Kalman Filter (KF) is an optimal linear state prediction algorithm, with applications in fields as diverse as engineering, economics, robotics, and space exploration. Here, we develop an extension of the KF, called a Pathspace Kalman Filter (PKF) which allows us to a) dynamically track the uncertainties associated with the underlying data and prior knowledge, and b) take as input an entire trajectory and an underlying mechanistic model, and using a Bayesian methodology quantify the different sources of uncertainty. An application of this algorithm is to automatically detect temporal windows where the internal mechanistic model deviates from the data in a time-dependent manner. First, we provide theorems characterizing the convergence of the PKF algorithm. Then, we numerically demonstrate that the PKF outperforms conventional KF methods on a synthetic dataset lowering the mean-squared-error by several orders of magnitude. Finally, we apply this method to biological time-course dataset involving over 1.8 million gene expression measurements.

Optimal control of stochastic nonlinear dynamical systems is a major challenge in the domain of robot learning. Given the intractability of the global control problem, state-of-the-art algorithms focus on approximate sequential optimization techniques, that heavily rely on heuristics for regularization in order to achieve stable convergence. By building upon the duality between inference and control, we develop the view of Optimal Control as Input Estimation, devising a probabilistic stochastic optimal control formulation that iteratively infers the optimal input distributions by minimizing an upper bound of the control cost. Inference is performed through Expectation Maximization and message passing on a probabilistic graphical model of the dynamical system, and time-varying linear Gaussian feedback controllers are extracted from the joint state-action distribution. This perspective incorporates uncertainty quantification, effective initialization through priors, and the principled regularization inherent to the Bayesian treatment. Moreover, it can be shown that for deterministic linearized systems, our framework derives the maximum entropy linear quadratic optimal control law. We provide a complete and detailed derivation of our probabilistic approach and highlight its advantages in comparison to other deterministic and probabilistic solvers.