Title: Approximate Probabilistic Inference with Composed Flows. Abstract: We study the problem of probabilistic inference on the joint distribution defined by a normalizing flow model. Given a pre-trained flow model $p(\boldsymbol{x})$, we wish to estimate $p(\boldsymbol{x}_2 \mid \boldsymbol{x}_1)$ for some arbitrary partitioning of the variables $\boldsymbol{x} = (\boldsymbol{x}_1, \boldsymbol{x}_2)$. We first show that this task is computationally hard for a large class of flow models.  Motivated by this hardness result, we propose a framework for $\textit{approximate}$ probabilistic inference. Specifically, our method trains a new generative model with the property that its composition with the given model approximates the target conditional distribution. By parametrizing this new distribution as another flow model, we can efficiently train it using variational inference and also handle conditioning under arbitrary differentiable transformations. Since the resulting approximate posterior remains a flow, it offers exact likelihood evaluation, inversion, and efficient sampling. We provide an extensive empirical evidence showcasing the flexibility of our method on a variety of inference tasks with applications to inverse problems. We also experimentally demonstrate that our approach is comparable to simple MCMC baselines in terms of sample quality. Further, we explain the failure of naively applying variational inference and show that our method does not suffer from the same issue.