We consider the inverse problem of reconstructing the posterior measure over the trajec- tories of a diffusion process from discrete time observations and continuous time constraints. We cast the problem in a Bayesian framework and derive approximations to the posterior distributions of single time marginals using variational approximate inference. We then show how the approximation can be extended to a wide class of discrete-state Markov jump pro- cesses by making use of the chemical Langevin equation. Our empirical results show that the proposed method is computationally efficient and provides good approximations for these classes of inverse problems.
@cite_2 and @cite_4 propose a continuous time extension of the popular unscented transformation in to obtain Gaussian state space approximations in SDE models with time-only dependent diffusion terms and both non-linear non-Gaussian discrete and continuous time observation. In the authors compare these approaches to the variational method in which they then use to improve on their smoothing estimates. In a recent work @cite_12 present a mean-field variational approximation where they approximate the posterior process with a set of independent univariate Gaussian processes (factorised approximation). The considered model has polynomial drift terms and state-independent diffusion terms and the observations are at discrete time-points. Due to a clever parameterisations (piecewise polynomials) of the mean and the variance function of the variational approximation the dimensionality of the state can scale to thousands.
@cite_4 @cite_12 @cite_2
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This paper is concerned with Bayesian optimal filtering and smoothing of non-linear continuous-discrete state space models, where the state dynamics are modeled with non-linear Ito-type stochastic differential equations, and measurements are obtained at discrete time instants from a non-linear measurement model with Gaussian noise. We first show how the recently developed sigma-point approximations as well as the multi-dimensional Gauss-Hermite quadrature and cubature approximations can be applied to classical continuous-discrete Gaussian filtering. We then derive two types of new Gaussian approximation based smoothers for continuous-discrete models and apply the numerical methods to the smoothers. We also show how the latter smoother can be efficiently implemented by including one additional cross-covariance differential equation to the filter prediction step. The performance of the methods is tested in a simulated application. This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies. This article considers the application of the unscented transformation to approximate fixed-interval optimal smoothing of continuous-time non-linear stochastic dynamic systems. The proposed methodology can be applied to systems, where the dynamics can be modeled with non-linear stochastic differential equations and the noise corrupted measurements are obtained continuously or at discrete times. The smoothing algorithm is based on computing the continuous-time limit of the recently proposed unscented Rauch-Tung-Striebel smoother, which is an approximate optimal smoothing algorithm for discrete-time stochastic dynamic systems.