Title: Greedy Bayesian Posterior Approximation with Deep Ensembles. Abstract: Ensembles of independently trained neural networks are a state-of-the-art approach to estimate predictive uncertainty in Deep Learning, and can be interpreted as an approximation of the posterior distribution via a mixture of delta functions. The training of ensembles relies on non-convexity of the loss landscape and random initialization of their individual members, making the resulting posterior approximation uncontrolled. This paper proposes a novel and principled method to tackle this limitation, minimizing an $f$-divergence between the true posterior and a kernel density estimator in a function space. We analyze this objective from a combinatorial point of view, and show that it is submodular with respect to mixture components for any $f$. Subsequently, we consider the problem of greedy ensemble construction, and from the marginal gain of the total objective, we derive a novel diversity term for ensemble methods. The performance of our approach is demonstrated on computer vision out-of-distribution detection benchmarks in a range of architectures trained on multiple datasets. The source code of our method is made publicly available.