Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations

Abstract

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.

Variational Inference for Stochastic Differential Equations

“…Note that in the context of optimal control, − ln φ(x, t) is the value function (or optimal cost to go) obeying the Hamilton-Jacobi-Bellmann equation, see ref. [19], which is equivalent to Equation (26). To illustrate this result for a simple case, let us assume that we start the process at y 0 .…”

“…One can show that this results in the end condition φ(x, T ) = δ(x − y 1 ). For t < T , we then have U = 0 and the partial differential equation (PDE) (26) reduces to the Kolmogorov backward equation. [1] This has the solution where p T −t (x |x) is the transition density of going from x to x in the time T − t for the process (1).…”

“…A great advantage of the variational method comes from the fact that one has the freedom to tune the complexity of the approximation to the complexity of the problem and the available computational resources. For example, by applying an additional mean field approximation (for diagonal diffusion matrices D) for the components of the vector X t , it is possible to perform inference on the unobserved state X t for large systems of coupled nonlinear SDE, [26] where a fraction of the dimensions of X t are not observed. The mean field approximation leads to effective 1D problems, where A(t) and S(t) are diagonal matrices.…”

“…The difference to Equation (27) of ref. [26], is due to a typo in this equation: The term s 2 i (t) in the denominator should be replaced by s i (t). From the minimization of the functional (32) with fixed boundary conditions m(0) = y 0 , m(T ) = y 1 together with S(0) = S(T ) = 0, we can recover the mean and variance for the OU-bridge process with drift (30).…”

Learning Stochastic Differential Equations With Gaussian Processes Without Gradient Matching

“…There has also been interest on estimating non-parametric SDE drift and diffusion functions from data using the general Bayesian formalism [3,4]. With linear drift approximations the state distribution turns out to be a Gaussian, which can be solved with variational smoothing algorithm [5] or by variational mean field approximation [6]. Non-linear drifts and diffusions are predominantly modelled with Gaussian processes [3,7,4], which are a family of Bayesian kernel methods [8].…”