# Laplace based approximate posterior inference for differential equation models

arXiv: https://arxiv.org/abs/1607.07203 · HTML: https://mlsanity.com/paper/1607.07203/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2016.

## Built on by (most cited first, 5 of 5)

- [Fenrir: Physics-Enhanced Regression for Initial Value Problems](https://mlsanity.com/paper/2202.01287/) (2022)
- [Robust and Scalable SDE Learning: A Functional Perspective](https://mlsanity.com/paper/2110.05167/) (2021)
- [Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations](https://mlsanity.com/paper/2306.05566/) (2023)
- [Diffusion Tempering Improves Parameter Estimation with Probabilistic Integrators for Ordinary Differential Equations](https://mlsanity.com/paper/2402.12231/) (2024)
- [Uncertainty Informed Optimal Resource Allocation with Gaussian Process based Bayesian Inference](https://mlsanity.com/paper/2307.00032/) (2023)

## Abstract (arXiv)

Ordinary differential equations are arguably the most popular and useful mathematical tool for describing physical and biological processes in the real world. Often, these physical and biological processes are observed with errors, in which case the most natural way to model such data is via regression where the mean function is defined by an ordinary differential equation believed to provide an understanding of the underlying process. These regression based dynamical models are called differential equation models. Parameter inference from differential equation models poses computational challenges mainly due to the fact that analytic solutions to most differential equations are not available. In this paper, we propose an approximation method for obtaining the posterior distribution of parameters in differential equation models. The approximation is done in two steps. In the first step, the solution of a differential equation is approximated by the general one-step method which is a class of numerical methods for ordinary differential equations including the Euler and the Runge-Kutta procedures; in the second step, nuisance parameters are marginalized using Laplace approximation. The proposed Laplace approximated posterior gives a computationally fast alternative to the full Bayesian computational scheme (such as Markov Chain Monte Carlo) and produces more accurate and stable estimators than the popular smoothing methods (called collocation methods) based on frequentist procedures. For a theoretical support of the proposed method, we prove that the Laplace approximated posterior converges to the actual posterior under certain conditions and analyze the relation between the order of numerical error and its Laplace approximation. The proposed method is tested on simulated data sets and compared with the other existing methods.