A short rant: Multiclass logistic regression and conditional random fields (CRF) are the same thing. This comes to a surprise to many people because CRFs tend to be surrounded by additional "stuff."
Understanding this very basic connection not only deepens our understanding, but also suggests a method for testing complex CRF code.
Multiclass logistic regression is simple. The goal is to predict the correct label \(y^*\) from handful of labels \(\mathcal{Y}\) given the observation \(x\) based on features \(\phi(x,y)\). Training this model typically requires computing the gradient:
where
At testtime, we often take the highestscoring label under the model.
A conditional random field is exactly multiclass logistic regression. The only difference is that the sums (\(Z(x)\) and \(\sum_{y \in \mathcal{Y}} p(yx) \phi(x,y)\)) and the argmax \(\widehat{y}(x)\) are inefficient to compute naively (i.e., by bruteforce enumeration). This point is often lost when people first learn about CRFs. Some people never make this connection.
Bruteforce enumeration is a very useful method for testing complex dynamic programming procedures for computing the sums and the argmax on relatively small examples. Don't just copy code for dynamic programming out of a paper! Test it!
Here's some stuff you'll see once we start talking about CRFs:

Inference algorithms (e.g., Viterbi decoding, forwardbackward, Junction tree)

Graphical models (factor graphs, Bayes nets, Markov random fields)

Model templates (i.e., repeated feature functions)
In the logistic regression case, we'd never use the term "inference" to describe the "sum" and "max" over a handful of categories. Once we move to a structured label space, this term gets throw around. (BTW, this isn't "statistical inference," just algorithms to compute sum and max over \(\mathcal{Y}\).)
Graphical models establish a notation and structural properties which allow efficient inference—things like cycles and treewidth.
Model templating is the only essential trick to move from logistic regression to a CRF. Templating "solves" the problem that not all training examples have the same "size"—the set of outputs \(\mathcal{Y}(x)\) now depends on \(x\). A model template specifies how to compute the features for an entire output, by looking at interactions between subsets of variables.
where \(\alpha\) is a labeled subset of variables often called a factor and \(\boldsymbol{y}_\alpha\) is the subvector containing values of variables \(\alpha\). Basically, the feature function \(\phi\) gets to look at some subset of the variables being predicted \(y\) and the entire input \(x\). The ability to look at more of \(y\) allows the model to make more coherent predictions.
Anywho, it's often useful to take a step back and think about what you are trying to compute instead of how you're computing it. In this post, this allowed us see the similarity between logistic regression and CRFs even though they seem quite different.