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Most applications of variational inference employ\nsimple families of posterior approximations in order to allow for efficient\ninference, focusing on mean-field or other simple structured approximations.\nThis restriction has a significant impact on the quality of inferences made\nusing variational methods. We introduce a new approach for specifying flexible,\narbitrarily complex and scalable approximate posterior distributions. Our\napproximations are distributions constructed through a normalizing flow,\nwhereby a simple initial density is transformed into a more complex one by\napplying a sequence of invertible transformations until a desired level of\ncomplexity is attained. We use this view of normalizing flows to develop\ncategories of finite and infinitesimal flows and provide a unified view of\napproaches for constructing rich posterior approximations. 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