State estimation for nonlinear state space models is a challenging task. Existing assimilation methodologies predominantly assume Gaussian posteriors on physical space, where the actual posteriors become inevitably non-Gaussian. We propose Deep Bayesian Filtering (DBF) for data assimilation on nonlinear state space models (SSMs). DBF constructs new latent variables h_t on a new latent ("fancy") space and assimilates observations o_t. By (i) constraining the state transition on fancy space to be linear and (ii) learning a Gaussian inverse observation operator q(h_t|o_t), posteriors always remain Gaussian for DBF. Quite distinctively, the structured design of posteriors provides an analytic formula for the recursive computation of posteriors without accumulating Monte-Carlo sampling errors over time steps. DBF seeks the Gaussian inverse observation operators q(h_t|o_t) and other latent SSM parameters (e.g., dynamics matrix) by maximizing the evidence lower bound. Experiments show that DBF outperforms model-based approaches and latent assimilation methods in various tasks and conditions. I will also present possible extensions to this methodology to apply DBF to climate data assimilation.