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The best way to combine the results of deep learning with standard 3D reconstruction pipelines remains an open problem. While systems that pass the output of traditional multi-view stereo approaches to a network for regularisation or refinement currently seem to get the best results, it may be preferable to treat deep neural networks as separate components whose results can be probabilistically fused into geometry-based systems. Unfortunately, the error models required to do this type of fusion are not well understood, with many different approaches being put forward. Recently, a few systems have achieved good results by having their networks predict probability distributions rather than single values. We propose using this approach to fuse a learned single-view depth prior into a standard 3D reconstruction system. Our system is capable of incrementally producing dense depth maps for a set of keyframes. We train a deep neural network to predict discrete, nonparametric probability distributions for the depth of each pixel from a single image. We then fuse this "probability volume" with another probability volume based on the photometric consistency between subsequent frames and the keyframe image. We argue that combining the probability volumes from these two sources will result in a volume that is better conditioned. To extract depth maps from the volume, we minimise a cost function that includes a regularisation term based on network predicted surface normals and occlusion boundaries. Through a series of experiments, we demonstrate that each of these components improves the overall performance of the system.