## 3 CRF Applications and Challenges  
In a typical CRF graphical model, the class labels at the graph node given the observed data are assumed to obey a predefined dependence structure. A Markov dependence structure (see below) is one for which the class label at node \(i\) depends on the class labels in the neighborhood of node \(i\). Additionally, the class label at the node \(i\) depends on \"compatibility\" of feature vectors corresponding to the node \(i\) and its neighbors in the neighborhood. Natural language processing extracts syntax from text, often to aid search and to find related documents. The response variable, \(Y\), is a part-of-speech tag and the predictors, \(X\), include information about the word at a given position, such as its identity and memberships in domain-specific lexicons. One common goal in computer vision is to classify regions in an image, and as Figure 1 suggests, the \(Y\) labels near a given pixel, \(i\), influence our beliefs about \(Y_{i}\), as is captured in the neighborhood structure in Equation (1). A particular example of Equation (1) for 2-dimensional image data is  
\[P\left(Y=\left|\left|X;\theta\right)=\dfrac{1}{Z}\exp\left(\sum_{i\in S}v^{T} \psi_{i}(y_{i})x_{i}+\sum_{i\in S}\sum_{j\in N_{i}}v^{T}\psi_{i}(y)x_{i}x_{j}\right.\right) \tag{4}\]  
In Equation (4), \(\psi_{i}(y_{i})=\left(1,y_{i}\right)\), a two-component vector and \(\psi_{i}(y)=\left(1,\left|\left.y_{i}-y_{j}\right.\right|\right)\), also a 2-component vector, and the \(x_{i}\) are all two-dimensional. The model parameter vector \(\theta=\left(w,v\right)\).The application for Equation (4) in [50,51] was image segmentation, which partitions or clusters an image into homogeneous pixels or pixel patches such as \"land\" or \"water\". One well-known MRF that has many applications, including image denoising, is the Ising model [58,61,62]. In the image denoising context, the interaction portion of Equation (2), \(\sum_{i\in S}\sum_{j\in N_{i}}\psi_{i}\left(y_{i},y_{j},X,\theta\right)\) can be regarded as being a prior probability for the true \(Y\) values. Such a prior probability can enforce a tendency for neighboring \(Y\) values to be similar or dissimilar, depending on the application. The term \(\sum_{i\in S}\varphi_{i}\left(y_{i},X,\theta\right)\) in Equation (2) can be regarded as the likelihood, choosing, for example, \(\psi_{i}=\exp\left\{-\theta(y_{i}-x_{i})\right\}^{2}\), where \(y_{i}=x_{i}+e_{i}\) with \(x_{i}\) the true image value, \(e_{i}\), the measurement errors, and neighboring \(y_{i}\), predicted to be similar or dissimilar by choice of \(\psi_{i}\left(y_{i},y_{j},X,\theta\right)\). Despite the computational issues described in Section 2, CRFs have proven to perform well for many applications. For example, [8] demonstrates good CRF performance compared to other pattern recognition methods such as a Gaussian mixture model or support vector machines [63] in labeling pixel patches as natural or manmade. Furthermore, a few papers, such as [50,51], show how well the various approximation schemes work in limited settings with experimental data. Aside from the need for better and faster approximate inference methods for CRFs, another open issue is choosing a model structure by using prior knowledge about the domain or by using trial and error with different structures. In the context of hidden Gibbs random fields, [64] recently developed a method to choose the dependency structure in Gibbs fields that relies on approximate Bayesian computation. Another way to assess candidate model structure (such as function forms or number of connections used in Equation (1)) is empirical classification accuracy. Choosing the model structure involves choosing the number of layers in the model, the structure of each layer, as well as the number of states of the latent variables in the hidden layers. Naturally, the probabilistic inference is an unavoidable component in estimating changes the model structure.