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 $O\left(N\right)$ 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.

• Received 14 June 2017
• Revised 18 May 2018

DOI:https://doi.org/10.1103/PhysRevE.98.012136