Stochastic differential equations (SDEs) arise naturally in many scientific and industrial disciplines, e.g., due to the interaction of a system with some environment. Inverse problems such as the inference of the model parameters of an SDE are of paramount importance. I describe the Backward Filtering Forward Guiding paradigm, capable of solving this task based on trajectories observed according to some observation scheme, suitable for neural SDEs, and scalable to high-dimensional systems.
In this virtual poster, I present the MitosisStochasticDiffEq.jl package (see https://github.com/mschauer/MitosisStochasticDiffEq.jl) for filtering and guiding of SDEs. Possible fields of application for these tools range from the control of the time evolution of the states of an SDE to the inference of the model parameters.
MitosisStochasticDiffEq.jl implements the automatic Backward Filtering Forward Guiding (BFFG) paradigm [1,2] for programmable inference on latent states and model parameters in models described by stochastic differential equations. We start from a generative model that describes how the stochastic process evolves forward and how (potentially noisy and indirect) observations are generated from the process. In Backward Filtering Forward Guiding the information provided by those observations is backpropagated through the model to transform the generative (forward) model into a pre-conditional model guided by the data. This pre-conditioned SDE model approximates the actual (intractable) conditional model with known likelihood-ratio between the two.
Having this guided generative model at hand allows one to sample efficiently latent states and parameters conditional on observations. Since the BFFG paradigm can be formulated in terms of a set of transformation rules, it can be straightforwardly incorporated into a probabilistic programming context. The MitosisStochasticDiffEq package is based on the SciML ecosystem and thus it automatically integrates high-performing and specialized SDE solvers, sensitivity analysis tools, as well as distributed, multithreaded, and GPU parallel ensemble simulations.
I demonstrate the workflow on the challenging study of parameter inference on a stochastic trait evolution model on a phylogenetic tree, which may be used to model the evolutionary relationships between biological species.
Frank van der Meulen, Delft Institute of Applied Mathematics (DIAM), Delft University of Technology
Moritz Schauer, Chalmers University of Technology and University of Gothenburg
 Marcin Mider, Moritz Schauer, Frank van der Meulen (2020): Continuous-discrete smoothing of diffusions. arxiv:1712.03807.
 Frank van der Meulen, Moritz Schauer (2020): Automatic Backward Filtering Forward Guiding for Markov processes and graphical models. arXiv:2010.03509.