# Fast Bayesian inference of the multivariate Ornstein-Uhlenbeck process

## TL;DR

This paper introduces an efficient Bayesian approach for estimating parameters of the multivariate Ornstein-Uhlenbeck process from discrete data, enabling detailed analysis of systems like colloidal particles in optical traps.

## Contribution

It presents an $O(N)$ Bayesian method using exact likelihoods for parameter estimation and model comparison in multivariate Ornstein-Uhlenbeck processes, including applications to physical systems.

## Key findings

- Efficient $O(N)$ Bayesian estimation of drift and diffusion matrices.
- Explicit maximum a posteriori estimates with standard errors.
- Bayesian model comparison between Kramers and Smoluchowski limits.

## Abstract

The multivariate Ornstein-Uhlenbeck process is used in many branches of science and engineering to describe the regression of a system to its stationary mean. Here we present an $O(N)$ Bayesian method to estimate the drift and diffusion matrices of the process from $N$ discrete observations of a sample path. We use exact likelihoods, expressed in terms of four sufficient statistic matrices, to derive explicit maximum a posteriori parameter estimates and their standard errors. We apply the method to the Brownian harmonic oscillator, a bivariate Ornstein-Uhlenbeck process, to jointly estimate its mass, damping, and stiffness and to provide Bayesian estimates of the correlation functions and power spectral densities. We present a Bayesian model comparison procedure, embodying Ockham's razor, to guide a data-driven choice between the Kramers and Smoluchowski limits of the oscillator. These provide novel methods of analyzing the inertial motion of colloidal particles in optical traps.