# Dirichlet Process Prior for Student's t Graph Variational Autoencoders
## 4 Method  
### DP-St Gvae  
In this section, we will introduce the heavy-tailed priors based on Bayesian nonparametric methods in detail, which is DP construction for St distribution priors. Utilizing the  
Figure 1: Several probability distribution function curves of common probability distribution. They share the same mean and scale parameters, and there are two St distribution with dof parameter 5 and 50 respectively. Figure 2: Several cumulative distribution function curves of common probability distribution. They share the same mean and scale parameters, and there are two St distribution with dof parameter 5 and 50, respectively. augment of the prior distribution complexity, latent variables can be stimulated to learn more features, and more expressive latent representation can be obtained to reduce the reconstruction error. Considering the St distribution as the prior distribution of latent variables, the bilateral tail is thicker, which can cover more latent features with higher probability, compared with the most commonly used Gaussian distribution. As the features increase, we have to allow higher dimensional latent representation to pertain to the nodes. Therefore, we hope to use DP to generate the mean parameter \(\mu\) and scale parameter \(\sigma\) of the St distribution. This method allows us to infer which factors have significant effects, and the dimensions of latent variables. For the last parameter, dof parameter \(v>0\) controls the tail shape of the pdf curve of St distribution, with the increase of dof \(v\), the curve of the latent variables changes gradually from the heavy-tailed Cauchy distribution to the Gaussian distribution, which makes us find the curve that fits the observed data most in the procedure of learning the dof parameter. In Figure 3, we show the framework of the DP-St GVAE model in detail. First, we assume that the observed data, adjacency matrix \(\mathbf{A}\in\mathbb{R}^{\mathbf{N}\times\mathbf{N}}\) and attribute matrix \(\mathbf{X}\in\mathbb{R}^{N\times\mathbf{C}}\), are used as the inputs of encoder to generate the latent variables \(\mathbf{Z}\in\mathbb{R}^{N\times K}\), where the \(N\) observations come from a graph undirected attributed graph \(\mathbf{G}(\mathbf{V},\mathbf{E},\mathbf{X})\). \(\mathbf{V}\) is the node set \(\{v_{1},v_{2},...,v_{N}\}\), and \(\mathbf{E}\) is a unweighted edge set, \(e_{ij}=1\) means there is a undirected link between node \(v_{i}\) and \(v_{j}\). We put a St distribution over the latent space, and place the outputs of GVAE's encoder over the mean, scale, and dof parameters of St distribution:  
\[q_{\phi}(\mathbf{Z}\mid\mathbf{A},\mathbf{X})=St(\mathbf{f}_{1,\phi}(\mathbf{A },\mathbf{X}),\mathbf{f}_{2,\phi}(\mathbf{A},\mathbf{X}),\mathbf{f}_{3,\phi}( \mathbf{A},\mathbf{X})), \tag{4}\]  
where \(\mathbf{f}_{1,\phi},\mathbf{f}_{2,\phi},\mathbf{f}_{3,\phi}\) denote the neural networks with parameter \(\phi\). Second, we replace the standard Gaussian prior with a DP-St distribution. The DP is composed of a discrete base distribution \(G_{0}\) and a concentration parameter. To explicitly illustrate the base distribution \(G_{0}\), the stick-breaking process is a good choice. It generates unbounded sticks \(b_{nk}\in[0,1]\) independently from a beta distribution:  
\[p(b_{nk}\mid\beta)=Beta(b_{nk}\mid 1,\beta). \tag{5}\]  
From the procedure above, we obtain an infinite dimensional vector with components \(\pi_{k}(\mathbf{b})=b_{k}\prod\limits_{l=1}^{k-1}(1-b_{l-1})\), which is used as the initialization of mean and scale parameters in St prior of each node, and the dof parameter is initialed from a Uniform distribution:  
Figure 3: The architecture of the DP-St GVAE. The encoder is a graph convolutional neural network that integrates the network structure \(A\) with node feature \(X\) and, by extension, generates the node representation \(Z\). The decoder reconstructs a graph from the inner product of the node representation \(Z\).