Efficient approximate inference in large Hybrid Networks (graphical models with discrete and continuous variables) is one of the major unsolved problems in machine learning, and insight into good solutions would be beneficial in advancing the application of sophisticated machine learning to a wide range of real-world problems. Such research would benefit potentially applications in Speech Recognition, Visual Object Tracking and Machine Vision, Robotics, Music Scene Analysis, Analysis of complex Times series, understanding and modelling complex computer networks, Condition monitoring, and other complex phenomena. This theory challenge specifically addresses a central component area of PASCAL, namely Bayesian Statistics and statistical modelling, and is also related to the other central areas of Computational Learning, Statistical Physics and Optimisation techniques. One aim of this challenge is to bring together leading researchers in graphical models and related areas to develop and improve on existing methods for tackling the fundamental intractability in HNs. We do not believe that there will necessarily emerge a single best approach, although we would expect that successes in one application area should be transferable to related areas. Many leading machine learning researches are currently working on applications that involve HNs, and we invite participants to suggest their own applications. Ideally, this would be in the form of a dataset along the lines of PASCAL.