将一下三块内容结合起来看比较好：
条件随机场的精彩总结性论述，更多详细信息可以参阅连接文章：http://homepages.inf.ed.ac.uk/csutton/publications/crftut-fnt.pdf
Much work in learning with graphical models, especially in statistical natural-language processing, has focused on generative models that explicitly attempt to model a joint probability distribution p(y,x) over inputs and outputs. Although this approach has advantages, it also has important limitations. Not only can the dimensionality of x be very large, but the features may have complex dependencies, so constructing a probability distribution over them is difficult. Modeling the dependencies among inputs can lead to intractable models, but ignoring them can lead to reduced performance.
A solution to this problem is a discriminative approach, similar to that taken in classifiers such as logistic regression. Here we model the conditional distribution p(y|x) directly, which is all that is needed for classification. This is the approach taken by conditional random fields (CRFs). CRFs are essentially a way of combining the advantages of discriminative classification and graphical modeling, combining the ability to compactly model multivariate outputs y with the ability to leverage a large number of input features x for prediction. The advantage to a conditional model is that dependencies that involve only variables in x play no role in the conditional model, so that an accurate conditional
李航-统计学习-线性链条件随机场
李航的书中对线性链条件随机场定义为如下所示，一开始难以理解，后来终于搞懂，式子中，索引i代表输出序列的位置索引，k代表两个输出序列之间的转移特征索引，即这种转移可能有多个特征，l代表每个输出序列位置对应的特征索引，即序列中每个节点也可能有多个特征。t和s只取1或0表征的是此特征是否存在，前面的lamda和mu表征的是每个特征的权重，这就和下面的神经网络联系在了一起。k个特征表征的是输出维度，每个维度的标量数值通过此处的权重来表征。
youtue -- https://www.youtube.com/watch?v=PGBlyKtfB74&list=PL6Xpj9I5qXYEcOhn7TqghAJ6NAPrNmUBH&index=19
下图为上面视频中的截图，利用神经网络来获取节点特征向量，利用矩阵V表征状态间的转移特征，李航的书中也有类似的矩阵表示，但是书中的矩阵与此处的矩阵有所不同，具体可以参考原书。
最后CRF是判别模型，概率计算采用前向后向算法，训练采用极大似然估计训练，预测采用维特比算法。目前对书中算法的细节没有仔细研读，等有时间再细看。
