# Approximate Bayes learning of stochastic differential equations

## TL;DR

This paper presents a nonparametric Gaussian process-based method for estimating drift and diffusion functions in stochastic differential equations, utilizing an approximate EM algorithm for sparse data and latent dynamics.

## Contribution

It introduces a novel Gaussian process framework combined with an approximate EM algorithm for efficient estimation of SDE parameters from dense and sparse observations.

## Key findings

- Effective estimation of SDE functions from dense data
- Handling of sparse observations via approximate EM
- Use of Gaussian processes for flexible modeling

## Abstract

We introduce a nonparametric approach for estimating drift and diffusion functions in systems of stochastic differential equations from observations of the state vector. Gaussian processes are used as flexible models for these functions and estimates are calculated directly from dense data sets using Gaussian process regression. We also develop an approximate expectation maximization algorithm to deal with the unobserved, latent dynamics between sparse observations. The posterior over states is approximated by a piecewise linearized process of the Ornstein-Uhlenbeck type and the maximum a posteriori estimation of the drift is facilitated by a sparse Gaussian process approximation.