A path-integral approach to Bayesian inference for inverse problems using the semiclassical approximation

We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (infinite dimensional) coefficient functions from ordinary or partial differential equations (ODE, PDE), a problem which is typically ill-posed. Regularization of these problems using $L^2$ function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem -- namely whether the subjective choice of regularization is compatible with prior knowledge. Using path-integral formalism, Bayesian inference can be explored through various perturbative techniques, such as the semiclassical approximation, which we use in this manuscript. Perturbative path-integral approaches, while offering alternatives to computational approaches like Markov-Chain-Monte-Carlo (MCMC), also provide natural starting points for MCMC methods that can be used to refine approximations. In this manuscript, we illustrate a path-integral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theories involving the Poisson equation.Comment: Fixed some spelling errors and the author affiliations. This is the version accepted for publication by J Stat Phy

Parametric and non-parametric gradient matching for network inference:a comparison

Abstract Background Reverse engineering of gene regulatory networks from time series gene-expression data is a challenging problem, not only because of the vast sets of candidate interactions but also due to the stochastic nature of gene expression. We limit our analysis to nonlinear differential equation based inference methods. In order to avoid the computational cost of large-scale simulations, a two-step Gaussian process interpolation based gradient matching approach has been proposed to solve differential equations approximately. Results We apply a gradient matching inference approach to a large number of candidate models, including parametric differential equations or their corresponding non-parametric representations, we evaluate the network inference performance under various settings for different inference objectives. We use model averaging, based on the Bayesian Information Criterion (BIC), to combine the different inferences. The performance of different inference approaches is evaluated using area under the precision-recall curves. Conclusions We found that parametric methods can provide comparable, and often improved inference compared to non-parametric methods; the latter, however, require no kinetic information and are computationally more efficient

Bayesian Neural Controlled Differential Equations for Treatment Effect Estimation

Treatment effect estimation in continuous time is crucial for personalized medicine. However, existing methods for this task are limited to point estimates of the potential outcomes, whereas uncertainty estimates have been ignored. Needless to say, uncertainty quantification is crucial for reliable decision-making in medical applications. To fill this gap, we propose a novel Bayesian neural controlled differential equation (BNCDE) for treatment effect estimation in continuous time. In our BNCDE, the time dimension is modeled through a coupled system of neural controlled differential equations and neural stochastic differential equations, where the neural stochastic differential equations allow for tractable variational Bayesian inference. Thereby, for an assigned sequence of treatments, our BNCDE provides meaningful posterior predictive distributions of the potential outcomes. To the best of our knowledge, ours is the first tailored neural method to provide uncertainty estimates of treatment effects in continuous time. As such, our method is of direct practical value for promoting reliable decision-making in medicine

A Probabilistic State Space Model for Joint Inference from Differential Equations and Data

Mechanistic models with differential equations are a key component of scientific applications of machine learning. Inference in such models is usually computationally demanding, because it involves repeatedly solving the differential equation. The main problem here is that the numerical solver is hard to combine with standard inference techniques. Recent work in probabilistic numerics has developed a new class of solvers for ordinary differential equations (ODEs) that phrase the solution process directly in terms of Bayesian filtering. We here show that this allows such methods to be combined very directly, with conceptual and numerical ease, with latent force models in the ODE itself. It then becomes possible to perform approximate Bayesian inference on the latent force as well as the ODE solution in a single, linear complexity pass of an extended Kalman filter / smoother - that is, at the cost of computing a single ODE solution. We demonstrate the expressiveness and performance of the algorithm by training, among others, a non-parametric SIRD model on data from the COVID-19 outbreak.Comment: 12 pages (+ 5 pages appendix), 7 figures. In: Advances in Neural Information Processing Systems (NeurIPS 2021