We determine asteroid orbital elements from optical astrometric observations using Bayesian a priori and a posteriori probability densities. Assuming Gaussian noise statistics and linearizing the orbit determination problem (linear approximation), a Gaussian a posteriori probability density results for the orbital elements. We develop a Monte Carlo method for the rigorous a posteriori probability density, study non-Gaussian noise and practical orbit determination methods, and describe the difficulties arising from correlated noise and systematic errors. We predict asteroid orbital positions and velocities using the linear approximation and the Monte Carlo approach and give analytically several orbital variances. The variance in the semimajor axis is particularly important: through the mean motion uncertainty, it determines the long-term rate of orbital degradation. In addition, we develop metrics to characterize the overall orbital quality and describe optimization of follow-up observations. Examples are the Jupiter Trojan asteroid (5209) 1989 CW 1, the main-belt asteroid (263) Dresda, the Mars Trojan asteroid (5261) Eureka, and the Apollo asteroid (4179) Toutatis.