhimg.com/v2-5ad6c3eef7b75aeaeaebbba_r.jpg&&&/figure&&br&说说其中重要的几个。&br&控制台：用于调整NAS的各种设置。&br&&br&&figure&&img src=&https://pic1.zhimg.com/v2-cd818087bff8_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic1.zhimg.com/v2-cd818087bff8_r.jpg&&&/figure&Music Station数字音乐库：可以播放NAS中存储的音乐，既可以推送到本地，也可以直接让NAS通过外接声卡输出。&br&&br&&figure&&img src=&https://pic3.zhimg.com/v2-98f002a02d79de6a1db03_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic3.zhimg.com/v2-98f002a02d79de6a1db03_r.jpg&&&/figure&Photo Station照片时光机：可以显示在NAS中存储的照片和视频，只要有网，随时随地查看。&br&&br&&figure&&img src=&https://pic3.zhimg.com/v2-2bf6e1f2e3aa221e4b290ac74d5c17d7_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic3.zhimg.com/v2-2bf6e1f2e3aa221e4b290ac74d5c17d7_r.jpg&&&/figure&File Station文件总管：直接管理NAS中存储的文件，可以上传下载，移动复制删除，也可以随时打开或者播放。&br&&br&&figure&&img src=&https://pic2.zhimg.com/v2-d92feba39ac69de01a39aca0c89ab111_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic2.zhimg.com/v2-d92feba39ac69de01a39aca0c89ab111_r.jpg&&&/figure&备份管理中心：可以选择使用各种方式对你的数据进行备份。&br&&br&&figure&&img src=&https://pic4.zhimg.com/v2-fd2a94dccc0_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic4.zhimg.com/v2-fd2a94dccc0_r.jpg&&&/figure&Qsync Central Station：这是我最喜欢也是最常用的功能，就是强大的同步。类似金山快盘或者坚果云这种网盘的同步，你可以在Windows系统、Mac系统、安卓系统、iOS系统等各种设备上安装客户端实现和网盘同步一样的功能，而且所有数据都在自己的服务器上，再也不怕网盘动不动就删你的文件。实在是网盘关闭后的必备良药啊。这一部分在第二篇中还要更详细的讲解。&br&&br&&figure&&img src=&https://pic4.zhimg.com/v2-75b47cf7939c_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic4.zhimg.com/v2-75b47cf7939c_r.jpg&&&/figure&APP Center：现在的NAS也是类似iOS和安卓系统一样有APP商店，直接点击安装，非常方便。各种功能的APP一应具全：同步备份、下载、娱乐、监控、系统工具、教育、家庭自动化等等～&br&&br&&figure&&img src=&https://pic1.zhimg.com/v2-4db6d21b0c64d7deb647_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic1.zhimg.com/v2-4db6d21b0c64d7deb647_r.jpg&&&/figure&Download Station：挂机下载，直接把需要下载资源的URL或者种子文件添加进来，NAS就会一刻不停的替你下载，除了常见的HTTP和FTP下载，还可以支持QQ下载、迅雷、磁力链接、快车下载等各种类型的下载URL。&br&&figure&&img src=&https://pic3.zhimg.com/v2-46636b2cab8bfd7bf2e7b6f4c6346c99_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic3.zhimg.com/v2-46636b2cab8bfd7bf2e7b6f4c6346c99_r.jpg&&&/figure&当然，以上说的还只是常用功能，还有好多功能大家可以慢慢研究。&br&&br&&br&&b&3 谁需要NAS&/b&&br&前面说过，NAS很便宜。&br&最便宜的一盘位（仅能插一块硬盘）NAS价格都在一千以内了，当然，笔者并不推荐一盘位的，因为一盘位很多功能受限制。家用的话，两盘位或者四盘位是比较理想的选择，当然对容量要求不高，那么两盘位是比较理想的。国内做NAS比较著名的厂商就是群晖和QNAP（威联通），笔者使用的两台NAS都是QNAP产品（因为同一厂家的产品可以方便的互换硬盘或者同步数据），这两个品牌的家用两盘位NAS产品的价格都在2000左右，性价比还是很高的。&br&什么？你觉得NAS会很丑？&br&话不多说，直接上图，这是笔者用过了两款NAS，分别是TS－212P和TAS－268，颜值自己看吧。&br&&figure&&img src=&https://pic4.zhimg.com/v2-0cd1c1ffad7f03dda4132d4_b.jpg& data-rawwidth=&2383& data-rawheight=&2415& class=&origin_image zh-lightbox-thumb& width=&2383& data-original=&https://pic4.zhimg.com/v2-0cd1c1ffad7f03dda4132d4_r.jpg&&&/figure&&figure&&img src=&https://pic4.zhimg.com/v2-cb6acf4aec20ce9c772e06_b.jpg& data-rawwidth=&2048& data-rawheight=&3025& class=&origin_image zh-lightbox-thumb& width=&2048& data-original=&https://pic4.zhimg.com/v2-cb6acf4aec20ce9c772e06_r.jpg&&&/figure&&br&公平起见，也上台群晖的产品图片。&br&&br&&figure&&img src=&https://pic3.zhimg.com/v2-224af295f9a6cc2ad569f4fcc67c0811_b.jpg& data-rawwidth=&1024& data-rawheight=&1024& class=&origin_image zh-lightbox-thumb& width=&1024& data-original=&https://pic3.zhimg.com/v2-224af295f9a6cc2ad569f4fcc67c0811_r.jpg&&&/figure&&figure&&img src=&https://pic2.zhimg.com/v2-0b1fcf44ec32b1aadf79ed357aa240dd_b.jpg& data-rawwidth=&1024& data-rawheight=&1024& class=&origin_image zh-lightbox-thumb& width=&1024& data-original=&https://pic2.zhimg.com/v2-0b1fcf44ec32b1aadf79ed357aa240dd_r.jpg&&&/figure&&br&下面说说谁特别需要NAS。&br&（1）网盘用户，存储了大量数据，网盘关闭了，数据转存到NAS才是出路。&br&（2）使用多台手机、笔记本、平板、台式机的用户，使用NAS可以方便的实现数据共享，相当于所有的设备都有了一块巨大的硬盘。&br&（3）对同步有要求的用户。大家都知道，网盘的同步是非常好用的，但是网盘本身不安全，随时可能关闭，且数据安全也不容易保证，但是存放在自己的NAS里，多台设备通过NAS同步则完全没有问题。&br&（4）家中人数较多，需要共享存储而不想抱着移动硬盘插来插去的。这个就不多说了。&br&（5）存储私密数据，网盘你懂的，8秒教育片你也懂得，老司机必备。&br&根据功能继续想把，我目前想到了这些～&/p&&br&&b&4 回答一些评论问题&/b&&br&文章发表以后，评论区讨论的也挺热，所以在此一并回答一下评论区的焦点问题。&br&（1）最简单的问题：NAS和云盘什么区别？本质区别就是NAS是你的数据在你的服务器你的硬盘上你说了算，而云盘是你的数据在别人服务器上别人硬盘上别人说了算，参考百度云删除用户数据和8秒教育片。&br&（2）关于网络问题，使用NAS的话肯定要架设家用千兆内网，千兆内网这个名词一听上去好像特复杂，其实很简单啦，买个支持千兆有线和802.11AC无线的宽带路由器就搞定了，这种类型的路由器价格一般在400元左右或以上，当然现在的家庭，手机笔记本平板电视特别多，没有个高性能路由器有不行。&br&（3）关于NAS的DIY，这个的确是可以，我当时也想过，GEN8也是一个很不错的平台，但是对于多数用户而言，门槛要高一些，除了自己搭建硬件平台控制功耗以外，还要自己安装服务器端的操作系统和各种应用软件，不如直接入一台NAS，软件硬件都有了，而且界面良好，说明书详细，上手非常容易。&br&（4）有朋友提小米的带硬盘的路由器，还有可以插硬盘的那种路由器，怎么说呢，如果你对数据可靠性没什么要求，仅仅存点电影电视剧，也是可以的，它相当于一个简易的1盘位NAS，不过功能和性能上比专业NAS差不少，当然数据可靠性差的就更多了，这个在下一篇文章中会继续讨论。&br&（5）对于宽带的上传限速，这个就得在选择宽带的时候注意一下，尽量选择上传速度快的，当然，这主要影响到你在家庭以外的网络访问NAS的速度，其实多数情况下，NAS和普通网盘或云盘在外网用起来体验是差不多的。&br&如有疑问，欢迎在评论区或者私信提问，我会及时回答并更新此文。&br&&br&&p&如果你还是觉得NAS太复杂，对你来说没必要，那么可以看看我以前写的这篇文章：&/p&&p&&a href=&https://zhuanlan.zhihu.com/p/& class=&internal&&U盘变私人云——Magic Disc魔碟网盘 - 家+智能 - 知乎专栏&/a&&br&&br&&/p&&p&入门篇就到这里了，下一篇为进阶篇：&/p&&p&&a href=&https://zhuanlan.zhihu.com/p/& class=&internal&&网盘纷纷关闭的时候，让我们来说说NAS——进阶篇 - 家+智能 - 知乎专栏&/a&&/p&&p&为方便阅读，文章会在微信公众号发表，欢迎关注“家+智能”公众号：homeandsmart&/p&&p&新浪微博：Henix_Sun&/p&
此文的灵感来自360云盘关闭。NAS（网络附加存储），其实就是存储服务器。 在这个网盘纷纷关闭的萧瑟的深秋，我觉得有必要聊聊这个话题。 这个东西，对我来说是一个偶然，但是用了之后，成了必然。 最初，我的MacBook air是128G的，我还喜欢拍照用RAW格式，…
&figure&&img data-rawheight=&1920& src=&https://pic4.zhimg.com/v2-acbd13afd87_b.jpg& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic4.zhimg.com/v2-acbd13afd87_r.jpg&&&/figure&&br&&br&硕士在读，方向机器学习，也是刚入门，目前在啃这本神经网络和深度学习，三百多页，目前看了1/3，觉得干货满满，预计最多还要两周看完。想看明白不需要太多的数学功底，只要求自己清楚一些机器学习的术语和基本阅读能力，更棒的是，作者能够教你一行一行的教你敲代码（看完我只想说，原来神经网络的代码可以写的这么简洁）。&br&&br&&br&就拿BP神经网络来说吧，之前大概了解神经网络是个什么东西，也知道神经网络能通过反向传播来训练网络参数，但不懂反向传播到底是怎么工作的。这本书好就好在作者能够图文并茂，从最基本的求导给你展示反向传播是怎么训练参数。并且后面从交叉墒，softmax 两个个方面阐述了选用不同的cost function会如何帮助训练过程跳出瓶颈（相对于最小二乘法），而且还实验多次给你阐述过拟合是怎么产生的，我们有什么好的办法去避免过拟合…&br&&br&&br&半夜手机码字太累了，需要交流可以加微信，大家一起进步。&br&&br&&br&链接：&br&&br&&a href=&//link.zhihu.com/?target=http%3A//neuralnetworksanddeeplearning.com& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&neuralnetworksanddeeplearning.com&/span&&span class=&invisible&&&/span&&/a&
硕士在读，方向机器学习，也是刚入门，目前在啃这本神经网络和深度学习，三百多页，目前看了1/3，觉得干货满满，预计最多还要两周看完。想看明白不需要太多的数学功底，只要求自己清楚一些机器学习的术语和基本阅读能力，更棒的是，作者能够教你一行一行的…
&figure&&img src=&https://pic4.zhimg.com/v2-ed364c69aed8cbdd3c255_b.jpg& data-rawwidth=&600& data-rawheight=&279& class=&origin_image zh-lightbox-thumb& width=&600& data-original=&https://pic4.zhimg.com/v2-ed364c69aed8cbdd3c255_r.jpg&&&/figure&&h2&一、图模型&/h2&&p&我们可以将概率分布对应成图模型。首先定义随机变量集合&img src=&https://www.zhihu.com/equation?tex=V%3DX+%5Ccup+Y& alt=&V=X \cup Y& eeimg=&1&&，其中&img src=&https://www.zhihu.com/equation?tex=X& alt=&X& eeimg=&1&&表示&b&输入变量&/b&（观测变量）集合，&img src=&https://www.zhihu.com/equation?tex=Y& alt=&Y& eeimg=&1&&表示&b&输出变量&/b&（预测变量）集合，以词性标注问题（POS）为例，输入变量是句子中的每一个单词，输出变量是每个单词对应的词性标记。&/p&&p&我们记&img src=&https://www.zhihu.com/equation?tex=x_A& alt=&x_A& eeimg=&1&&是&img src=&https://www.zhihu.com/equation?tex=X& alt=&X& eeimg=&1&&的一个子集，&img src=&https://www.zhihu.com/equation?tex=y_A& alt=&y_A& eeimg=&1&&是&img src=&https://www.zhihu.com/equation?tex=Y& alt=&Y& eeimg=&1&&的一个子集，函数&img src=&https://www.zhihu.com/equation?tex=%5CPsi_A+%28x_A%2C+y_A%29& alt=&\Psi_A (x_A, y_A)& eeimg=&1&&是对应&img src=&https://www.zhihu.com/equation?tex=x_A& alt=&x_A& eeimg=&1&&和&img src=&https://www.zhihu.com/equation?tex=y_A& alt=&y_A& eeimg=&1&&的&b&兼容函数&/b&（compatibility function），因子&img src=&https://www.zhihu.com/equation?tex=F+%3D+%5C%7B+%5CPsi_A+%5C%7D& alt=&F = \{ \Psi_A \}& eeimg=&1&&表示所有兼容函数的集合，在本文中，假设所有兼容函数都有如下的形式，&img src=&https://www.zhihu.com/equation?tex=%5CPsi_A%28x_A%2C+y_A%29+%3D+%5Cexp+%5Cleft%5C%7B+%5Csum_k+%5Ctheta_%7BAk%7D+f_%7BAk%7D%28x_A%2C+y_A%29+%5Cright%5C%7D& alt=&\Psi_A(x_A, y_A) = \exp \left\{ \sum_k \theta_{Ak} f_{Ak}(x_A, y_A) \right\}& eeimg=&1&&，其中&img src=&https://www.zhihu.com/equation?tex=f_%7BAk%7D%28x_A%2Cy_A%29& alt=&f_{Ak}(x_A,y_A)& eeimg=&1&&是关于&img src=&https://www.zhihu.com/equation?tex=x_A& alt=&x_A& eeimg=&1&&和&img src=&https://www.zhihu.com/equation?tex=y_A& alt=&y_A& eeimg=&1&&的特征函数，&img src=&https://www.zhihu.com/equation?tex=%5Ctheta_%7BAk%7D& alt=&\theta_{Ak}& eeimg=&1&&是参数。定义判断函数&img src=&https://www.zhihu.com/equation?tex=%5Cbm%7B1%7D+%5C%7B+%5Ccdot+%5C%7D& alt=&\bm{1} \{ \cdot \}& eeimg=&1&&，如果括号里的值为真则函数值为1，否则为0。&/p&&p&对于特定的兼容函数集合，我们可以将输入变量、输出变量的联合概率分布对应成一个无向图模型&img src=&https://www.zhihu.com/equation?tex=G%3D%28V%2CF%29& alt=&G=(V,F)& eeimg=&1&&的形式，其中&img src=&https://www.zhihu.com/equation?tex=V& alt=&V& eeimg=&1&&表示图模型中的顶点，对应概率分布中的每一个输入/输出变量，&img src=&https://www.zhihu.com/equation?tex=F& alt=&F& eeimg=&1&&表示图模型中由不同顶点组成的关联，对应概率分布中的每一个兼容函数。如Equ1，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Bequation%7D%0Ap%28x%2Cy%29+%3D+%5Cprod_%7BA%7D+%5CPsi_A+%28x_A%7Cy_A%29+%3D+%5Cprod_A+%5Cexp+%5Cleft%5C%7B+%5Csum_k+%5Ctheta_%7BAk%7D+f_%7BAk%7D%28x_A%2C+y_A%29+%5Cright%5C%7D%0A%5Cend%7Bequation%7D& alt=&\begin{equation}
p(x,y) = \prod_{A} \Psi_A (x_A|y_A) = \prod_A \exp \left\{ \sum_k \theta_{Ak} f_{Ak}(x_A, y_A) \right\}
\end{equation}& eeimg=&1&&&p&接下来以朴素贝叶斯为例，对这个框架进行深入理解。&/p&&h2&二、朴素贝叶斯&/h2&朴素贝叶斯模型需要建模的是输入变量和输出变量的&b&联合概率分布&/b&&img src=&https://www.zhihu.com/equation?tex=p%28y%2Cx_1%2Cx_2%2C%5Ccdots%2Cx_N%29& alt=&p(y,x_1,x_2,\cdots,x_N)& eeimg=&1&&，有N个输入变量，每个输入变量表示一个特征，有1个输出变量，输出变量表示类别。以文本分类问题为例，输入变量是词袋中的每个单词，输入变量是文档类别。基于输入变量条件独立的假设，即任意两个输入变量之间是相互独立的，因此联合概率分布可以写成&img src=&https://www.zhihu.com/equation?tex=p%28y%2Cx_1%2C%5Ccdots%2Cx_N%29%3Dp%28y%29%5Cprod_%7Bi%3D1%7D%5E%7BN%7D+p%28x_i%7Cy%29+& alt=&p(y,x_1,\cdots,x_N)=p(y)\prod_{i=1}^{N} p(x_i|y) & eeimg=&1&&，对应的无向图模型如下。&p&&figure&&img src=&https://pic1.zhimg.com/v2-b050746baf36b9f7e85e1a8db94f4043_b.jpg& data-rawwidth=&579& data-rawheight=&302& class=&origin_image zh-lightbox-thumb& width=&579& data-original=&https://pic1.zhimg.com/v2-b050746baf36b9f7e85e1a8db94f4043_r.jpg&&&/figure&按照如下的对应关系，则可以将朴素贝叶斯模型的定义适配到上述Equ1的框架中。令，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0AF+%3D+%26%5C%7B%26+%5C%7B+f_%7B01%7D%28y%29%2C+%5Ccdots%2C+f_%7B0M%7D%28y%29+%5C%7D%2C+%5C%5C%0A%26%26+%5C%7B+f_%7B11%7D%28x_1%2Cy%29%2C+%5Ccdots%2C+f_%7B1M%7D%28x_1%2Cy%29+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+f_%7BN1%7D%28x_N%2Cy%29%2C+%5Ccdots%2C+f_%7BNM%7D%28x_N%2Cy%29+%5C%7D+%5C+%5C+%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
F = &\{& \{ f_{01}(y), \cdots, f_{0M}(y) \}, \\
&& \{ f_{11}(x_1,y), \cdots, f_{1M}(x_1,y) \}, \\
&& \cdots \\
&& \{ f_{N1}(x_N,y), \cdots, f_{NM}(x_N,y) \} \ \ \}
\end{eqnarray}& eeimg=&1&&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5CTheta+%3D+%26%5C%7B%26+%5C%7B+%5Ctheta_%7B01%7D%2C+%5Ccdots%2C+%5Ctheta_%7B0M%7D+%5C%7D%2C+%5C%5C%0A%26%26+%5C%7B+%5Ctheta_%7B11%7D%2C+%5Ccdots%2C+%5Ctheta_%7B1M%7D+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+%5Ctheta_%7BN1%7D%2C+%5Ccdots%2C+%5Ctheta_%7BNM%7D+%5C%7D+%5C+%5C+%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\Theta = &\{& \{ \theta_{01}, \cdots, \theta_{0M} \}, \\
&& \{ \theta_{11}, \cdots, \theta_{1M} \}, \\
&& \cdots \\
&& \{ \theta_{N1}, \cdots, \theta_{NM} \} \ \ \}
\end{eqnarray}& eeimg=&1&&&br&&p&其中，有如下关系&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%26%26+f_%7B0k%7D%28y%29+%3D%5Cbm%7B1%7D%5C%7B+y%3Dc_k+%5C%7D%2C+f_%7Bik%7D%28x_i%2Cy%29+%3D%5Cbm%7B1%7D%5C%7B+y%3Dc_k+%5C%7D+%5C%5C%0A%26%26+%5Ctheta_%7B0k%7D+%3D+%5Clog+p%28y%3Dc_k%29%2C+%5Ctheta_%7Bik%7D+%3D+%5Clog+p%28x_i%7Cy%3Dc_k%29+%5C%5C%0A%26%26+%5CPsi_i%28x_i%2Cy%29+%3D+%5Cexp+%5Cleft%5C%7B+%5Csum%5EM_%7Bk%3D1%7D+%5Ctheta_%7Bik%7D+f_%7Bik%7D%28x_i%2C+y%29+%5Cright%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
&& f_{0k}(y) =\bm{1}\{ y=c_k \}, f_{ik}(x_i,y) =\bm{1}\{ y=c_k \} \\
&& \theta_{0k} = \log p(y=c_k), \theta_{ik} = \log p(x_i|y=c_k) \\
&& \Psi_i(x_i,y) = \exp \left\{ \sum^M_{k=1} \theta_{ik} f_{ik}(x_i, y) \right\}
\end{eqnarray}& eeimg=&1&&&br&&p&根据这样的兼容函数和参数，带入到Equ1中，得到&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0Ap%28y%2Cx_1%2C%5Ccdots%2Cx_N%29+%26%3D%26+%5Cprod_%7Bi%3D0%7D%5EN+%5CPsi_i%28x_i%2Cy%29+%5C%5C%0A%26%3D%26+%5Cprod_%7Bi%3D0%7D%5EN+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5EM+%5Ctheta_%7Bik%7D+f_%7Bik%7D%28x_i%2Cy%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+%5Cexp+%5Cleft%5C%7B+%5Clog+p%28y%29+%5Cright%5C%7D+%5Cprod_%7Bi%3D1%7D%5EN+%5Cexp+%5Cleft%5C%7B+%5Clog+p%28x_i%7Cy%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+p%28y%29+%5Cprod_%7Bi%3D1%7D%5EN+%5Clog+p%28x_i%7Cy%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
p(y,x_1,\cdots,x_N) &=& \prod_{i=0}^N \Psi_i(x_i,y) \\
&=& \prod_{i=0}^N \exp \left\{ \sum_{k=1}^M \theta_{ik} f_{ik}(x_i,y) \right\} \\
&=& \exp \left\{ \log p(y) \right\} \prod_{i=1}^N \exp \left\{ \log p(x_i|y) \right\} \\
&=& p(y) \prod_{i=1}^N \log p(x_i|y)
\end{eqnarray}& eeimg=&1&&&br&&p&综上说述，按照给定的&img src=&https://www.zhihu.com/equation?tex=%5CPsi+& alt=&\Psi & eeimg=&1&&和&img src=&https://www.zhihu.com/equation?tex=%5CTheta+& alt=&\Theta & eeimg=&1&&就可以推导出朴素贝叶斯模型的公式，也就是说朴素贝叶斯模型可以转化成无向图模型。接下来推导出隐马尔科夫模型对应的无向图模型。&/p&&h2&三、隐马尔科夫模型&/h2&&p&隐马尔科夫模型需要建模的是输入变量和输出变量的&b&联合概率分布&/b&&img src=&https://www.zhihu.com/equation?tex=p%28y_1%2C+%5Ccdots%2C+y_T+%2C+x_1%2C+%5Ccdots%2C+x_T%29& alt=&p(y_1, \cdots, y_T , x_1, \cdots, x_T)& eeimg=&1&&，T个输入变量，表示每一个观测值，T个输出变量，表示每一个状态值。以词性标注问题为例，输入变量是句子中的每一个单词，输出变量是每个单词对应的词性标记。隐马尔科夫过程假设当前状态只由上一个状态决定，而与历史无关，并且当前观测只由当前状态决定，因此联合概率分布可以写成&img src=&https://www.zhihu.com/equation?tex=p%28y_1%2C+%5Ccdots%2C+y_T+%2C+x_1%2C+%5Ccdots%2C+x_T%29%3D%5Cprod_%7Bt%3D1%7D%5ET+p%28y_t%7Cy_%7Bt-1%7D%29p%28x_t%7Cy_t%29& alt=&p(y_1, \cdots, y_T , x_1, \cdots, x_T)=\prod_{t=1}^T p(y_t|y_{t-1})p(x_t|y_t)& eeimg=&1&&，对应的无向图模型如下。&br&&/p&&figure&&img src=&https://pic3.zhimg.com/v2-c586fbf39abdb_b.jpg& data-rawwidth=&693& data-rawheight=&304& class=&origin_image zh-lightbox-thumb& width=&693& data-original=&https://pic3.zhimg.com/v2-c586fbf39abdb_r.jpg&&&/figure&&p&按照如下的对应关系，则可以将朴素贝叶斯模型的定义适配到上述Equ1的框架中。令，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0AF+%3D+%26%5C%7B%26+%5C%7B+f_%7B11%7D%5E1%28y_t%3Ds_1%2Cy_%7Bt-1%7D%3Ds_1%29%2C+%5Ccdots%2C+f%5E1_%7B1M%7D%28y_t%3Ds_1%2Cy_%7Bt-1%7D%3Ds_M%29+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+f_%7BM1%7D%5E1%28y_t%3Ds_M%2Cy_%7Bt-1%7D%3Ds_1%29%2C+%5Ccdots%2C+f%5E1_%7BMM%7D%28y_t%3Ds_M%2Cy_%7Bt-1%7D%3Ds_M%29+%5C%7D%2C+%5C%5C%0A%26%26+%5C%7B+f_%7B11%7D%5E2+%28x_t%3Do_1%2Cy_t%3Ds_1%29%2C+%5Ccdots%2C+f%5E2_%7B1M%7D%28x_t%3Do_1%2Cy_t%3Ds_M%29+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+f_%7BN1%7D%5E2+%28x_t%3Do_N%2Cy_t%3Ds_1%29%2C+%5Ccdots%2C+f%5E2_%7BNM%7D%28x_t%3Do_N%2Cy_t%3Ds_M%29+%5C%7D+%5C+%5C+%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
F = &\{& \{ f_{11}^1(y_t=s_1,y_{t-1}=s_1), \cdots, f^1_{1M}(y_t=s_1,y_{t-1}=s_M) \}, \\
&& \cdots \\
&& \{ f_{M1}^1(y_t=s_M,y_{t-1}=s_1), \cdots, f^1_{MM}(y_t=s_M,y_{t-1}=s_M) \}, \\
&& \{ f_{11}^2 (x_t=o_1,y_t=s_1), \cdots, f^2_{1M}(x_t=o_1,y_t=s_M) \}, \\
&& \cdots \\
&& \{ f_{N1}^2 (x_t=o_N,y_t=s_1), \cdots, f^2_{NM}(x_t=o_N,y_t=s_M) \} \ \ \}
\end{eqnarray}& eeimg=&1&&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5CTheta+%3D+%26%5C%7B%26+%5C%7B+%5Ctheta%5E1_%7B11%7D%2C+%5Ccdots%2C+%5Ctheta%5E1_%7B1M%7D+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+%5Ctheta%5E1_%7BM1%7D%2C+%5Ccdots%2C+%5Ctheta%5E1_%7BMM%7D+%5C%7D%2C+%5C%5C%0A%26%26+%5C%7B+%5Ctheta%5E2_%7B11%7D%2C+%5Ccdots%2C+%5Ctheta%5E2_%7B1M%7D+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+%5Ctheta%5E2_%7BN1%7D%2C+%5Ccdots%2C+%5Ctheta%5E2_%7BNM%7D+%5C%7D+%5C+%5C+%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\Theta = &\{& \{ \theta^1_{11}, \cdots, \theta^1_{1M} \}, \\
&& \cdots \\
&& \{ \theta^1_{M1}, \cdots, \theta^1_{MM} \}, \\
&& \{ \theta^2_{11}, \cdots, \theta^2_{1M} \}, \\
&& \cdots \\
&& \{ \theta^2_{N1}, \cdots, \theta^2_{NM} \} \ \ \}
\end{eqnarray}& eeimg=&1&&&br&&p&为了方便记录，我们将上述的特征函数&img src=&https://www.zhihu.com/equation?tex=f_%7Bij%7D%5E1%28y_t%2C+y_%7Bt-1%7D%29%2Cf%5E2_%7Boi%7D%28y_t%2Cx_t%29& alt=&f_{ij}^1(y_t, y_{t-1}),f^2_{oi}(y_t,x_t)& eeimg=&1&&和参数&img src=&https://www.zhihu.com/equation?tex=%5Ctheta%5E1_%7Bij%7D%2C+%5Ctheta%5E2_%7Boi%7D& alt=&\theta^1_{ij}, \theta^2_{oi}& eeimg=&1&&写成一种形式，即特征函数写成&img src=&https://www.zhihu.com/equation?tex=f_k%28y_t%2Cy_%7Bt-1%7D%2Cx_t%29& alt=&f_k(y_t,y_{t-1},x_t)& eeimg=&1&&，参数写成&img src=&https://www.zhihu.com/equation?tex=%5Ctheta_k& alt=&\theta_k& eeimg=&1&&。则，有如下关系，K表示所有特征函数的个数，也就是所有参数的个数。&br&&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%0A%26%26+f_%7Bij%7D%5E1%28y_t%3Ds_i%2Cy_%7Bt-1%7D%3Ds_j%29%3D%5Cbm%7B1%7D%5C%7By_t%3Ds_i%5C%7D%5Cbm%7B1%7D%5C%7By_%7Bt-1%7D%3Ds_j%5C%7D%2C+%5C%5C%0A%26%26+f_%7Bli%7D%5E2%28x_t%3Do_l%2Cy_t%3Ds_i%29%3D%5Cbm%7B1%7D%5C%7Bx_t%3Do_l%5C%7D%5Cbm%7B1%7D%5C%7By_t%3Ds_i%5C%7D%2C+%5C%5C%0A%26%26+%5Ctheta%5E1_%7Bij%7D+%3D+%5Clog+p%28y_t%3Ds_i+%7C+y_%7Bt-1%7D%3Ds_j%29%2C+%5C%5C%0A%26%26+%5Ctheta%5E2_%7Bli%7D+%3D+%5Clog+p%28x_t%3Do_l+%7C+y_t%3Ds_i%29+%5C%5C%0A%26%26+%5CPsi_t%28y_t%2C+y_%7By-1%7D%2C+x_t%29+%3D+%5Cexp+%5Cleft%5C%7B+%5Csum%5EK_%7Bk%3D1%7D+%5Ctheta_%7Bk%7D+f_%7Bk%7D%28y_t%2C+y_%7Bt-1%7D%2C+x_i%29+%5Cright%5C%7D+%3D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bs_i+%5Cin+S%7D+%5Csum_%7Bs_j+%5Cin+S%7D+%5Ctheta%5E1_%7Bij%7D+f%5E1_%7Bij%7D%28y_t%3Ds_i%2C+y_%7Bt-1%7D%3Ds_j%29+%2B+%5Csum_%7Bo_l+%5Cin+O%7D+%5Csum_%7Bs_i+%5Cin+S%7D+%5Ctheta%5E2_%7Bli%7D+f%5E2_%7Bli%7D%28x_t%3Do_l%2C+y_t%3Ds_i%29+%5Cright%5C%7D%0A%5Cend%7Beqnarray%7D+& alt=&\begin{eqnarray}
&& f_{ij}^1(y_t=s_i,y_{t-1}=s_j)=\bm{1}\{y_t=s_i\}\bm{1}\{y_{t-1}=s_j\}, \\
&& f_{li}^2(x_t=o_l,y_t=s_i)=\bm{1}\{x_t=o_l\}\bm{1}\{y_t=s_i\}, \\
&& \theta^1_{ij} = \log p(y_t=s_i | y_{t-1}=s_j), \\
&& \theta^2_{li} = \log p(x_t=o_l | y_t=s_i) \\
&& \Psi_t(y_t, y_{y-1}, x_t) = \exp \left\{ \sum^K_{k=1} \theta_{k} f_{k}(y_t, y_{t-1}, x_i) \right\} = \exp \left\{ \sum_{s_i \in S} \sum_{s_j \in S} \theta^1_{ij} f^1_{ij}(y_t=s_i, y_{t-1}=s_j) + \sum_{o_l \in O} \sum_{s_i \in S} \theta^2_{li} f^2_{li}(x_t=o_l, y_t=s_i) \right\}
\end{eqnarray} & eeimg=&1&&&br&&p&根据这样的兼容函数和参数，带入到Equ1中，得到，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0Ap%28y_1%2C+%5Ccdots%2C+y_T%2C+x_1%2C%5Ccdots%2Cx_T%29+%26%3D%26+%5Cprod_%7Bi%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Cprod_%7Bt%3D1%7D%5ET+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bs_i+%5Cin+S%7D+%5Csum_%7Bs_j+%5Cin+S%7D+%5Ctheta%5E1_%7Bij%7D+f%5E1_%7Bij%7D%28y_t%3Ds_i%2C+y_%7Bt-1%7D%3Ds_j%29+%2B+%5Csum_%7Bo_l+%5Cin+O%7D+%5Csum_%7Bs_i+%5Cin+S%7D+%5Ctheta%5E2_%7Bli%7D+f%5E2_%7Bli%7D%28x_t%3Do_l%2C+y_t%3Ds_i%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+%5Cprod_%7Bt%3D1%7D%5ET+%5Cexp+%5Cleft%5C%7B+%5Clog+p%28y_t%7Cy_%7Bt-1%7D%29+%2B+%5Clog+p%28x_t%7Cy_t%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+%5Cprod_%7Bt%3D1%7D%5ET+p%28y_t%7Cy_%7Bt-1%7D%29+p%28x_t%7Cy_t%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
p(y_1, \cdots, y_T, x_1,\cdots,x_T) &=& \prod_{i=1}^T \Psi_t(y_t, y_{t-1}, x_t) \\
&=& \prod_{t=1}^T \exp \left\{ \sum_{s_i \in S} \sum_{s_j \in S} \theta^1_{ij} f^1_{ij}(y_t=s_i, y_{t-1}=s_j) + \sum_{o_l \in O} \sum_{s_i \in S} \theta^2_{li} f^2_{li}(x_t=o_l, y_t=s_i) \right\} \\
&=& \prod_{t=1}^T \exp \left\{ \log p(y_t|y_{t-1}) + \log p(x_t|y_t) \right\} \\
&=& \prod_{t=1}^T p(y_t|y_{t-1}) p(x_t|y_t)
\end{eqnarray}& eeimg=&1&&&br&&p&综上说述，按照给定的&img src=&https://www.zhihu.com/equation?tex=%5CPsi+& alt=&\Psi & eeimg=&1&&和&img src=&https://www.zhihu.com/equation?tex=%5CTheta+& alt=&\Theta & eeimg=&1&&就可以推导出隐马尔科夫模型的公式，也就是说隐马尔科夫模型可以转化成无向图模型。接下来推导出条件随机场模型对应的无向图模型。&/p&&h2&四、条件随机场&/h2&条件随机场模型需要建模的是输入变量和输出变量的&b&条件概率分布&/b&&img src=&https://www.zhihu.com/equation?tex=p%28y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T%29& alt=&p(y_1, \cdots, y_T | x_1, \cdots, x_T)& eeimg=&1&&，因此对应的无向图模型如下，&br&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0Ap%28y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C%5Ccdots%2Cx_T%29+%26%3D%26+%5Cfrac%7B+%5Cprod_%7Bi%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%7D%7B%5Csum_y++%5Cprod_%7Bi%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
p(y_1, \cdots, y_T | x_1,\cdots,x_T) &=& \frac{ \prod_{i=1}^T \Psi_t(y_t, y_{t-1}, x_t) }{\sum_y
\prod_{i=1}^T \Psi_t(y_t, y_{t-1}, x_t) }
\end{eqnarray}& eeimg=&1&&&br&&p&条件随机场模型和隐马尔科夫模型一样，重点解决三个问题：&/p&&ol&&li&&b&概率问题&/b&：给定模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&、观测值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%2C+%5Ccdots%2C+x_T+%5C%7D& alt=&\{ x_1, \cdots, x_T \}& eeimg=&1&&和状态值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+y_1%2C+%5Ccdots%2C+y_T+%5C%7D& alt=&\{ y_1, \cdots, y_T \}& eeimg=&1&&的前提下，计算条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T )& eeimg=&1&&。&/li&&li&&b&推测问题&/b&：给定模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&和观测值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%2C+%5Ccdots%2C+x_T+%5C%7D& alt=&\{ x_1, \cdots, x_T \}& eeimg=&1&&，计算最大条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T )& eeimg=&1&&对应的状态值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+y_1%2C+%5Ccdots%2C+y_T+%5C%7D& alt=&\{ y_1, \cdots, y_T \}& eeimg=&1&&。&/li&&li&&b&训练问题&/b&：给定训练数据&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%5E%7B%28i%29%7D%2C+%5Ccdots%2C+x_%7BT_i%7D%5E%7B%28i%29%7D%2C+y_1%5E%7B%28i%29%7D%2C+%5Ccdots%2C+y_%7BT_i%7D%5E%7B%28i%29%7D+%5C%7D_%7Bi%3D1%7D%5EN& alt=&\{ x_1^{(i)}, \cdots, x_{T_i}^{(i)}, y_1^{(i)}, \cdots, y_{T_i}^{(i)} \}_{i=1}^N& eeimg=&1&&的前提下，最大化条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T )& eeimg=&1&&为目标，训练模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&。&/li&&/ol&本文接下来介绍这三个问题。&br&&h2&五、概率问题&/h2&&p&给定模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&、观测值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%2C+%5Ccdots%2C+x_T+%5C%7D& alt=&\{ x_1, \cdots, x_T \}& eeimg=&1&&和状态值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+y_1%2C+%5Ccdots%2C+y_T+%5C%7D& alt=&\{ y_1, \cdots, y_T \}& eeimg=&1&&的前提下，计算条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T )& eeimg=&1&&。&br&&/p&&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29+%3D+%5Cfrac%7B+%5Cprod_%7Bt%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%7D%7B+%5Csum_y+%5Cprod_%7Bt%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%7D& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T ) = \frac{ \prod_{t=1}^T \Psi_t(y_t, y_{t-1}, x_t) }{ \sum_y \prod_{t=1}^T \Psi_t(y_t, y_{t-1}, x_t) }& eeimg=&1&&&br&&p&根据已知的数据，带入即可求出分子，这个过程很容易，但是难点在分母，将分母展开写，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Csum_y+%5Cprod_%7Bt%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%3D+%5Csum_%7By_T+%5Cin+S%7D+%5Csum_%7By_%7BT-1%7D+%5Cin+S%7D+%5CPsi_T%28y_T%2C+y_%7BT-1%7D%2C+x_T%29+%5Csum_%7By_%7BT-2%7D+%5Cin+S%7D+%5CPsi_%7BT-1%7D%28y_%7BT-1%7D%2C+y_%7BT-2%7D%2C+x_%7BT-1%7D%29+%5Ccdots+%5Csum_%7By_1+%5Cin+S%7D+%5CPsi_2%28y_2%2C+y_1%2C+x_2%29+%5CPsi_1%28y_1%2C+y_0%2C+s_1%29& alt=&\sum_y \prod_{t=1}^T \Psi_t(y_t, y_{t-1}, x_t) = \sum_{y_T \in S} \sum_{y_{T-1} \in S} \Psi_T(y_T, y_{T-1}, x_T) \sum_{y_{T-2} \in S} \Psi_{T-1}(y_{T-1}, y_{T-2}, x_{T-1}) \cdots \sum_{y_1 \in S} \Psi_2(y_2, y_1, x_2) \Psi_1(y_1, y_0, s_1)& eeimg=&1&&&br&&p&设状态总数为S，则计算分母过程中，乘法次数为&img src=&https://www.zhihu.com/equation?tex=%28T-1%29+%5Ccdot+S%5ET& alt=&(T-1) \cdot S^T& eeimg=&1&&，加法次数为&img src=&https://www.zhihu.com/equation?tex=%5Cfrac%7BS%5ET%7D%7B2%7D& alt=&\frac{S^T}{2}& eeimg=&1&&，当S和T很大时，用这种暴力遍历的方法几乎无法计算，因此需要设计一种复杂度更低的方法计算分母。由于在计算过程中，很多&img src=&https://www.zhihu.com/equation?tex=+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29& alt=& \Psi_t(y_t, y_{t-1}, x_t)& eeimg=&1&&被重复计算，因此想到使用动态规划的方法，将问题分解成子问题，迭代计算。&/p&&p&令&img src=&https://www.zhihu.com/equation?tex=%5Calpha_%7Bt%7D%28s_i%29+%3D+p%28x_1%2C+%5Ccdots%2C+x_t%2C+y_t%3Ds_i%29& alt=&\alpha_{t}(s_i) = p(x_1, \cdots, x_t, y_t=s_i)& eeimg=&1&&为每个子问题，则分母可以表示为，&/p&&p&&img src=&https://www.zhihu.com/equation?tex=p%28x_1%2C+%5Ccdots%2C+x_T%29+%3D+%5Csum_%7Bs_i+%5Cin+S%7D+p%28x_1%2C+%5Ccdots%2C+x_T%2C+y_T+%3D+s_i%29+%3D+%5Csum_%7Bs_i+%5Cin+S%7D+%5Calpha_T%28s_i%29& alt=&p(x_1, \cdots, x_T) = \sum_{s_i \in S} p(x_1, \cdots, x_T, y_T = s_i) = \sum_{s_i \in S} \alpha_T(s_i)& eeimg=&1&&，而子问题的递推关系如下，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5Calpha_t%28s_i%29+%26%3D%26+p%28x_1%2C+%5Ccdots%2C+x_t%2C+y_t%3Ds_i%29+%3D+%5Csum_%7By_1+%5Cin+S%2C+%5Ccdots%2C+y_%7Bt-1%7D+%5Cin+S%7D+p%28x_1%2C+%5Ccdots%2C+x_%7Bt-1%7D%2C+y_1%2C+%5Ccdots%2C+y_%7Bt-1%7D%2C+x_t%2C+y_t%3Ds_i%29+%5C%5C%0A%26%3D%26+%5Csum_%7By_1+%5Cin+S%2C+%5Ccdots%2C+y_%7Bt-1%7D+%5Cin+S%7D+%5Cprod_%7Bt%27%3D1%7D%5E%7Bt-1%7D+%5CPsi_%7Bt%27%7D%28y_%7Bt%27%7D%2C+y_%7Bt%27-1%7D%2C+x_%7Bt%27%7D%29+%5CPsi_t%28s_i%2C+y_%7Bt-1%7D%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Csum_%7Bs_j+%5Cin+S%7D+%5Csum_%7By_1+%5Cin+S%2C+%5Ccdots%2C+y_%7Bt-2%7D+%5Cin+S%7D+%5Cprod_%7Bt%27%3D1%7D%5E%7Bt-2%7D+%5CPsi_%7Bt%27%7D%28y_%7Bt%27%7D%2C+y_%7Bt%27-1%7D%2C+x_%7Bt%27%7D%29+%5CPsi_%7Bt-1%7D%28s_j%2C+y_%7Bt-2%7D%2C+x_%7Bt-1%7D%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Csum_%7Bs_j+%5Cin+S%7D+%5Calpha_%7Bt-1%7D%28s_j%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\alpha_t(s_i) &=& p(x_1, \cdots, x_t, y_t=s_i) = \sum_{y_1 \in S, \cdots, y_{t-1} \in S} p(x_1, \cdots, x_{t-1}, y_1, \cdots, y_{t-1}, x_t, y_t=s_i) \\
&=& \sum_{y_1 \in S, \cdots, y_{t-1} \in S} \prod_{t'=1}^{t-1} \Psi_{t'}(y_{t'}, y_{t'-1}, x_{t'}) \Psi_t(s_i, y_{t-1}, x_t) \\
&=& \sum_{s_j \in S} \sum_{y_1 \in S, \cdots, y_{t-2} \in S} \prod_{t'=1}^{t-2} \Psi_{t'}(y_{t'}, y_{t'-1}, x_{t'}) \Psi_{t-1}(s_j, y_{t-2}, x_{t-1}) \Psi_t(s_i, s_j, x_t) \\
&=& \sum_{s_j \in S} \alpha_{t-1}(s_j) \Psi_t(s_i, s_j, x_t)
\end{eqnarray}& eeimg=&1&&&br&&p&并且，当&img src=&https://www.zhihu.com/equation?tex=t%3D1& alt=&t=1& eeimg=&1&&时，有&img src=&https://www.zhihu.com/equation?tex=%5Calpha_1%28s_i%29+%3D+p%28x_1%2C+y_1%3Ds_i%29+%3D+%5CPsi_1%28s_i%2C+y_0%2C+x_1%29& alt=&\alpha_1(s_i) = p(x_1, y_1=s_i) = \Psi_1(s_i, y_0, x_1)& eeimg=&1&&，因此，整个概率问题求解过程的递推关系表达式为，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%0Ainit+%26%3A%26+%5Calpha_1%28s_i%29+%3D+%5CPsi_1%28s_i%2C+y_0%2C+x_1%29+%5C%5C%0Arecursive+%26%3A%26+%5Calpha_t%28s_i%29+%3D+%5Csum_%7Bs_j+%5Cin+S%7D+%5Calpha_%7Bt-1%7D%28s_j%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29+%5C%5C%0Aend+%26%3A%26+p%28x_1%2C+%5Ccdots%2C+x_T%29+%3D+%5Csum_%7Bs_i+%5Cin+S%7D%5Calpha_T%28s_i%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
init &:& \alpha_1(s_i) = \Psi_1(s_i, y_0, x_1) \\
recursive &:& \alpha_t(s_i) = \sum_{s_j \in S} \alpha_{t-1}(s_j) \Psi_t(s_i, s_j, x_t) \\
end &:& p(x_1, \cdots, x_T) = \sum_{s_i \in S}\alpha_T(s_i)
\end{eqnarray}& eeimg=&1&&&br&&p&设状态总数为S，则使用动态规划算法计算分母过程中，乘法次数为&img src=&https://www.zhihu.com/equation?tex=S%28S-1%29& alt=&S(S-1)& eeimg=&1&&，加法次数为&img src=&https://www.zhihu.com/equation?tex=S%5E2& alt=&S^2& eeimg=&1&&，大大提升了计算效率。&/p&&h2&六、推测问题&/h2&&p&给定模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&和观测值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%2C+%5Ccdots%2C+x_T+%5C%7D& alt=&\{ x_1, \cdots, x_T \}& eeimg=&1&&，计算最大条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+y_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, y_T )& eeimg=&1&&对应的状态值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+y_1%2C+%5Ccdots%2C+y_T+%5C%7D& alt=&\{ y_1, \cdots, y_T \}& eeimg=&1&&，&img src=&https://www.zhihu.com/equation?tex=y%5E%2A+%3D+%5Cmax_%7By_1%2C+%5Ccdots%2C+y_T%7D+p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T%29& alt=&y^* = \max_{y_1, \cdots, y_T} p( y_1, \cdots, y_T | x_1, \cdots, x_T)& eeimg=&1&&。&/p&&p&同样使用动态规划的思想，将其分解成子问题，设，&/p&&p&&img src=&https://www.zhihu.com/equation?tex=%5Cdelta_t%28s_i%29+%3D+%5Cmax_%7By_1%2C+%5Ccdots%2C+y_%7Bt-1%7D%7D+p%28y_1%2C+%5Ccdots%2C+y_%7Bt-1%7D%2C+y_t%3Ds_i+%7Cx_1%2C+%5Ccdots%2C+x_t%29& alt=&\delta_t(s_i) = \max_{y_1, \cdots, y_{t-1}} p(y_1, \cdots, y_{t-1}, y_t=s_i |x_1, \cdots, x_t)& eeimg=&1&&，则，原始问题可以表示为，&br&&/p&&p&&img src=&https://www.zhihu.com/equation?tex=y%5E%2A+%3D+%5Cmax_%7By_1%2C+%5Ccdots%2C+y_T%7D+p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T%29+%3D+%5Cmax_%7Bs_i+%5Cin+S%7D+%5Cmax_%7By_1%2C+%5Ccdots%2C+y_T%7D+p%28+y_1%2C+%5Ccdots%2C+y_T%3Ds_i+%7C+x_1%2C+%5Ccdots%2C+x_T%29+%3D+%5Cmax_%7Bs_i+%5Cin+S%7D+%5Cdelta_T%28s_i%29& alt=&y^* = \max_{y_1, \cdots, y_T} p( y_1, \cdots, y_T | x_1, \cdots, x_T) = \max_{s_i \in S} \max_{y_1, \cdots, y_T} p( y_1, \cdots, y_T=s_i | x_1, \cdots, x_T) = \max_{s_i \in S} \delta_T(s_i)& eeimg=&1&&，而子问题具有如下的递推关系，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5Cdelta_t%28s_i%29+%26%3D%26%5Cmax_%7By_1%2C+%5Ccdots%2C+y_%7Bt-1%7D%7D+p%28y_1%2C+%5Ccdots%2C+y_%7Bt-1%7D%2C+y_t%3Ds_i+%7Cx_1%2C+%5Ccdots%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Cmax_%7Bs_j+%5Cin+S%7D++%5Cmax_%7By_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%7D+p%28y_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%2C+y_%7Bt-1%7D%3Ds_j%2C+y_t%3Ds_i+%7Cx_1%2C+%5Ccdots%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Cmax_%7Bs_j+%5Cin+S%7D+%5Cmax_%7By_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%7D+p%28y_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%2C+y_%7Bt-1%7D%3Ds_j+%7C+x_1%2C+%5Ccdots%2C+x_%7Bt-1%7D%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Cmax_%7Bs_j+%5Cin+S%7D+%5Cdelta_%7Bt-1%7D%28s_j%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\delta_t(s_i) &=&\max_{y_1, \cdots, y_{t-1}} p(y_1, \cdots, y_{t-1}, y_t=s_i |x_1, \cdots, x_t) \\
&=& \max_{s_j \in S}
\max_{y_1, \cdots, y_{t-2}} p(y_1, \cdots, y_{t-2}, y_{t-1}=s_j, y_t=s_i |x_1, \cdots, x_t) \\
&=& \max_{s_j \in S} \max_{y_1, \cdots, y_{t-2}} p(y_1, \cdots, y_{t-2}, y_{t-1}=s_j | x_1, \cdots, x_{t-1}) \Psi_t(s_i, s_j, x_t) \\
&=& \max_{s_j \in S} \delta_{t-1}(s_j) \Psi_t(s_i, s_j, x_t)
\end{eqnarray}& eeimg=&1&&&br&&p&并且，当&img src=&https://www.zhihu.com/equation?tex=t%3D0& alt=&t=0& eeimg=&1&&时，有&img src=&https://www.zhihu.com/equation?tex=%5Cdelta_0%28s_i%29+%3D+%5Csum_%7Bs_i+%5Cin+S%7D+p%28y_0%29+%3D1& alt=&\delta_0(s_i) = \sum_{s_i \in S} p(y_0) =1& eeimg=&1&&，因此，整个概率问题求解过程的递推关系表达式为，&br&&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%0Ainit+%26%3A%26+%5Cdelta_0%28s_i%29+%3D+1+%5C%5C%0Arecursive+%26%3A%26+%5Cdelta_t%28s_i%29+%3D+%5Cmax_%7Bs_j+%5Cin+S%7D+%5Cdelta_%7Bt-1%7D%28s_j%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29+%5C%5C%0Aend+%26%3A%26+y%5E%2A+%3D+%5Cmax_%7Bs_i+%5Cin+S%7D%5Cdelta_T%28s_i%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
init &:& \delta_0(s_i) = 1 \\
recursive &:& \delta_t(s_i) = \max_{s_j \in S} \delta_{t-1}(s_j) \Psi_t(s_i, s_j, x_t) \\
end &:& y^* = \max_{s_i \in S}\delta_T(s_i)
\end{eqnarray}& eeimg=&1&&&br&&p&使用动态规划算法求解推测问题，大大提升了计算效率，类似于隐马尔科夫模型中的Viterbi算法。&/p&&h2&七、训练问题&/h2&&p&给定训练数据&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%5E%7B%28i%29%7D%2C+%5Ccdots%2C+x_%7BT_i%7D%5E%7B%28i%29%7D%2C+y_1%5E%7B%28i%29%7D%2C+%5Ccdots%2C+y_%7BT_i%7D%5E%7B%28i%29%7D+%5C%7D_%7Bi%3D1%7D%5EN& alt=&\{ x_1^{(i)}, \cdots, x_{T_i}^{(i)}, y_1^{(i)}, \cdots, y_{T_i}^{(i)} \}_{i=1}^N& eeimg=&1&&的前提下，最大化条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T )& eeimg=&1&&为目标，训练模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&。&br&&/p&&p&根据极大似然估计，对数据集取似然对数，目标函数为，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5Clog+L%28%5Ctheta%29+%26%3D%26+%5Csum_%7Bi%3D1%7D%5EN+p%28y%5E%7B%28i%29%7D+%7C+x%5E%7B%28i%29%7D%29+%5C%5C%0A%26%3D%26+%5Csum_%7Bi%3D1%7D%5EN+%5Clog+%5Cfrac%7B+%5Cprod_%7Bt%3D1%7D%5E%7BT_i%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D+%0A%2C+x_t%5E%7B%28i%29%7D%29+%5Cright%5C%7D+%7D%7B+%5Csum_y+%5Cprod_%7Bt%3D1%7D%5E%7BT_i%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D+%0A%2C+x_t%5E%7B%28i%29%7D%29+%5Cright%5C%7D+%7D+%5C%5C%0A%26%3D%26+%5Csum_%7Bi%3D1%7D%5EN+%5Csum_%7Bt%3D1%7D%5E%7BT_i%7D+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D+%0A%2C+x_t%5E%7B%28i%29%7D%29+-+%5Csum_%7Bi%3D1%7D%5EN+%5Clog+%5Csum_y+%5Cprod_%7Bt%3D1%7D%5E%7BT_i%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D+%0A%2C+x_t%5E%7B%28i%29%7D%29+%5Cright%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\log L(\theta) &=& \sum_{i=1}^N p(y^{(i)} | x^{(i)}) \\
&=& \sum_{i=1}^N \log \frac{ \prod_{t=1}^{T_i} \exp \left\{ \sum_{k=1}^K \theta_k f_k(y_t^{(i)}, y_{t-1}^{(i)}
, x_t^{(i)}) \right\} }{ \sum_y \prod_{t=1}^{T_i} \exp \left\{ \sum_{k=1}^K \theta_k f_k(y_t^{(i)}, y_{t-1}^{(i)}
, x_t^{(i)}) \right\} } \\
&=& \sum_{i=1}^N \sum_{t=1}^{T_i} \sum_{k=1}^K \theta_k f_k(y_t^{(i)}, y_{t-1}^{(i)}
, x_t^{(i)}) - \sum_{i=1}^N \log \sum_y \prod_{t=1}^{T_i} \exp \left\{ \sum_{k=1}^K \theta_k f_k(y_t^{(i)}, y_{t-1}^{(i)}
, x_t^{(i)}) \right\}
\end{eqnarray}& eeimg=&1&&&br&&p&令，&img src=&https://www.zhihu.com/equation?tex=Z%28x%5E%7B%28i%29%7D%29+%3D+%5Csum_y+%5Cprod_%7Bt%3D1%7D%5E%7BT_i%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D+%0A%2C+x_t%5E%7B%28i%29%7D%29+%5Cright%5C%7D+%3D+%5Csum_%7By_T%7D+%5Csum_%7By_%7BT-1%7D%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_T%2C+y_%7BT-1%7D%2C+x_T%29+%5Cright%5C%7D+%5Ccdots+%5Csum_%7By_1%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_2%2C+y_%7B1%7D%2C+x_2%29+%5Cright%5C%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_1%2C+y_0%2C+x_1%29+%5Cright%5C%7D& alt=&Z(x^{(i)}) = \sum_y \prod_{t=1}^{T_i} \exp \left\{ \sum_{k=1}^K \theta_k f_k(y_t^{(i)}, y_{t-1}^{(i)}
, x_t^{(i)}) \right\} = \sum_{y_T} \sum_{y_{T-1}} \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_T, y_{T-1}, x_T) \right\} \cdots \sum_{y_1} \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_2, y_{1}, x_2) \right\} \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_1, y_0, x_1) \right\}& eeimg=&1&&对&img src=&https://www.zhihu.com/equation?tex=%5Clog+L%28%5Ctheta%29& alt=&\log L(\theta)& eeimg=&1&&求导数，可得，&img src=&https://www.zhihu.com/equation?tex=%5Cfrac%7B%5Cpartial+%5Clog+L%28%5Ctheta%29%7D%7B%5Cpartial+%5Ctheta_k%7D+%3D++%5Csum_%7Bi%3D1%7D%5EN+%5Csum_%7Bt%3D1%7D%5E%7BT_i%7D+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D%2C+x_t%5E%7B%28i%29%7D%29+-+%5Cfrac%7B+%5Cpartial+Z%28x%5E%7B%28i%29%7D%29%7D%7B+%5Cpartial+%5Ctheta_k%7D& alt=&\frac{\partial \log L(\theta)}{\partial \theta_k} =
\sum_{i=1}^N \sum_{t=1}^{T_i} f_k(y_t^{(i)}, y_{t-1}^{(i)}, x_t^{(i)}) - \frac{ \partial Z(x^{(i)})}{ \partial \theta_k}& eeimg=&1&&，难点在第二项，我们单独对第二项进行推导，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5Cfrac%7B+%5Cpartial+%5Clog+Z%28x%5E%7B%28i%29%7D%29%7D%7B+%5Cpartial+%5Ctheta_k%7D+%26%3D%26+%5Cfrac%7B1%7D%7BZ%28x%5E%7B%28i%29%7D%29%7D+%5Ccdot+%5Cfrac%7B%5Cpartial+Z%28x%5E%7B%28i%29%7D%29%7D%7B+%5Cpartial+%5Ctheta_k%7D+%3D+%5Csum_%7By_1%2C+%5Ccdots%2C+y_T%7D+p%28y_1%2C+%5Ccdots%2C+y_T%7Cx%5E%7B%28i%29%7D%29+%5Csum_%7Bt%3D1%7D%5E%7BT_i%7D+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\frac{ \partial \log Z(x^{(i)})}{ \partial \theta_k} &=& \frac{1}{Z(x^{(i)})} \cdot \frac{\partial Z(x^{(i)})}{ \partial \theta_k} = \sum_{y_1, \cdots, y_T} p(y_1, \cdots, y_T|x^{(i)}) \sum_{t=1}^{T_i} f_k(y_t, y_{t-1}, x_t)
\end{eqnarray}& eeimg=&1&&&br&&p&我们重点来理解第二个等号是怎么得到的，为了简单起见，我们先假设状态集合为&img src=&https://www.zhihu.com/equation?tex=%5C%7B+s_1%2C+s_2+%5C%7D& alt=&\{ s_1, s_2 \}& eeimg=&1&&，再去掉上标，由于&img src=&https://www.zhihu.com/equation?tex=Z%28x%29& alt=&Z(x)& eeimg=&1&&外部是对每一条路径&img src=&https://www.zhihu.com/equation?tex=%3Cy_1%2C+y_2%2C+%5Ccdots%2C+y_T%3E& alt=&&y_1, y_2, \cdots, y_T&& eeimg=&1&&的求和，那么对其求导则可以单独对每一项求导再加和，对每一项单独求导可写成，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5Cnabla%5E%7B%28y_1%2C+%5Ccdots%2C+y_T%29%7D_%7B%5Ctheta%7D+%26%3D%26+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+%5Ctheta_k%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_T%2C+y_%7BT-1%7D%2C+x_T%29+%5Cright%5C%7D+%5Ccdots+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_2%2C+y_%7B1%7D%2C+x_2%29+%5Cright%5C%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_1%2C+y_0%2C+x_1%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+%5Ctheta_k%7D+%5Cexp+%5Cleft%5C%7B+%0A%5Csum_%7Bt%3D1%7D%5ET+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+%5Cexp+%5Cleft%5C%7B+%0A%5Csum_%7Bt%3D1%7D%5ET+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%5Cright%5C%7D+%5Csum_%7Bt%3D1%7D%5ET+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\nabla^{(y_1, \cdots, y_T)}_{\theta} &=& \frac{\partial}{\partial \theta_k} \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_T, y_{T-1}, x_T) \right\} \cdots \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_2, y_{1}, x_2) \right\} \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_1, y_0, x_1) \right\} \\
&=& \frac{\partial}{\partial \theta_k} \exp \left\{
\sum_{t=1}^T \sum_{k=1}^K \theta_k f_k(y_t, y_{t-1}, x_t) \right\} \\
&=& \exp \left\{
\sum_{t=1}^T \sum_{k=1}^K \theta_k f_k(y_t, y_{t-1}, x_t) \right\} \sum_{t=1}^T f_k(y_t, y_{t-1}, x_t)
\end{eqnarray}& eeimg=&1&&&br&&p&那么，对这些导数求和，就可以得到&img src=&https://www.zhihu.com/equation?tex=%5Clog+Z%28x%5E%7B%28i%29%7D%29& alt=&\log Z(x^{(i)})& eeimg=&1&&的导数，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cfrac%7B+%5Cpartial+%5Clog+Z%28x%29%7D%7B+%5Cpartial+%5Ctheta_k%7D+%3D+%5Csum_%7By_1%2C+%5Ccdots%2C+y_T%7D+%5Cfrac%7B+%5Cexp+%5Cleft%5C%7B+%0A%5Csum_%7Bt%3D1%7D%5ET+%5Csum_k%5EK+%5Ctheta_k+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%5Cright%5C%7D+%7D%7B+%5Csum_y+%5Cexp+%5Cleft%5C%7B+%0A%5Csum_%7Bt%3D1%7D%5ET+%5Csum_k%5EK+%5Ctheta_k+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%5Cright%5C%7D+%7D+%5Csum_%7Bt%3D1%7D%5ET+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%3D+%5Csum_%7By_1%2C+%5Ccdots%2C+y_T%7D+p%28y_1%2C+%5Ccdots%2C+y_T%7Cx%29+%5Csum_%7Bt%3D1%7D%5ET+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29& alt=&\frac{ \partial \log Z(x)}{ \partial \theta_k} = \sum_{y_1, \cdots, y_T} \frac{ \exp \left\{
\sum_{t=1}^T \sum_k^K \theta_k f_k(y_t, y_{t-1}, x_t) \right\} }{ \sum_y \exp \left\{
\sum_{t=1}^T \sum_k^K \theta_k f_k(y_t, y_{t-1}, x_t) \right\} } \sum_{t=1}^T f_k(y_t, y_{t-1}, x_t) = \sum_{y_1, \cdots, y_T} p(y_1, \cdots, y_T|x) \sum_{t=1}^T f_k(y_t, y_{t-1}, x_t)& eeimg=&1&&&br&&p&对于每一个&img src=&https://www.zhihu.com/equation?tex=p%28y_1%2C+%5Ccdots%2C+y_T%7Cx%29& alt=&p(y_1, \cdots, y_T|x)& eeimg=&1&&，会乘以，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Csum_%7Bt%3D1%7D%5ET+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%3D+f_k%28y_1%2C+y_0%2C+x_1%29+%2B+%5Ccdots+%2B+f_k%28y_T%2C+y_%7BT-1%7D%2C+x_T%29& alt=&\sum_{t=1}^T f_k(y_t, y_{t-1}, x_t) = f_k(y_1, y_0, x_1) + \cdots + f_k(y_T, y_{T-1}, x_T)& eeimg=&1&&&p&那么反过来，对于每一个&img src=&https://www.zhihu.com/equation?tex=f_k%28y_t+%3D+s%2C+y_%7Bt-1%7D+%3D+s%27%2C+x_t%29& alt=&f_k(y_t = s, y_{t-1} = s', x_t)& eeimg=&1&&，会乘以，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Csum_%7By_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%2C+y_%7Bt%2B1%7D%2C+y_T%7D+p%28y_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%2C+y_t%3Ds%2C+y_%7Bt-1%7D%3Ds%27%2C+y_%7Bt%2B1%7D%2C+y_T+%7C+x%29+%3D+p%28s%2C+s%27+%7C+x%29& alt=&\sum_{y_1, \cdots, y_{t-2}, y_{t+1}, y_T} p(y_1, \cdots, y_{t-2}, y_t=s, y_{t-1}=s', y_{t+1}, y_T | x) = p(s, s' | x)& eeimg=&1&}
一夜山边去，江山一夜归。
山风春草色，秋水夜声深。
何事同相见，应知旧子人。
何当不相见，何处见江边。
一叶生云里，春风出竹堂。
何时有相访，不得在君心。
&/code&&/pre&&/div&&p&&b&13、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_13/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&序列分类问题详解&/a&&/b&&/p&&p&一类数列为等差数列，一类数列为随机数列，使用RNN，达到99%的分类准确率。&/p&&p&&b&14、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_14/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&词的向量表示：word2vec与词嵌入&/a&&/b&&/p&&p&词嵌入在各种自然语言处理问题时经常用到，这章使用Skip-gram方法训练一个词嵌入，并进行简单的可视化。&/p&&figure&&img src=&https://pic1.zhimg.com/v2-a0d297de65c2df4b1a20_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1736& data-rawheight=&608& class=&origin_image zh-lightbox-thumb& width=&1736& data-original=&https://pic1.zhimg.com/v2-a0d297de65c2df4b1a20_r.jpg&&&/figure&&p&&b&15、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_15/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&在TensorFlow中进行时间序列预测&/a&&/b&&/p&&p&用TensorFlow来解决时间序列预测问题，预测方法包括传统的AR模型和LSTM模型。&/p&&p&预测效果如下（红色部分为预测值）：&/p&&figure&&img src=&https://pic3.zhimg.com/v2-061f118aeee53ff8209c74a_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1500& data-rawheight=&500& class=&origin_image zh-lightbox-thumb& width=&1500& data-original=&https://pic3.zhimg.com/v2-061f118aeee53ff8209c74a_r.jpg&&&/figure&&p&&b&16、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_16/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&神经网络机器翻译技术&/a&&/b&&/p&&p&这一章会训练一个把英文翻译成中文的模型。&/p&&p&例如，输入英文：It is learned that china has on many occasions previously made known its stand on its relations with the g-8.&/p&&p&可以翻译为中文：据悉，中国曾多次表明同八国集团之间的关系。&/p&&p&使用的代码为TensorFlow NMT。&/p&&p&&b&17、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_17/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&看图说话：将图像转换为文字&/a&&/b&&/p&&p&给定一张图片：&/p&&figure&&img src=&https://pic2.zhimg.com/v2-3adff6ec008e302bc100f122d2431965_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1648& data-rawheight=&712& class=&origin_image zh-lightbox-thumb& width=&1648& data-original=&https://pic2.zhimg.com/v2-3adff6ec008e302bc100f122d2431965_r.jpg&&&/figure&&p&输出一句描述它的话，如&a man riding a wave on top of a surfboard .&&/p&&p&&b&四、强化学习基础与深度强化学习&/b&&/p&&p&&b&18、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_18/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&强化学习入门之Q Learning&/a&&/b&&/p&&p&借助一个走迷宫问题，来讲解强化学习基础算法 —— Q Learning。&/p&&p&&b&19、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_19/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&强化学习入门之SARSA算法&/a&&/b&&/p&&p&解决的问题和上一章类似，不过使用的算法是另一种强化学习基础算法SARSA。&/p&&p&&b&20、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_20/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&深度强化学习：Deep Q Learning&/a&&/b&&/p&&p&训练一个DQN，来玩打砖块游戏。&/p&&figure&&img src=&https://pic4.zhimg.com/v2-4a02b2504aeaebc_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&2144& data-rawheight=&854& class=&origin_image zh-lightbox-thumb& width=&2144& data-original=&https://pic4.zhimg.com/v2-4a02b2504aeaebc_r.jpg&&&/figure&&p&&b&21、&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples/blob/master/chapter_21/README.md& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&策略梯度（Policy Gradient）算法&/a&&/b&&/p&&p&训练一个策略梯度网络，来玩Cartpole。&/p&&figure&&img src=&https://pic2.zhimg.com/v2-5fb3abbbf27d4f58af41_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1728& data-rawheight=&676& class=&origin_image zh-lightbox-thumb& width=&1728& data-original=&https://pic2.zhimg.com/v2-5fb3abbbf27d4f58af41_r.jpg&&&/figure&&h2&&b&为什么写这本书&/b&&/h2&&p&去年6月份，电子工业出版社的一位编辑朋友找到我，希望能够帮忙编写一本以项目合集为形式的深度学习入门书。当时没有怎么犹豫就答应下来了，一方面，市面上确实缺少一本这样以实践案例组成的入门书，另一方面，我个人也觉得，&b&实践是入门深度学习的的极佳方式，原因如下：&/b&&/p&&ul&&li&在学习的初期，这种方式具有趣味感、成就感，比较容易坚持&/li&&li&深度学习需要深厚的数理基础，对于初学者来说不是很友好，另外TensorFlow这个框架也不是很容易学习。使用具体例子来理解更加直观有效。&/li&&li&深度学习更偏向是一门“实验科学”。很多深度学习算法都存在一些无法用理论解释的trick和训练技巧，这些东西如果不是自己去经历一次是很难有直观感受的。&/li&&/ul&&p&另外，在这个专栏中，我曾经写过很多比较有意思的深度学习实践项目。这本书是在专栏的基础上，添加了更多的项目，整理成册，希望提供一个比较完整的深度学习入门案例合集。&/p&&h2&&b&最后&/b&&/h2&&p&最后大家如果对这本书有兴趣的话，可以猛戳下面的购买链接购买：&/p&&ul&&li&京东（&a href=&http://link.zhihu.com/?target=https%3A//item.jd.com/.html& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&https://&/span&&span class=&visible&&item.jd.com/.ht&/span&&span class=&invisible&&ml&/span&&span class=&ellipsis&&&/span&&/a&）&/li&&li&天猫（&a href=&http://link.zhihu.com/?target=https%3A//detail.tmall.com/item.htm%3Fid%3D& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&https://&/span&&span class=&visible&&detail.tmall.com/item.h&/span&&span class=&invisible&&tm?id=&/span&&span class=&ellipsis&&&/span&&/a&）&/li&&li&当当（&a href=&http://link.zhihu.com/?target=http%3A//product.dangdang.com/.html& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&product.dangdang.com/25&/span&&span class=&invisible&&245282.html&/span&&span class=&ellipsis&&&/span&&/a&）&/li&&li&亚马逊（&a href=&http://link.zhihu.com/?target=https%3A//www.amazon.cn/dp/B07BF39RHQ& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&https://www.&/span&&span class=&visible&&amazon.cn/dp/B07BF39RHQ&/span&&span class=&invisible&&&/span&&/a&）&/li&&/ul&&p&也欢迎随时访问本书的Github：&a href=&http://link.zhihu.com/?target=https%3A//github.com/hzy46/Deep-Learning-21-Examples& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&hzy46/Deep-Learning-21-Examples&/a& 吐槽、提issue，在这里谢谢各位啦，比心！&/p&
首先感谢大家还关注这个专栏，前段时间一直在写一本书，题为《21个项目玩转深度学习——基于TensorFlow的实践详解》。简单地介绍一下，这本书的定位是深度学习+TensorFlow的入门实验书，希望提供一些在TensorFlow下常用的、并且真正能work的实践项目，方便…
&figure&&img src=&https://pic2.zhimg.com/v2-678fe1ce18bdfd_b.jpg& data-rawwidth=&1482& data-rawheight=&1080& class=&origin_image zh-lightbox-thumb& width=&1482& data-original=&https://pic2.zhimg.com/v2-678fe1ce18bdfd_r.jpg&&&/figure&此文的灵感来自360云盘关闭。&p&&figure&&img src=&https://pic1.zhimg.com/v2-ec1e44faf51_b.jpg& data-rawwidth=&800& data-rawheight=&450& class=&origin_image zh-lightbox-thumb& width=&800& data-original=&https://pic1.zhimg.com/v2-ec1e44faf51_r.jpg&&&/figure&NAS（网络附加存储），其实就是存储服务器。&br&在这个网盘纷纷关闭的萧瑟的深秋，我觉得有必要聊聊这个话题。&br&这个东西，对我来说是一个偶然，但是用了之后，成了必然。&br&最初，我的MacBook air是128G的，我还喜欢拍照用RAW格式，你说怎么就这么命苦？&br&移动硬盘倒是可以，但是我有三台笔记本、两个平板、三个手机，这个可怎么办？&br&于是，遇到NAS这个名词，脑海中有个声音说，来一台吧，然后来了一台。&br&&figure&&img src=&https://pic3.zhimg.com/v2-c66f1c2cd748a977adc25bd3_b.jpg& data-rawwidth=&2011& data-rawheight=&1718& class=&origin_image zh-lightbox-thumb& width=&2011& data-original=&https://pic3.zhimg.com/v2-c66f1c2cd748a977adc25bd3_r.jpg&&&/figure&最初使用它的作用主要是扩展MBA的存储，那时候我都是用iPhoto修图，有了NAS之后，再也不担心MBA的存储了，同时，NAS可以作为苹果TimeMachine的备份盘，让苹果时光机发挥作用；而且可以多个设备共用，就像网盘客户端。&br& 不乱说了，系统一点讲吧。本篇是入门篇，主要讲一些简单的应用，高级应用将在下一篇里讲解。&br&哦，对了，还有一句要说的就是：这玩意不贵，门槛很低。&br&&br&&br&&b&1 NAS能做什么&/b&&br&NAS作为一台存储服务器，其实就是一台配置不高但是存储空间巨大，功耗很低可以长时间运行的电脑。它的主要功能就是存储，形成家庭或者办公室的数据中心，所以最主要的功能就是存储空间的共享，然后就是在此基础上附加的一些功能：数据同步（个人云）、照片共享、音乐库、影音库、下载、数据备份等功能，一项一项说说吧。&/p&&p&&figure&&img src=&https://pic4.zhimg.com/v2-bf23cdf0a37ea139a84a6_b.jpg& data-rawwidth=&1024& data-rawheight=&966& class=&origin_image zh-lightbox-thumb& width=&1024& data-original=&https://pic4.zhimg.com/v2-bf23cdf0a37ea139a84a6_r.jpg&&&/figure&&br&&b&共享存储：&/b&你所有的设备，只要可以和NAS连接，就可以访问NAS中存储的数据，所以你可以在手机、笔记本、平板、智能电视等等各种设备上直接访问NAS，就像使用本地数据一样使用NAS中存储的数据，所以这是一件相当方便的事。&br&比如你把拍摄的照片存在NAS里，然后你用你的MBA修图，修好后的图片可以直接在平板和智能电视观看，是不是很方便？而且，假如你的NAS总共有4T空间，则相当于你的所有设备都有了这4T的空间。&br&&b&多媒体共享&/b&：这一部分包括照片、音乐和视频，存在NAS中的多媒体文件可以直接以你想要的方式呈现到你希望呈现的设备。比如你在看NAS上的电影、你女朋友可以查看NAS上存储的照片，你出差在外，可以直接网页登录到家中的NAS上，欣赏你存储在NAS里的音乐。&br&&b&挂机下载：&/b&这个很简单，因为NAS就是一台电脑，不过就是功耗低，一直开着而已，所以你想下载的东西可以直接丢给它，让它给你下载就是了，反正它有的是时间。&br&&b&数据备份：&/b&NAS都具备强大的数据备份功能，确保数据安全，当然，人家NAS就是干这个的。你可以对你的数据进行RAID的保护，也可以设置备份任务直接备份到网盘，也可以将重要数据自动备份到移动硬盘中，反正就是可以保证你的数据非常安全，这一部分内容很多，到下一篇里面会详细讲解，你所要知道的就是，这玩意存储可靠性非常高，有三四层楼那么高。&br&除了这些功能，很多NAS还有一些特色功能，比如说直接解码高清电影，相当于备具了硬盘播放机的功能；还有可以直接外接HIFI声卡作为音源；因为一直开着，所以可以作为自己的网站服务器，把自己的个人主页挂在上面；还可以作为监控摄像头的存储盘，直接存储摄像头录制的视频；有的还带有Android系统，兼容各种安卓APP，还可以打植物大战僵尸～～哎，太多了，一时半会说不完。&br&这么多功能，是不是用起来很麻烦？我当初也是这么想，后来了解了一下就不这么想了，下面就说说NAS的用法。&br&&br&&b&2 NAS怎么用&/b&&br&话不多说，直接上图，这就是一台NAS的管理界面。&br&这个界面怎么来的？网页登录上去就是了，既可以在家中登录，也可以在任何一个网络能够联通的地方登录。&br&&br&这是QNAP（著名NAS厂家）的NAS管理页面，系统名称叫QTS，一看就明白了，用起来非常简单。&br&&figure&&img src=&https://pic3.zhimg.com/v2-5ad6c3eef7b75aeaeaebbba_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic3.zhimg.com/v2-5ad6c3eef7b75aeaeaebbba_r.jpg&&&/figure&&br&说说其中重要的几个。&br&控制台：用于调整NAS的各种设置。&br&&br&&figure&&img src=&https://pic1.zhimg.com/v2-cd818087bff8_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic1.zhimg.com/v2-cd818087bff8_r.jpg&&&/figure&Music Station数字音乐库：可以播放NAS中存储的音乐，既可以推送到本地，也可以直接让NAS通过外接声卡输出。&br&&br&&figure&&img src=&https://pic3.zhimg.com/v2-98f002a02d79de6a1db03_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic3.zhimg.com/v2-98f002a02d79de6a1db03_r.jpg&&&/figure&Photo Station照片时光机：可以显示在NAS中存储的照片和视频，只要有网，随时随地查看。&br&&br&&figure&&img src=&https://pic3.zhimg.com/v2-2bf6e1f2e3aa221e4b290ac74d5c17d7_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic3.zhimg.com/v2-2bf6e1f2e3aa221e4b290ac74d5c17d7_r.jpg&&&/figure&File Station文件总管：直接管理NAS中存储的文件，可以上传下载，移动复制删除，也可以随时打开或者播放。&br&&br&&figure&&img src=&https://pic2.zhimg.com/v2-d92feba39ac69de01a39aca0c89ab111_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic2.zhimg.com/v2-d92feba39ac69de01a39aca0c89ab111_r.jpg&&&/figure&备份管理中心：可以选择使用各种方式对你的数据进行备份。&br&&br&&figure&&img src=&https://pic4.zhimg.com/v2-fd2a94dccc0_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic4.zhimg.com/v2-fd2a94dccc0_r.jpg&&&/figure&Qsync Central Station：这是我最喜欢也是最常用的功能，就是强大的同步。类似金山快盘或者坚果云这种网盘的同步，你可以在Windows系统、Mac系统、安卓系统、iOS系统等各种设备上安装客户端实现和网盘同步一样的功能，而且所有数据都在自己的服务器上，再也不怕网盘动不动就删你的文件。实在是网盘关闭后的必备良药啊。这一部分在第二篇中还要更详细的讲解。&br&&br&&figure&&img src=&https://pic4.zhimg.com/v2-75b47cf7939c_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic4.zhimg.com/v2-75b47cf7939c_r.jpg&&&/figure&APP Center：现在的NAS也是类似iOS和安卓系统一样有APP商店，直接点击安装，非常方便。各种功能的APP一应具全：同步备份、下载、娱乐、监控、系统工具、教育、家庭自动化等等～&br&&br&&figure&&img src=&https://pic1.zhimg.com/v2-4db6d21b0c64d7deb647_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic1.zhimg.com/v2-4db6d21b0c64d7deb647_r.jpg&&&/figure&Download Station：挂机下载，直接把需要下载资源的URL或者种子文件添加进来，NAS就会一刻不停的替你下载，除了常见的HTTP和FTP下载，还可以支持QQ下载、迅雷、磁力链接、快车下载等各种类型的下载URL。&br&&figure&&img src=&https://pic3.zhimg.com/v2-46636b2cab8bfd7bf2e7b6f4c6346c99_b.jpg& data-rawwidth=&1440& data-rawheight=&900& class=&origin_image zh-lightbox-thumb& width=&1440& data-original=&https://pic3.zhimg.com/v2-46636b2cab8bfd7bf2e7b6f4c6346c99_r.jpg&&&/figure&当然，以上说的还只是常用功能，还有好多功能大家可以慢慢研究。&br&&br&&br&&b&3 谁需要NAS&/b&&br&前面说过，NAS很便宜。&br&最便宜的一盘位（仅能插一块硬盘）NAS价格都在一千以内了，当然，笔者并不推荐一盘位的，因为一盘位很多功能受限制。家用的话，两盘位或者四盘位是比较理想的选择，当然对容量要求不高，那么两盘位是比较理想的。国内做NAS比较著名的厂商就是群晖和QNAP（威联通），笔者使用的两台NAS都是QNAP产品（因为同一厂家的产品可以方便的互换硬盘或者同步数据），这两个品牌的家用两盘位NAS产品的价格都在2000左右，性价比还是很高的。&br&什么？你觉得NAS会很丑？&br&话不多说，直接上图，这是笔者用过了两款NAS，分别是TS－212P和TAS－268，颜值自己看吧。&br&&figure&&img src=&https://pic4.zhimg.com/v2-0cd1c1ffad7f03dda4132d4_b.jpg& data-rawwidth=&2383& data-rawheight=&2415& class=&origin_image zh-lightbox-thumb& width=&2383& data-original=&https://pic4.zhimg.com/v2-0cd1c1ffad7f03dda4132d4_r.jpg&&&/figure&&figure&&img src=&https://pic4.zhimg.com/v2-cb6acf4aec20ce9c772e06_b.jpg& data-rawwidth=&2048& data-rawheight=&3025& class=&origin_image zh-lightbox-thumb& width=&2048& data-original=&https://pic4.zhimg.com/v2-cb6acf4aec20ce9c772e06_r.jpg&&&/figure&&br&公平起见，也上台群晖的产品图片。&br&&br&&figure&&img src=&https://pic3.zhimg.com/v2-224af295f9a6cc2ad569f4fcc67c0811_b.jpg& data-rawwidth=&1024& data-rawheight=&1024& class=&origin_image zh-lightbox-thumb& width=&1024& data-original=&https://pic3.zhimg.com/v2-224af295f9a6cc2ad569f4fcc67c0811_r.jpg&&&/figure&&figure&&img src=&https://pic2.zhimg.com/v2-0b1fcf44ec32b1aadf79ed357aa240dd_b.jpg& data-rawwidth=&1024& data-rawheight=&1024& class=&origin_image zh-lightbox-thumb& width=&1024& data-original=&https://pic2.zhimg.com/v2-0b1fcf44ec32b1aadf79ed357aa240dd_r.jpg&&&/figure&&br&下面说说谁特别需要NAS。&br&（1）网盘用户，存储了大量数据，网盘关闭了，数据转存到NAS才是出路。&br&（2）使用多台手机、笔记本、平板、台式机的用户，使用NAS可以方便的实现数据共享，相当于所有的设备都有了一块巨大的硬盘。&br&（3）对同步有要求的用户。大家都知道，网盘的同步是非常好用的，但是网盘本身不安全，随时可能关闭，且数据安全也不容易保证，但是存放在自己的NAS里，多台设备通过NAS同步则完全没有问题。&br&（4）家中人数较多，需要共享存储而不想抱着移动硬盘插来插去的。这个就不多说了。&br&（5）存储私密数据，网盘你懂的，8秒教育片你也懂得，老司机必备。&br&根据功能继续想把，我目前想到了这些～&/p&&br&&b&4 回答一些评论问题&/b&&br&文章发表以后，评论区讨论的也挺热，所以在此一并回答一下评论区的焦点问题。&br&（1）最简单的问题：NAS和云盘什么区别？本质区别就是NAS是你的数据在你的服务器你的硬盘上你说了算，而云盘是你的数据在别人服务器上别人硬盘上别人说了算，参考百度云删除用户数据和8秒教育片。&br&（2）关于网络问题，使用NAS的话肯定要架设家用千兆内网，千兆内网这个名词一听上去好像特复杂，其实很简单啦，买个支持千兆有线和802.11AC无线的宽带路由器就搞定了，这种类型的路由器价格一般在400元左右或以上，当然现在的家庭，手机笔记本平板电视特别多，没有个高性能路由器有不行。&br&（3）关于NAS的DIY，这个的确是可以，我当时也想过，GEN8也是一个很不错的平台，但是对于多数用户而言，门槛要高一些，除了自己搭建硬件平台控制功耗以外，还要自己安装服务器端的操作系统和各种应用软件，不如直接入一台NAS，软件硬件都有了，而且界面良好，说明书详细，上手非常容易。&br&（4）有朋友提小米的带硬盘的路由器，还有可以插硬盘的那种路由器，怎么说呢，如果你对数据可靠性没什么要求，仅仅存点电影电视剧，也是可以的，它相当于一个简易的1盘位NAS，不过功能和性能上比专业NAS差不少，当然数据可靠性差的就更多了，这个在下一篇文章中会继续讨论。&br&（5）对于宽带的上传限速，这个就得在选择宽带的时候注意一下，尽量选择上传速度快的，当然，这主要影响到你在家庭以外的网络访问NAS的速度，其实多数情况下，NAS和普通网盘或云盘在外网用起来体验是差不多的。&br&如有疑问，欢迎在评论区或者私信提问，我会及时回答并更新此文。&br&&br&&p&如果你还是觉得NAS太复杂，对你来说没必要，那么可以看看我以前写的这篇文章：&/p&&p&&a href=&https://zhuanlan.zhihu.com/p/& class=&internal&&U盘变私人云——Magic Disc魔碟网盘 - 家+智能 - 知乎专栏&/a&&br&&br&&/p&&p&入门篇就到这里了，下一篇为进阶篇：&/p&&p&&a href=&https://zhuanlan.zhihu.com/p/& class=&internal&&网盘纷纷关闭的时候，让我们来说说NAS——进阶篇 - 家+智能 - 知乎专栏&/a&&/p&&p&为方便阅读，文章会在微信公众号发表，欢迎关注“家+智能”公众号：homeandsmart&/p&&p&新浪微博：Henix_Sun&/p&
此文的灵感来自360云盘关闭。NAS（网络附加存储），其实就是存储服务器。 在这个网盘纷纷关闭的萧瑟的深秋，我觉得有必要聊聊这个话题。 这个东西，对我来说是一个偶然，但是用了之后，成了必然。 最初，我的MacBook air是128G的，我还喜欢拍照用RAW格式，…
&figure&&img data-rawheight=&1920& src=&https://pic4.zhimg.com/v2-acbd13afd87_b.jpg& data-rawwidth=&1080& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic4.zhimg.com/v2-acbd13afd87_r.jpg&&&/figure&&br&&br&硕士在读，方向机器学习，也是刚入门，目前在啃这本神经网络和深度学习，三百多页，目前看了1/3，觉得干货满满，预计最多还要两周看完。想看明白不需要太多的数学功底，只要求自己清楚一些机器学习的术语和基本阅读能力，更棒的是，作者能够教你一行一行的教你敲代码（看完我只想说，原来神经网络的代码可以写的这么简洁）。&br&&br&&br&就拿BP神经网络来说吧，之前大概了解神经网络是个什么东西，也知道神经网络能通过反向传播来训练网络参数，但不懂反向传播到底是怎么工作的。这本书好就好在作者能够图文并茂，从最基本的求导给你展示反向传播是怎么训练参数。并且后面从交叉墒，softmax 两个个方面阐述了选用不同的cost function会如何帮助训练过程跳出瓶颈（相对于最小二乘法），而且还实验多次给你阐述过拟合是怎么产生的，我们有什么好的办法去避免过拟合…&br&&br&&br&半夜手机码字太累了，需要交流可以加微信，大家一起进步。&br&&br&&br&链接：&br&&br&&a href=&//link.zhihu.com/?target=http%3A//neuralnetworksanddeeplearning.com& class=& external& target=&_blank& rel=&nofollow noreferrer&&&span class=&invisible&&http://&/span&&span class=&visible&&neuralnetworksanddeeplearning.com&/span&&span class=&invisible&&&/span&&/a&
硕士在读，方向机器学习，也是刚入门，目前在啃这本神经网络和深度学习，三百多页，目前看了1/3，觉得干货满满，预计最多还要两周看完。想看明白不需要太多的数学功底，只要求自己清楚一些机器学习的术语和基本阅读能力，更棒的是，作者能够教你一行一行的…
&figure&&img src=&https://pic4.zhimg.com/v2-ed364c69aed8cbdd3c255_b.jpg& data-rawwidth=&600& data-rawheight=&279& class=&origin_image zh-lightbox-thumb& width=&600& data-original=&https://pic4.zhimg.com/v2-ed364c69aed8cbdd3c255_r.jpg&&&/figure&&h2&一、图模型&/h2&&p&我们可以将概率分布对应成图模型。首先定义随机变量集合&img src=&https://www.zhihu.com/equation?tex=V%3DX+%5Ccup+Y& alt=&V=X \cup Y& eeimg=&1&&，其中&img src=&https://www.zhihu.com/equation?tex=X& alt=&X& eeimg=&1&&表示&b&输入变量&/b&（观测变量）集合，&img src=&https://www.zhihu.com/equation?tex=Y& alt=&Y& eeimg=&1&&表示&b&输出变量&/b&（预测变量）集合，以词性标注问题（POS）为例，输入变量是句子中的每一个单词，输出变量是每个单词对应的词性标记。&/p&&p&我们记&img src=&https://www.zhihu.com/equation?tex=x_A& alt=&x_A& eeimg=&1&&是&img src=&https://www.zhihu.com/equation?tex=X& alt=&X& eeimg=&1&&的一个子集，&img src=&https://www.zhihu.com/equation?tex=y_A& alt=&y_A& eeimg=&1&&是&img src=&https://www.zhihu.com/equation?tex=Y& alt=&Y& eeimg=&1&&的一个子集，函数&img src=&https://www.zhihu.com/equation?tex=%5CPsi_A+%28x_A%2C+y_A%29& alt=&\Psi_A (x_A, y_A)& eeimg=&1&&是对应&img src=&https://www.zhihu.com/equation?tex=x_A& alt=&x_A& eeimg=&1&&和&img src=&https://www.zhihu.com/equation?tex=y_A& alt=&y_A& eeimg=&1&&的&b&兼容函数&/b&（compatibility function），因子&img src=&https://www.zhihu.com/equation?tex=F+%3D+%5C%7B+%5CPsi_A+%5C%7D& alt=&F = \{ \Psi_A \}& eeimg=&1&&表示所有兼容函数的集合，在本文中，假设所有兼容函数都有如下的形式，&img src=&https://www.zhihu.com/equation?tex=%5CPsi_A%28x_A%2C+y_A%29+%3D+%5Cexp+%5Cleft%5C%7B+%5Csum_k+%5Ctheta_%7BAk%7D+f_%7BAk%7D%28x_A%2C+y_A%29+%5Cright%5C%7D& alt=&\Psi_A(x_A, y_A) = \exp \left\{ \sum_k \theta_{Ak} f_{Ak}(x_A, y_A) \right\}& eeimg=&1&&，其中&img src=&https://www.zhihu.com/equation?tex=f_%7BAk%7D%28x_A%2Cy_A%29& alt=&f_{Ak}(x_A,y_A)& eeimg=&1&&是关于&img src=&https://www.zhihu.com/equation?tex=x_A& alt=&x_A& eeimg=&1&&和&img src=&https://www.zhihu.com/equation?tex=y_A& alt=&y_A& eeimg=&1&&的特征函数，&img src=&https://www.zhihu.com/equation?tex=%5Ctheta_%7BAk%7D& alt=&\theta_{Ak}& eeimg=&1&&是参数。定义判断函数&img src=&https://www.zhihu.com/equation?tex=%5Cbm%7B1%7D+%5C%7B+%5Ccdot+%5C%7D& alt=&\bm{1} \{ \cdot \}& eeimg=&1&&，如果括号里的值为真则函数值为1，否则为0。&/p&&p&对于特定的兼容函数集合，我们可以将输入变量、输出变量的联合概率分布对应成一个无向图模型&img src=&https://www.zhihu.com/equation?tex=G%3D%28V%2CF%29& alt=&G=(V,F)& eeimg=&1&&的形式，其中&img src=&https://www.zhihu.com/equation?tex=V& alt=&V& eeimg=&1&&表示图模型中的顶点，对应概率分布中的每一个输入/输出变量，&img src=&https://www.zhihu.com/equation?tex=F& alt=&F& eeimg=&1&&表示图模型中由不同顶点组成的关联，对应概率分布中的每一个兼容函数。如Equ1，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Bequation%7D%0Ap%28x%2Cy%29+%3D+%5Cprod_%7BA%7D+%5CPsi_A+%28x_A%7Cy_A%29+%3D+%5Cprod_A+%5Cexp+%5Cleft%5C%7B+%5Csum_k+%5Ctheta_%7BAk%7D+f_%7BAk%7D%28x_A%2C+y_A%29+%5Cright%5C%7D%0A%5Cend%7Bequation%7D& alt=&\begin{equation}
p(x,y) = \prod_{A} \Psi_A (x_A|y_A) = \prod_A \exp \left\{ \sum_k \theta_{Ak} f_{Ak}(x_A, y_A) \right\}
\end{equation}& eeimg=&1&&&p&接下来以朴素贝叶斯为例，对这个框架进行深入理解。&/p&&h2&二、朴素贝叶斯&/h2&朴素贝叶斯模型需要建模的是输入变量和输出变量的&b&联合概率分布&/b&&img src=&https://www.zhihu.com/equation?tex=p%28y%2Cx_1%2Cx_2%2C%5Ccdots%2Cx_N%29& alt=&p(y,x_1,x_2,\cdots,x_N)& eeimg=&1&&，有N个输入变量，每个输入变量表示一个特征，有1个输出变量，输出变量表示类别。以文本分类问题为例，输入变量是词袋中的每个单词，输入变量是文档类别。基于输入变量条件独立的假设，即任意两个输入变量之间是相互独立的，因此联合概率分布可以写成&img src=&https://www.zhihu.com/equation?tex=p%28y%2Cx_1%2C%5Ccdots%2Cx_N%29%3Dp%28y%29%5Cprod_%7Bi%3D1%7D%5E%7BN%7D+p%28x_i%7Cy%29+& alt=&p(y,x_1,\cdots,x_N)=p(y)\prod_{i=1}^{N} p(x_i|y) & eeimg=&1&&，对应的无向图模型如下。&p&&figure&&img src=&https://pic1.zhimg.com/v2-b050746baf36b9f7e85e1a8db94f4043_b.jpg& data-rawwidth=&579& data-rawheight=&302& class=&origin_image zh-lightbox-thumb& width=&579& data-original=&https://pic1.zhimg.com/v2-b050746baf36b9f7e85e1a8db94f4043_r.jpg&&&/figure&按照如下的对应关系，则可以将朴素贝叶斯模型的定义适配到上述Equ1的框架中。令，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0AF+%3D+%26%5C%7B%26+%5C%7B+f_%7B01%7D%28y%29%2C+%5Ccdots%2C+f_%7B0M%7D%28y%29+%5C%7D%2C+%5C%5C%0A%26%26+%5C%7B+f_%7B11%7D%28x_1%2Cy%29%2C+%5Ccdots%2C+f_%7B1M%7D%28x_1%2Cy%29+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+f_%7BN1%7D%28x_N%2Cy%29%2C+%5Ccdots%2C+f_%7BNM%7D%28x_N%2Cy%29+%5C%7D+%5C+%5C+%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
F = &\{& \{ f_{01}(y), \cdots, f_{0M}(y) \}, \\
&& \{ f_{11}(x_1,y), \cdots, f_{1M}(x_1,y) \}, \\
&& \cdots \\
&& \{ f_{N1}(x_N,y), \cdots, f_{NM}(x_N,y) \} \ \ \}
\end{eqnarray}& eeimg=&1&&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5CTheta+%3D+%26%5C%7B%26+%5C%7B+%5Ctheta_%7B01%7D%2C+%5Ccdots%2C+%5Ctheta_%7B0M%7D+%5C%7D%2C+%5C%5C%0A%26%26+%5C%7B+%5Ctheta_%7B11%7D%2C+%5Ccdots%2C+%5Ctheta_%7B1M%7D+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+%5Ctheta_%7BN1%7D%2C+%5Ccdots%2C+%5Ctheta_%7BNM%7D+%5C%7D+%5C+%5C+%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\Theta = &\{& \{ \theta_{01}, \cdots, \theta_{0M} \}, \\
&& \{ \theta_{11}, \cdots, \theta_{1M} \}, \\
&& \cdots \\
&& \{ \theta_{N1}, \cdots, \theta_{NM} \} \ \ \}
\end{eqnarray}& eeimg=&1&&&br&&p&其中，有如下关系&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%26%26+f_%7B0k%7D%28y%29+%3D%5Cbm%7B1%7D%5C%7B+y%3Dc_k+%5C%7D%2C+f_%7Bik%7D%28x_i%2Cy%29+%3D%5Cbm%7B1%7D%5C%7B+y%3Dc_k+%5C%7D+%5C%5C%0A%26%26+%5Ctheta_%7B0k%7D+%3D+%5Clog+p%28y%3Dc_k%29%2C+%5Ctheta_%7Bik%7D+%3D+%5Clog+p%28x_i%7Cy%3Dc_k%29+%5C%5C%0A%26%26+%5CPsi_i%28x_i%2Cy%29+%3D+%5Cexp+%5Cleft%5C%7B+%5Csum%5EM_%7Bk%3D1%7D+%5Ctheta_%7Bik%7D+f_%7Bik%7D%28x_i%2C+y%29+%5Cright%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
&& f_{0k}(y) =\bm{1}\{ y=c_k \}, f_{ik}(x_i,y) =\bm{1}\{ y=c_k \} \\
&& \theta_{0k} = \log p(y=c_k), \theta_{ik} = \log p(x_i|y=c_k) \\
&& \Psi_i(x_i,y) = \exp \left\{ \sum^M_{k=1} \theta_{ik} f_{ik}(x_i, y) \right\}
\end{eqnarray}& eeimg=&1&&&br&&p&根据这样的兼容函数和参数，带入到Equ1中，得到&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0Ap%28y%2Cx_1%2C%5Ccdots%2Cx_N%29+%26%3D%26+%5Cprod_%7Bi%3D0%7D%5EN+%5CPsi_i%28x_i%2Cy%29+%5C%5C%0A%26%3D%26+%5Cprod_%7Bi%3D0%7D%5EN+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5EM+%5Ctheta_%7Bik%7D+f_%7Bik%7D%28x_i%2Cy%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+%5Cexp+%5Cleft%5C%7B+%5Clog+p%28y%29+%5Cright%5C%7D+%5Cprod_%7Bi%3D1%7D%5EN+%5Cexp+%5Cleft%5C%7B+%5Clog+p%28x_i%7Cy%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+p%28y%29+%5Cprod_%7Bi%3D1%7D%5EN+%5Clog+p%28x_i%7Cy%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
p(y,x_1,\cdots,x_N) &=& \prod_{i=0}^N \Psi_i(x_i,y) \\
&=& \prod_{i=0}^N \exp \left\{ \sum_{k=1}^M \theta_{ik} f_{ik}(x_i,y) \right\} \\
&=& \exp \left\{ \log p(y) \right\} \prod_{i=1}^N \exp \left\{ \log p(x_i|y) \right\} \\
&=& p(y) \prod_{i=1}^N \log p(x_i|y)
\end{eqnarray}& eeimg=&1&&&br&&p&综上说述，按照给定的&img src=&https://www.zhihu.com/equation?tex=%5CPsi+& alt=&\Psi & eeimg=&1&&和&img src=&https://www.zhihu.com/equation?tex=%5CTheta+& alt=&\Theta & eeimg=&1&&就可以推导出朴素贝叶斯模型的公式，也就是说朴素贝叶斯模型可以转化成无向图模型。接下来推导出隐马尔科夫模型对应的无向图模型。&/p&&h2&三、隐马尔科夫模型&/h2&&p&隐马尔科夫模型需要建模的是输入变量和输出变量的&b&联合概率分布&/b&&img src=&https://www.zhihu.com/equation?tex=p%28y_1%2C+%5Ccdots%2C+y_T+%2C+x_1%2C+%5Ccdots%2C+x_T%29& alt=&p(y_1, \cdots, y_T , x_1, \cdots, x_T)& eeimg=&1&&，T个输入变量，表示每一个观测值，T个输出变量，表示每一个状态值。以词性标注问题为例，输入变量是句子中的每一个单词，输出变量是每个单词对应的词性标记。隐马尔科夫过程假设当前状态只由上一个状态决定，而与历史无关，并且当前观测只由当前状态决定，因此联合概率分布可以写成&img src=&https://www.zhihu.com/equation?tex=p%28y_1%2C+%5Ccdots%2C+y_T+%2C+x_1%2C+%5Ccdots%2C+x_T%29%3D%5Cprod_%7Bt%3D1%7D%5ET+p%28y_t%7Cy_%7Bt-1%7D%29p%28x_t%7Cy_t%29& alt=&p(y_1, \cdots, y_T , x_1, \cdots, x_T)=\prod_{t=1}^T p(y_t|y_{t-1})p(x_t|y_t)& eeimg=&1&&，对应的无向图模型如下。&br&&/p&&figure&&img src=&https://pic3.zhimg.com/v2-c586fbf39abdb_b.jpg& data-rawwidth=&693& data-rawheight=&304& class=&origin_image zh-lightbox-thumb& width=&693& data-original=&https://pic3.zhimg.com/v2-c586fbf39abdb_r.jpg&&&/figure&&p&按照如下的对应关系，则可以将朴素贝叶斯模型的定义适配到上述Equ1的框架中。令，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0AF+%3D+%26%5C%7B%26+%5C%7B+f_%7B11%7D%5E1%28y_t%3Ds_1%2Cy_%7Bt-1%7D%3Ds_1%29%2C+%5Ccdots%2C+f%5E1_%7B1M%7D%28y_t%3Ds_1%2Cy_%7Bt-1%7D%3Ds_M%29+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+f_%7BM1%7D%5E1%28y_t%3Ds_M%2Cy_%7Bt-1%7D%3Ds_1%29%2C+%5Ccdots%2C+f%5E1_%7BMM%7D%28y_t%3Ds_M%2Cy_%7Bt-1%7D%3Ds_M%29+%5C%7D%2C+%5C%5C%0A%26%26+%5C%7B+f_%7B11%7D%5E2+%28x_t%3Do_1%2Cy_t%3Ds_1%29%2C+%5Ccdots%2C+f%5E2_%7B1M%7D%28x_t%3Do_1%2Cy_t%3Ds_M%29+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+f_%7BN1%7D%5E2+%28x_t%3Do_N%2Cy_t%3Ds_1%29%2C+%5Ccdots%2C+f%5E2_%7BNM%7D%28x_t%3Do_N%2Cy_t%3Ds_M%29+%5C%7D+%5C+%5C+%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
F = &\{& \{ f_{11}^1(y_t=s_1,y_{t-1}=s_1), \cdots, f^1_{1M}(y_t=s_1,y_{t-1}=s_M) \}, \\
&& \cdots \\
&& \{ f_{M1}^1(y_t=s_M,y_{t-1}=s_1), \cdots, f^1_{MM}(y_t=s_M,y_{t-1}=s_M) \}, \\
&& \{ f_{11}^2 (x_t=o_1,y_t=s_1), \cdots, f^2_{1M}(x_t=o_1,y_t=s_M) \}, \\
&& \cdots \\
&& \{ f_{N1}^2 (x_t=o_N,y_t=s_1), \cdots, f^2_{NM}(x_t=o_N,y_t=s_M) \} \ \ \}
\end{eqnarray}& eeimg=&1&&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5CTheta+%3D+%26%5C%7B%26+%5C%7B+%5Ctheta%5E1_%7B11%7D%2C+%5Ccdots%2C+%5Ctheta%5E1_%7B1M%7D+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+%5Ctheta%5E1_%7BM1%7D%2C+%5Ccdots%2C+%5Ctheta%5E1_%7BMM%7D+%5C%7D%2C+%5C%5C%0A%26%26+%5C%7B+%5Ctheta%5E2_%7B11%7D%2C+%5Ccdots%2C+%5Ctheta%5E2_%7B1M%7D+%5C%7D%2C+%5C%5C%0A%26%26+%5Ccdots+%5C%5C%0A%26%26+%5C%7B+%5Ctheta%5E2_%7BN1%7D%2C+%5Ccdots%2C+%5Ctheta%5E2_%7BNM%7D+%5C%7D+%5C+%5C+%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\Theta = &\{& \{ \theta^1_{11}, \cdots, \theta^1_{1M} \}, \\
&& \cdots \\
&& \{ \theta^1_{M1}, \cdots, \theta^1_{MM} \}, \\
&& \{ \theta^2_{11}, \cdots, \theta^2_{1M} \}, \\
&& \cdots \\
&& \{ \theta^2_{N1}, \cdots, \theta^2_{NM} \} \ \ \}
\end{eqnarray}& eeimg=&1&&&br&&p&为了方便记录，我们将上述的特征函数&img src=&https://www.zhihu.com/equation?tex=f_%7Bij%7D%5E1%28y_t%2C+y_%7Bt-1%7D%29%2Cf%5E2_%7Boi%7D%28y_t%2Cx_t%29& alt=&f_{ij}^1(y_t, y_{t-1}),f^2_{oi}(y_t,x_t)& eeimg=&1&&和参数&img src=&https://www.zhihu.com/equation?tex=%5Ctheta%5E1_%7Bij%7D%2C+%5Ctheta%5E2_%7Boi%7D& alt=&\theta^1_{ij}, \theta^2_{oi}& eeimg=&1&&写成一种形式，即特征函数写成&img src=&https://www.zhihu.com/equation?tex=f_k%28y_t%2Cy_%7Bt-1%7D%2Cx_t%29& alt=&f_k(y_t,y_{t-1},x_t)& eeimg=&1&&，参数写成&img src=&https://www.zhihu.com/equation?tex=%5Ctheta_k& alt=&\theta_k& eeimg=&1&&。则，有如下关系，K表示所有特征函数的个数，也就是所有参数的个数。&br&&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%0A%26%26+f_%7Bij%7D%5E1%28y_t%3Ds_i%2Cy_%7Bt-1%7D%3Ds_j%29%3D%5Cbm%7B1%7D%5C%7By_t%3Ds_i%5C%7D%5Cbm%7B1%7D%5C%7By_%7Bt-1%7D%3Ds_j%5C%7D%2C+%5C%5C%0A%26%26+f_%7Bli%7D%5E2%28x_t%3Do_l%2Cy_t%3Ds_i%29%3D%5Cbm%7B1%7D%5C%7Bx_t%3Do_l%5C%7D%5Cbm%7B1%7D%5C%7By_t%3Ds_i%5C%7D%2C+%5C%5C%0A%26%26+%5Ctheta%5E1_%7Bij%7D+%3D+%5Clog+p%28y_t%3Ds_i+%7C+y_%7Bt-1%7D%3Ds_j%29%2C+%5C%5C%0A%26%26+%5Ctheta%5E2_%7Bli%7D+%3D+%5Clog+p%28x_t%3Do_l+%7C+y_t%3Ds_i%29+%5C%5C%0A%26%26+%5CPsi_t%28y_t%2C+y_%7By-1%7D%2C+x_t%29+%3D+%5Cexp+%5Cleft%5C%7B+%5Csum%5EK_%7Bk%3D1%7D+%5Ctheta_%7Bk%7D+f_%7Bk%7D%28y_t%2C+y_%7Bt-1%7D%2C+x_i%29+%5Cright%5C%7D+%3D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bs_i+%5Cin+S%7D+%5Csum_%7Bs_j+%5Cin+S%7D+%5Ctheta%5E1_%7Bij%7D+f%5E1_%7Bij%7D%28y_t%3Ds_i%2C+y_%7Bt-1%7D%3Ds_j%29+%2B+%5Csum_%7Bo_l+%5Cin+O%7D+%5Csum_%7Bs_i+%5Cin+S%7D+%5Ctheta%5E2_%7Bli%7D+f%5E2_%7Bli%7D%28x_t%3Do_l%2C+y_t%3Ds_i%29+%5Cright%5C%7D%0A%5Cend%7Beqnarray%7D+& alt=&\begin{eqnarray}
&& f_{ij}^1(y_t=s_i,y_{t-1}=s_j)=\bm{1}\{y_t=s_i\}\bm{1}\{y_{t-1}=s_j\}, \\
&& f_{li}^2(x_t=o_l,y_t=s_i)=\bm{1}\{x_t=o_l\}\bm{1}\{y_t=s_i\}, \\
&& \theta^1_{ij} = \log p(y_t=s_i | y_{t-1}=s_j), \\
&& \theta^2_{li} = \log p(x_t=o_l | y_t=s_i) \\
&& \Psi_t(y_t, y_{y-1}, x_t) = \exp \left\{ \sum^K_{k=1} \theta_{k} f_{k}(y_t, y_{t-1}, x_i) \right\} = \exp \left\{ \sum_{s_i \in S} \sum_{s_j \in S} \theta^1_{ij} f^1_{ij}(y_t=s_i, y_{t-1}=s_j) + \sum_{o_l \in O} \sum_{s_i \in S} \theta^2_{li} f^2_{li}(x_t=o_l, y_t=s_i) \right\}
\end{eqnarray} & eeimg=&1&&&br&&p&根据这样的兼容函数和参数，带入到Equ1中，得到，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0Ap%28y_1%2C+%5Ccdots%2C+y_T%2C+x_1%2C%5Ccdots%2Cx_T%29+%26%3D%26+%5Cprod_%7Bi%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Cprod_%7Bt%3D1%7D%5ET+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bs_i+%5Cin+S%7D+%5Csum_%7Bs_j+%5Cin+S%7D+%5Ctheta%5E1_%7Bij%7D+f%5E1_%7Bij%7D%28y_t%3Ds_i%2C+y_%7Bt-1%7D%3Ds_j%29+%2B+%5Csum_%7Bo_l+%5Cin+O%7D+%5Csum_%7Bs_i+%5Cin+S%7D+%5Ctheta%5E2_%7Bli%7D+f%5E2_%7Bli%7D%28x_t%3Do_l%2C+y_t%3Ds_i%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+%5Cprod_%7Bt%3D1%7D%5ET+%5Cexp+%5Cleft%5C%7B+%5Clog+p%28y_t%7Cy_%7Bt-1%7D%29+%2B+%5Clog+p%28x_t%7Cy_t%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+%5Cprod_%7Bt%3D1%7D%5ET+p%28y_t%7Cy_%7Bt-1%7D%29+p%28x_t%7Cy_t%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
p(y_1, \cdots, y_T, x_1,\cdots,x_T) &=& \prod_{i=1}^T \Psi_t(y_t, y_{t-1}, x_t) \\
&=& \prod_{t=1}^T \exp \left\{ \sum_{s_i \in S} \sum_{s_j \in S} \theta^1_{ij} f^1_{ij}(y_t=s_i, y_{t-1}=s_j) + \sum_{o_l \in O} \sum_{s_i \in S} \theta^2_{li} f^2_{li}(x_t=o_l, y_t=s_i) \right\} \\
&=& \prod_{t=1}^T \exp \left\{ \log p(y_t|y_{t-1}) + \log p(x_t|y_t) \right\} \\
&=& \prod_{t=1}^T p(y_t|y_{t-1}) p(x_t|y_t)
\end{eqnarray}& eeimg=&1&&&br&&p&综上说述，按照给定的&img src=&https://www.zhihu.com/equation?tex=%5CPsi+& alt=&\Psi & eeimg=&1&&和&img src=&https://www.zhihu.com/equation?tex=%5CTheta+& alt=&\Theta & eeimg=&1&&就可以推导出隐马尔科夫模型的公式，也就是说隐马尔科夫模型可以转化成无向图模型。接下来推导出条件随机场模型对应的无向图模型。&/p&&h2&四、条件随机场&/h2&条件随机场模型需要建模的是输入变量和输出变量的&b&条件概率分布&/b&&img src=&https://www.zhihu.com/equation?tex=p%28y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T%29& alt=&p(y_1, \cdots, y_T | x_1, \cdots, x_T)& eeimg=&1&&，因此对应的无向图模型如下，&br&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0Ap%28y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C%5Ccdots%2Cx_T%29+%26%3D%26+%5Cfrac%7B+%5Cprod_%7Bi%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%7D%7B%5Csum_y++%5Cprod_%7Bi%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
p(y_1, \cdots, y_T | x_1,\cdots,x_T) &=& \frac{ \prod_{i=1}^T \Psi_t(y_t, y_{t-1}, x_t) }{\sum_y
\prod_{i=1}^T \Psi_t(y_t, y_{t-1}, x_t) }
\end{eqnarray}& eeimg=&1&&&br&&p&条件随机场模型和隐马尔科夫模型一样，重点解决三个问题：&/p&&ol&&li&&b&概率问题&/b&：给定模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&、观测值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%2C+%5Ccdots%2C+x_T+%5C%7D& alt=&\{ x_1, \cdots, x_T \}& eeimg=&1&&和状态值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+y_1%2C+%5Ccdots%2C+y_T+%5C%7D& alt=&\{ y_1, \cdots, y_T \}& eeimg=&1&&的前提下，计算条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T )& eeimg=&1&&。&/li&&li&&b&推测问题&/b&：给定模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&和观测值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%2C+%5Ccdots%2C+x_T+%5C%7D& alt=&\{ x_1, \cdots, x_T \}& eeimg=&1&&，计算最大条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T )& eeimg=&1&&对应的状态值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+y_1%2C+%5Ccdots%2C+y_T+%5C%7D& alt=&\{ y_1, \cdots, y_T \}& eeimg=&1&&。&/li&&li&&b&训练问题&/b&：给定训练数据&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%5E%7B%28i%29%7D%2C+%5Ccdots%2C+x_%7BT_i%7D%5E%7B%28i%29%7D%2C+y_1%5E%7B%28i%29%7D%2C+%5Ccdots%2C+y_%7BT_i%7D%5E%7B%28i%29%7D+%5C%7D_%7Bi%3D1%7D%5EN& alt=&\{ x_1^{(i)}, \cdots, x_{T_i}^{(i)}, y_1^{(i)}, \cdots, y_{T_i}^{(i)} \}_{i=1}^N& eeimg=&1&&的前提下，最大化条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T )& eeimg=&1&&为目标，训练模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&。&/li&&/ol&本文接下来介绍这三个问题。&br&&h2&五、概率问题&/h2&&p&给定模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&、观测值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%2C+%5Ccdots%2C+x_T+%5C%7D& alt=&\{ x_1, \cdots, x_T \}& eeimg=&1&&和状态值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+y_1%2C+%5Ccdots%2C+y_T+%5C%7D& alt=&\{ y_1, \cdots, y_T \}& eeimg=&1&&的前提下，计算条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T )& eeimg=&1&&。&br&&/p&&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29+%3D+%5Cfrac%7B+%5Cprod_%7Bt%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%7D%7B+%5Csum_y+%5Cprod_%7Bt%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%7D& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T ) = \frac{ \prod_{t=1}^T \Psi_t(y_t, y_{t-1}, x_t) }{ \sum_y \prod_{t=1}^T \Psi_t(y_t, y_{t-1}, x_t) }& eeimg=&1&&&br&&p&根据已知的数据，带入即可求出分子，这个过程很容易，但是难点在分母，将分母展开写，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Csum_y+%5Cprod_%7Bt%3D1%7D%5ET+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%3D+%5Csum_%7By_T+%5Cin+S%7D+%5Csum_%7By_%7BT-1%7D+%5Cin+S%7D+%5CPsi_T%28y_T%2C+y_%7BT-1%7D%2C+x_T%29+%5Csum_%7By_%7BT-2%7D+%5Cin+S%7D+%5CPsi_%7BT-1%7D%28y_%7BT-1%7D%2C+y_%7BT-2%7D%2C+x_%7BT-1%7D%29+%5Ccdots+%5Csum_%7By_1+%5Cin+S%7D+%5CPsi_2%28y_2%2C+y_1%2C+x_2%29+%5CPsi_1%28y_1%2C+y_0%2C+s_1%29& alt=&\sum_y \prod_{t=1}^T \Psi_t(y_t, y_{t-1}, x_t) = \sum_{y_T \in S} \sum_{y_{T-1} \in S} \Psi_T(y_T, y_{T-1}, x_T) \sum_{y_{T-2} \in S} \Psi_{T-1}(y_{T-1}, y_{T-2}, x_{T-1}) \cdots \sum_{y_1 \in S} \Psi_2(y_2, y_1, x_2) \Psi_1(y_1, y_0, s_1)& eeimg=&1&&&br&&p&设状态总数为S，则计算分母过程中，乘法次数为&img src=&https://www.zhihu.com/equation?tex=%28T-1%29+%5Ccdot+S%5ET& alt=&(T-1) \cdot S^T& eeimg=&1&&，加法次数为&img src=&https://www.zhihu.com/equation?tex=%5Cfrac%7BS%5ET%7D%7B2%7D& alt=&\frac{S^T}{2}& eeimg=&1&&，当S和T很大时，用这种暴力遍历的方法几乎无法计算，因此需要设计一种复杂度更低的方法计算分母。由于在计算过程中，很多&img src=&https://www.zhihu.com/equation?tex=+%5CPsi_t%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29& alt=& \Psi_t(y_t, y_{t-1}, x_t)& eeimg=&1&&被重复计算，因此想到使用动态规划的方法，将问题分解成子问题，迭代计算。&/p&&p&令&img src=&https://www.zhihu.com/equation?tex=%5Calpha_%7Bt%7D%28s_i%29+%3D+p%28x_1%2C+%5Ccdots%2C+x_t%2C+y_t%3Ds_i%29& alt=&\alpha_{t}(s_i) = p(x_1, \cdots, x_t, y_t=s_i)& eeimg=&1&&为每个子问题，则分母可以表示为，&/p&&p&&img src=&https://www.zhihu.com/equation?tex=p%28x_1%2C+%5Ccdots%2C+x_T%29+%3D+%5Csum_%7Bs_i+%5Cin+S%7D+p%28x_1%2C+%5Ccdots%2C+x_T%2C+y_T+%3D+s_i%29+%3D+%5Csum_%7Bs_i+%5Cin+S%7D+%5Calpha_T%28s_i%29& alt=&p(x_1, \cdots, x_T) = \sum_{s_i \in S} p(x_1, \cdots, x_T, y_T = s_i) = \sum_{s_i \in S} \alpha_T(s_i)& eeimg=&1&&，而子问题的递推关系如下，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5Calpha_t%28s_i%29+%26%3D%26+p%28x_1%2C+%5Ccdots%2C+x_t%2C+y_t%3Ds_i%29+%3D+%5Csum_%7By_1+%5Cin+S%2C+%5Ccdots%2C+y_%7Bt-1%7D+%5Cin+S%7D+p%28x_1%2C+%5Ccdots%2C+x_%7Bt-1%7D%2C+y_1%2C+%5Ccdots%2C+y_%7Bt-1%7D%2C+x_t%2C+y_t%3Ds_i%29+%5C%5C%0A%26%3D%26+%5Csum_%7By_1+%5Cin+S%2C+%5Ccdots%2C+y_%7Bt-1%7D+%5Cin+S%7D+%5Cprod_%7Bt%27%3D1%7D%5E%7Bt-1%7D+%5CPsi_%7Bt%27%7D%28y_%7Bt%27%7D%2C+y_%7Bt%27-1%7D%2C+x_%7Bt%27%7D%29+%5CPsi_t%28s_i%2C+y_%7Bt-1%7D%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Csum_%7Bs_j+%5Cin+S%7D+%5Csum_%7By_1+%5Cin+S%2C+%5Ccdots%2C+y_%7Bt-2%7D+%5Cin+S%7D+%5Cprod_%7Bt%27%3D1%7D%5E%7Bt-2%7D+%5CPsi_%7Bt%27%7D%28y_%7Bt%27%7D%2C+y_%7Bt%27-1%7D%2C+x_%7Bt%27%7D%29+%5CPsi_%7Bt-1%7D%28s_j%2C+y_%7Bt-2%7D%2C+x_%7Bt-1%7D%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Csum_%7Bs_j+%5Cin+S%7D+%5Calpha_%7Bt-1%7D%28s_j%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\alpha_t(s_i) &=& p(x_1, \cdots, x_t, y_t=s_i) = \sum_{y_1 \in S, \cdots, y_{t-1} \in S} p(x_1, \cdots, x_{t-1}, y_1, \cdots, y_{t-1}, x_t, y_t=s_i) \\
&=& \sum_{y_1 \in S, \cdots, y_{t-1} \in S} \prod_{t'=1}^{t-1} \Psi_{t'}(y_{t'}, y_{t'-1}, x_{t'}) \Psi_t(s_i, y_{t-1}, x_t) \\
&=& \sum_{s_j \in S} \sum_{y_1 \in S, \cdots, y_{t-2} \in S} \prod_{t'=1}^{t-2} \Psi_{t'}(y_{t'}, y_{t'-1}, x_{t'}) \Psi_{t-1}(s_j, y_{t-2}, x_{t-1}) \Psi_t(s_i, s_j, x_t) \\
&=& \sum_{s_j \in S} \alpha_{t-1}(s_j) \Psi_t(s_i, s_j, x_t)
\end{eqnarray}& eeimg=&1&&&br&&p&并且，当&img src=&https://www.zhihu.com/equation?tex=t%3D1& alt=&t=1& eeimg=&1&&时，有&img src=&https://www.zhihu.com/equation?tex=%5Calpha_1%28s_i%29+%3D+p%28x_1%2C+y_1%3Ds_i%29+%3D+%5CPsi_1%28s_i%2C+y_0%2C+x_1%29& alt=&\alpha_1(s_i) = p(x_1, y_1=s_i) = \Psi_1(s_i, y_0, x_1)& eeimg=&1&&，因此，整个概率问题求解过程的递推关系表达式为，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%0Ainit+%26%3A%26+%5Calpha_1%28s_i%29+%3D+%5CPsi_1%28s_i%2C+y_0%2C+x_1%29+%5C%5C%0Arecursive+%26%3A%26+%5Calpha_t%28s_i%29+%3D+%5Csum_%7Bs_j+%5Cin+S%7D+%5Calpha_%7Bt-1%7D%28s_j%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29+%5C%5C%0Aend+%26%3A%26+p%28x_1%2C+%5Ccdots%2C+x_T%29+%3D+%5Csum_%7Bs_i+%5Cin+S%7D%5Calpha_T%28s_i%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
init &:& \alpha_1(s_i) = \Psi_1(s_i, y_0, x_1) \\
recursive &:& \alpha_t(s_i) = \sum_{s_j \in S} \alpha_{t-1}(s_j) \Psi_t(s_i, s_j, x_t) \\
end &:& p(x_1, \cdots, x_T) = \sum_{s_i \in S}\alpha_T(s_i)
\end{eqnarray}& eeimg=&1&&&br&&p&设状态总数为S，则使用动态规划算法计算分母过程中，乘法次数为&img src=&https://www.zhihu.com/equation?tex=S%28S-1%29& alt=&S(S-1)& eeimg=&1&&，加法次数为&img src=&https://www.zhihu.com/equation?tex=S%5E2& alt=&S^2& eeimg=&1&&，大大提升了计算效率。&/p&&h2&六、推测问题&/h2&&p&给定模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&和观测值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%2C+%5Ccdots%2C+x_T+%5C%7D& alt=&\{ x_1, \cdots, x_T \}& eeimg=&1&&，计算最大条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+y_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, y_T )& eeimg=&1&&对应的状态值&img src=&https://www.zhihu.com/equation?tex=%5C%7B+y_1%2C+%5Ccdots%2C+y_T+%5C%7D& alt=&\{ y_1, \cdots, y_T \}& eeimg=&1&&，&img src=&https://www.zhihu.com/equation?tex=y%5E%2A+%3D+%5Cmax_%7By_1%2C+%5Ccdots%2C+y_T%7D+p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T%29& alt=&y^* = \max_{y_1, \cdots, y_T} p( y_1, \cdots, y_T | x_1, \cdots, x_T)& eeimg=&1&&。&/p&&p&同样使用动态规划的思想，将其分解成子问题，设，&/p&&p&&img src=&https://www.zhihu.com/equation?tex=%5Cdelta_t%28s_i%29+%3D+%5Cmax_%7By_1%2C+%5Ccdots%2C+y_%7Bt-1%7D%7D+p%28y_1%2C+%5Ccdots%2C+y_%7Bt-1%7D%2C+y_t%3Ds_i+%7Cx_1%2C+%5Ccdots%2C+x_t%29& alt=&\delta_t(s_i) = \max_{y_1, \cdots, y_{t-1}} p(y_1, \cdots, y_{t-1}, y_t=s_i |x_1, \cdots, x_t)& eeimg=&1&&，则，原始问题可以表示为，&br&&/p&&p&&img src=&https://www.zhihu.com/equation?tex=y%5E%2A+%3D+%5Cmax_%7By_1%2C+%5Ccdots%2C+y_T%7D+p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T%29+%3D+%5Cmax_%7Bs_i+%5Cin+S%7D+%5Cmax_%7By_1%2C+%5Ccdots%2C+y_T%7D+p%28+y_1%2C+%5Ccdots%2C+y_T%3Ds_i+%7C+x_1%2C+%5Ccdots%2C+x_T%29+%3D+%5Cmax_%7Bs_i+%5Cin+S%7D+%5Cdelta_T%28s_i%29& alt=&y^* = \max_{y_1, \cdots, y_T} p( y_1, \cdots, y_T | x_1, \cdots, x_T) = \max_{s_i \in S} \max_{y_1, \cdots, y_T} p( y_1, \cdots, y_T=s_i | x_1, \cdots, x_T) = \max_{s_i \in S} \delta_T(s_i)& eeimg=&1&&，而子问题具有如下的递推关系，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5Cdelta_t%28s_i%29+%26%3D%26%5Cmax_%7By_1%2C+%5Ccdots%2C+y_%7Bt-1%7D%7D+p%28y_1%2C+%5Ccdots%2C+y_%7Bt-1%7D%2C+y_t%3Ds_i+%7Cx_1%2C+%5Ccdots%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Cmax_%7Bs_j+%5Cin+S%7D++%5Cmax_%7By_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%7D+p%28y_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%2C+y_%7Bt-1%7D%3Ds_j%2C+y_t%3Ds_i+%7Cx_1%2C+%5Ccdots%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Cmax_%7Bs_j+%5Cin+S%7D+%5Cmax_%7By_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%7D+p%28y_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%2C+y_%7Bt-1%7D%3Ds_j+%7C+x_1%2C+%5Ccdots%2C+x_%7Bt-1%7D%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29+%5C%5C%0A%26%3D%26+%5Cmax_%7Bs_j+%5Cin+S%7D+%5Cdelta_%7Bt-1%7D%28s_j%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\delta_t(s_i) &=&\max_{y_1, \cdots, y_{t-1}} p(y_1, \cdots, y_{t-1}, y_t=s_i |x_1, \cdots, x_t) \\
&=& \max_{s_j \in S}
\max_{y_1, \cdots, y_{t-2}} p(y_1, \cdots, y_{t-2}, y_{t-1}=s_j, y_t=s_i |x_1, \cdots, x_t) \\
&=& \max_{s_j \in S} \max_{y_1, \cdots, y_{t-2}} p(y_1, \cdots, y_{t-2}, y_{t-1}=s_j | x_1, \cdots, x_{t-1}) \Psi_t(s_i, s_j, x_t) \\
&=& \max_{s_j \in S} \delta_{t-1}(s_j) \Psi_t(s_i, s_j, x_t)
\end{eqnarray}& eeimg=&1&&&br&&p&并且，当&img src=&https://www.zhihu.com/equation?tex=t%3D0& alt=&t=0& eeimg=&1&&时，有&img src=&https://www.zhihu.com/equation?tex=%5Cdelta_0%28s_i%29+%3D+%5Csum_%7Bs_i+%5Cin+S%7D+p%28y_0%29+%3D1& alt=&\delta_0(s_i) = \sum_{s_i \in S} p(y_0) =1& eeimg=&1&&，因此，整个概率问题求解过程的递推关系表达式为，&br&&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D+%0Ainit+%26%3A%26+%5Cdelta_0%28s_i%29+%3D+1+%5C%5C%0Arecursive+%26%3A%26+%5Cdelta_t%28s_i%29+%3D+%5Cmax_%7Bs_j+%5Cin+S%7D+%5Cdelta_%7Bt-1%7D%28s_j%29+%5CPsi_t%28s_i%2C+s_j%2C+x_t%29+%5C%5C%0Aend+%26%3A%26+y%5E%2A+%3D+%5Cmax_%7Bs_i+%5Cin+S%7D%5Cdelta_T%28s_i%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
init &:& \delta_0(s_i) = 1 \\
recursive &:& \delta_t(s_i) = \max_{s_j \in S} \delta_{t-1}(s_j) \Psi_t(s_i, s_j, x_t) \\
end &:& y^* = \max_{s_i \in S}\delta_T(s_i)
\end{eqnarray}& eeimg=&1&&&br&&p&使用动态规划算法求解推测问题，大大提升了计算效率，类似于隐马尔科夫模型中的Viterbi算法。&/p&&h2&七、训练问题&/h2&&p&给定训练数据&img src=&https://www.zhihu.com/equation?tex=%5C%7B+x_1%5E%7B%28i%29%7D%2C+%5Ccdots%2C+x_%7BT_i%7D%5E%7B%28i%29%7D%2C+y_1%5E%7B%28i%29%7D%2C+%5Ccdots%2C+y_%7BT_i%7D%5E%7B%28i%29%7D+%5C%7D_%7Bi%3D1%7D%5EN& alt=&\{ x_1^{(i)}, \cdots, x_{T_i}^{(i)}, y_1^{(i)}, \cdots, y_{T_i}^{(i)} \}_{i=1}^N& eeimg=&1&&的前提下，最大化条件概率&img src=&https://www.zhihu.com/equation?tex=p%28+y_1%2C+%5Ccdots%2C+y_T+%7C+x_1%2C+%5Ccdots%2C+x_T+%29& alt=&p( y_1, \cdots, y_T | x_1, \cdots, x_T )& eeimg=&1&&为目标，训练模型参数&img src=&https://www.zhihu.com/equation?tex=%5CTheta& alt=&\Theta& eeimg=&1&&。&br&&/p&&p&根据极大似然估计，对数据集取似然对数，目标函数为，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5Clog+L%28%5Ctheta%29+%26%3D%26+%5Csum_%7Bi%3D1%7D%5EN+p%28y%5E%7B%28i%29%7D+%7C+x%5E%7B%28i%29%7D%29+%5C%5C%0A%26%3D%26+%5Csum_%7Bi%3D1%7D%5EN+%5Clog+%5Cfrac%7B+%5Cprod_%7Bt%3D1%7D%5E%7BT_i%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D+%0A%2C+x_t%5E%7B%28i%29%7D%29+%5Cright%5C%7D+%7D%7B+%5Csum_y+%5Cprod_%7Bt%3D1%7D%5E%7BT_i%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D+%0A%2C+x_t%5E%7B%28i%29%7D%29+%5Cright%5C%7D+%7D+%5C%5C%0A%26%3D%26+%5Csum_%7Bi%3D1%7D%5EN+%5Csum_%7Bt%3D1%7D%5E%7BT_i%7D+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D+%0A%2C+x_t%5E%7B%28i%29%7D%29+-+%5Csum_%7Bi%3D1%7D%5EN+%5Clog+%5Csum_y+%5Cprod_%7Bt%3D1%7D%5E%7BT_i%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D+%0A%2C+x_t%5E%7B%28i%29%7D%29+%5Cright%5C%7D%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\log L(\theta) &=& \sum_{i=1}^N p(y^{(i)} | x^{(i)}) \\
&=& \sum_{i=1}^N \log \frac{ \prod_{t=1}^{T_i} \exp \left\{ \sum_{k=1}^K \theta_k f_k(y_t^{(i)}, y_{t-1}^{(i)}
, x_t^{(i)}) \right\} }{ \sum_y \prod_{t=1}^{T_i} \exp \left\{ \sum_{k=1}^K \theta_k f_k(y_t^{(i)}, y_{t-1}^{(i)}
, x_t^{(i)}) \right\} } \\
&=& \sum_{i=1}^N \sum_{t=1}^{T_i} \sum_{k=1}^K \theta_k f_k(y_t^{(i)}, y_{t-1}^{(i)}
, x_t^{(i)}) - \sum_{i=1}^N \log \sum_y \prod_{t=1}^{T_i} \exp \left\{ \sum_{k=1}^K \theta_k f_k(y_t^{(i)}, y_{t-1}^{(i)}
, x_t^{(i)}) \right\}
\end{eqnarray}& eeimg=&1&&&br&&p&令，&img src=&https://www.zhihu.com/equation?tex=Z%28x%5E%7B%28i%29%7D%29+%3D+%5Csum_y+%5Cprod_%7Bt%3D1%7D%5E%7BT_i%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D+%0A%2C+x_t%5E%7B%28i%29%7D%29+%5Cright%5C%7D+%3D+%5Csum_%7By_T%7D+%5Csum_%7By_%7BT-1%7D%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_T%2C+y_%7BT-1%7D%2C+x_T%29+%5Cright%5C%7D+%5Ccdots+%5Csum_%7By_1%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_2%2C+y_%7B1%7D%2C+x_2%29+%5Cright%5C%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_1%2C+y_0%2C+x_1%29+%5Cright%5C%7D& alt=&Z(x^{(i)}) = \sum_y \prod_{t=1}^{T_i} \exp \left\{ \sum_{k=1}^K \theta_k f_k(y_t^{(i)}, y_{t-1}^{(i)}
, x_t^{(i)}) \right\} = \sum_{y_T} \sum_{y_{T-1}} \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_T, y_{T-1}, x_T) \right\} \cdots \sum_{y_1} \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_2, y_{1}, x_2) \right\} \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_1, y_0, x_1) \right\}& eeimg=&1&&对&img src=&https://www.zhihu.com/equation?tex=%5Clog+L%28%5Ctheta%29& alt=&\log L(\theta)& eeimg=&1&&求导数，可得，&img src=&https://www.zhihu.com/equation?tex=%5Cfrac%7B%5Cpartial+%5Clog+L%28%5Ctheta%29%7D%7B%5Cpartial+%5Ctheta_k%7D+%3D++%5Csum_%7Bi%3D1%7D%5EN+%5Csum_%7Bt%3D1%7D%5E%7BT_i%7D+f_k%28y_t%5E%7B%28i%29%7D%2C+y_%7Bt-1%7D%5E%7B%28i%29%7D%2C+x_t%5E%7B%28i%29%7D%29+-+%5Cfrac%7B+%5Cpartial+Z%28x%5E%7B%28i%29%7D%29%7D%7B+%5Cpartial+%5Ctheta_k%7D& alt=&\frac{\partial \log L(\theta)}{\partial \theta_k} =
\sum_{i=1}^N \sum_{t=1}^{T_i} f_k(y_t^{(i)}, y_{t-1}^{(i)}, x_t^{(i)}) - \frac{ \partial Z(x^{(i)})}{ \partial \theta_k}& eeimg=&1&&，难点在第二项，我们单独对第二项进行推导，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5Cfrac%7B+%5Cpartial+%5Clog+Z%28x%5E%7B%28i%29%7D%29%7D%7B+%5Cpartial+%5Ctheta_k%7D+%26%3D%26+%5Cfrac%7B1%7D%7BZ%28x%5E%7B%28i%29%7D%29%7D+%5Ccdot+%5Cfrac%7B%5Cpartial+Z%28x%5E%7B%28i%29%7D%29%7D%7B+%5Cpartial+%5Ctheta_k%7D+%3D+%5Csum_%7By_1%2C+%5Ccdots%2C+y_T%7D+p%28y_1%2C+%5Ccdots%2C+y_T%7Cx%5E%7B%28i%29%7D%29+%5Csum_%7Bt%3D1%7D%5E%7BT_i%7D+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\frac{ \partial \log Z(x^{(i)})}{ \partial \theta_k} &=& \frac{1}{Z(x^{(i)})} \cdot \frac{\partial Z(x^{(i)})}{ \partial \theta_k} = \sum_{y_1, \cdots, y_T} p(y_1, \cdots, y_T|x^{(i)}) \sum_{t=1}^{T_i} f_k(y_t, y_{t-1}, x_t)
\end{eqnarray}& eeimg=&1&&&br&&p&我们重点来理解第二个等号是怎么得到的，为了简单起见，我们先假设状态集合为&img src=&https://www.zhihu.com/equation?tex=%5C%7B+s_1%2C+s_2+%5C%7D& alt=&\{ s_1, s_2 \}& eeimg=&1&&，再去掉上标，由于&img src=&https://www.zhihu.com/equation?tex=Z%28x%29& alt=&Z(x)& eeimg=&1&&外部是对每一条路径&img src=&https://www.zhihu.com/equation?tex=%3Cy_1%2C+y_2%2C+%5Ccdots%2C+y_T%3E& alt=&&y_1, y_2, \cdots, y_T&& eeimg=&1&&的求和，那么对其求导则可以单独对每一项求导再加和，对每一项单独求导可写成，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cbegin%7Beqnarray%7D%0A%5Cnabla%5E%7B%28y_1%2C+%5Ccdots%2C+y_T%29%7D_%7B%5Ctheta%7D+%26%3D%26+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+%5Ctheta_k%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_T%2C+y_%7BT-1%7D%2C+x_T%29+%5Cright%5C%7D+%5Ccdots+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_2%2C+y_%7B1%7D%2C+x_2%29+%5Cright%5C%7D+%5Cexp+%5Cleft%5C%7B+%5Csum_%7Bk%3D1%7D%5E%7BK%7D+%5Ctheta_k+f_k%28y_1%2C+y_0%2C+x_1%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+%5Ctheta_k%7D+%5Cexp+%5Cleft%5C%7B+%0A%5Csum_%7Bt%3D1%7D%5ET+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%5Cright%5C%7D+%5C%5C%0A%26%3D%26+%5Cexp+%5Cleft%5C%7B+%0A%5Csum_%7Bt%3D1%7D%5ET+%5Csum_%7Bk%3D1%7D%5EK+%5Ctheta_k+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%5Cright%5C%7D+%5Csum_%7Bt%3D1%7D%5ET+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29%0A%5Cend%7Beqnarray%7D& alt=&\begin{eqnarray}
\nabla^{(y_1, \cdots, y_T)}_{\theta} &=& \frac{\partial}{\partial \theta_k} \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_T, y_{T-1}, x_T) \right\} \cdots \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_2, y_{1}, x_2) \right\} \exp \left\{ \sum_{k=1}^{K} \theta_k f_k(y_1, y_0, x_1) \right\} \\
&=& \frac{\partial}{\partial \theta_k} \exp \left\{
\sum_{t=1}^T \sum_{k=1}^K \theta_k f_k(y_t, y_{t-1}, x_t) \right\} \\
&=& \exp \left\{
\sum_{t=1}^T \sum_{k=1}^K \theta_k f_k(y_t, y_{t-1}, x_t) \right\} \sum_{t=1}^T f_k(y_t, y_{t-1}, x_t)
\end{eqnarray}& eeimg=&1&&&br&&p&那么，对这些导数求和，就可以得到&img src=&https://www.zhihu.com/equation?tex=%5Clog+Z%28x%5E%7B%28i%29%7D%29& alt=&\log Z(x^{(i)})& eeimg=&1&&的导数，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Cfrac%7B+%5Cpartial+%5Clog+Z%28x%29%7D%7B+%5Cpartial+%5Ctheta_k%7D+%3D+%5Csum_%7By_1%2C+%5Ccdots%2C+y_T%7D+%5Cfrac%7B+%5Cexp+%5Cleft%5C%7B+%0A%5Csum_%7Bt%3D1%7D%5ET+%5Csum_k%5EK+%5Ctheta_k+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%5Cright%5C%7D+%7D%7B+%5Csum_y+%5Cexp+%5Cleft%5C%7B+%0A%5Csum_%7Bt%3D1%7D%5ET+%5Csum_k%5EK+%5Ctheta_k+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%5Cright%5C%7D+%7D+%5Csum_%7Bt%3D1%7D%5ET+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%3D+%5Csum_%7By_1%2C+%5Ccdots%2C+y_T%7D+p%28y_1%2C+%5Ccdots%2C+y_T%7Cx%29+%5Csum_%7Bt%3D1%7D%5ET+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29& alt=&\frac{ \partial \log Z(x)}{ \partial \theta_k} = \sum_{y_1, \cdots, y_T} \frac{ \exp \left\{
\sum_{t=1}^T \sum_k^K \theta_k f_k(y_t, y_{t-1}, x_t) \right\} }{ \sum_y \exp \left\{
\sum_{t=1}^T \sum_k^K \theta_k f_k(y_t, y_{t-1}, x_t) \right\} } \sum_{t=1}^T f_k(y_t, y_{t-1}, x_t) = \sum_{y_1, \cdots, y_T} p(y_1, \cdots, y_T|x) \sum_{t=1}^T f_k(y_t, y_{t-1}, x_t)& eeimg=&1&&&br&&p&对于每一个&img src=&https://www.zhihu.com/equation?tex=p%28y_1%2C+%5Ccdots%2C+y_T%7Cx%29& alt=&p(y_1, \cdots, y_T|x)& eeimg=&1&&，会乘以，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Csum_%7Bt%3D1%7D%5ET+f_k%28y_t%2C+y_%7Bt-1%7D%2C+x_t%29+%3D+f_k%28y_1%2C+y_0%2C+x_1%29+%2B+%5Ccdots+%2B+f_k%28y_T%2C+y_%7BT-1%7D%2C+x_T%29& alt=&\sum_{t=1}^T f_k(y_t, y_{t-1}, x_t) = f_k(y_1, y_0, x_1) + \cdots + f_k(y_T, y_{T-1}, x_T)& eeimg=&1&&&p&那么反过来，对于每一个&img src=&https://www.zhihu.com/equation?tex=f_k%28y_t+%3D+s%2C+y_%7Bt-1%7D+%3D+s%27%2C+x_t%29& alt=&f_k(y_t = s, y_{t-1} = s', x_t)& eeimg=&1&&，会乘以，&/p&&img src=&https://www.zhihu.com/equation?tex=%5Csum_%7By_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%2C+y_%7Bt%2B1%7D%2C+y_T%7D+p%28y_1%2C+%5Ccdots%2C+y_%7Bt-2%7D%2C+y_t%3Ds%2C+y_%7Bt-1%7D%3Ds%27%2C+y_%7Bt%2B1%7D%2C+y_T+%7C+x%29+%3D+p%28s%2C+s%27+%7C+x%29& alt=&\sum_{y_1, \cdots, y_{t-2}, y_{t+1}, y_T} p(y_1, \cdots, y_{t-2}, y_t=s, y_{t-1}=s', y_{t+1}, y_T | x) = p(s, s' | x)& eeimg=&1&}
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