Efficient total probability prediction via convex optimization and optimal transport

In this paper, we consider state space modeling for sequential continuous estimation. We consider the one-step prediction update, which transforms our previous belief state (posterior distribution of the previous state) to new belief state (posterior distribution of the current state). We demonstrate a recursive algorithm for updating the latent state at every time by avoiding intractable integral or Gaussian approximation. The construction of the desired map is pursued through the optimal transportation theory, and we demonstrate that for the large class of log-concave state transition functions, the one-step prediction problem for continuous hidden variable is solvable through convex optimization.