  Precise radial velocity measurements have led to the discovery of ~100
extrasolar planetary systems. We investigate the uncertainty in the orbital
solutions that have been fit to these observations. Understanding these
uncertainties will become more and more important as the discovery space for
extrasolar planets shifts to longer and longer periods. While detections of
short period planets can be rapidly refined, planets with long orbital periods
will require observations spanning decades to constrain the orbital parameters
precisely. Already in some cases, multiple distinct orbital solutions provide
similarly good fits, particularly in multiple planet systems. We present a
method for quantifying the uncertainties in orbital fits and addressing
specific questions directly from the observational data rather than relying on
best fit orbital solutions. This Markov chain Monte Carlo (MCMC) technique has
the advantage that it is well suited for the high dimensional parameter spaces
necessary for the multiple planet systems. We apply the MCMC technique to
several extrasolar planetary systems, assessing the uncertainties in orbital
elements for several systems. Our MCMC simulations demonstrate that for some
systems there are strong correlations between orbital parameters and/or
significant non-Gaussianities in parameter distributions, even though the
measurement errors are nearly Gaussian. Once these effects are considered the
actual uncertainties in orbital elements can be significantly larger or smaller
than the published uncertainties. We also present simple applications of our
methods such as predicting the times of possible transits for GJ 876.
