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The recent introduction of Graph Neural Networks (GNNs) and their growing popularity in the past few years has enabled the application of deep learning algorithms to non-Euclidean, graph-structured data. GNNs have achieved state-of-the-art results across an impressive array of graph-based machine learning problems. Nevertheless, despite their rapid pace of development, much of the work on GNNs has focused on graph classification and embedding techniques, largely ignoring regression tasks over graph data. In this paper, we develop a Graph Mixture Density Network (GraphMDN), which combines graph neural networks with mixture density network (MDN) outputs. By combining these techniques, GraphMDNs have the advantage of naturally being able to incorporate graph structured information into a neural architecture, as well as the ability to model multi-modal regression targets. As such, GraphMDNs are designed to excel on regression tasks wherein the data are graph structured, and target statistics are better represented by mixtures of densities rather than singular values (so-called ``inverse problems"). To demonstrate this, we extend an existing GNN architecture known as Semantic GCN (SemGCN) to a GraphMDN structure, and show results from the Human3.6M pose estimation task. The extended model consistently outperforms both GCN and MDN architectures on their own, with a comparable number of parameters.