  In this article, we develop a Bayesian approach to estimate parameters from
time traces that originate from an overdamped Brownian particle in a harmonic
potential, or Ornstein-Uhlenbeck process (OU). We show that least-square
fitting the autocorrelation function, which is often the standard way of
analyzing such data, is significantly underestimating the confidence intervals
of the fitted parameters. Here, we develop a rigorous maximum likelihood theory
that properly captures the underlying statistics. From the analytic solution,
we found that there exists an optimal measurement spacing ($\Delta t = 0.7968
\tau$) that maximizes the statistical accuracy of the estimate for the
decay-time $\tau$ of the process for a fixed number of samples $N$, which plays
a similar role than the Nyquist-Shannon theorem for the OU-process. In summary,
our results have strong implications for parameter estimation for processes
that result in a single exponential decay in the autocorrelation function. Our
analysis can directly be applied to single-component dynamic light scattering
experiments or optical trap calibration experiments.
