  Estimates for asteroid masses are based on their gravitational perturbations
on the orbits of other objects such as Mars, spacecraft, or other asteroids
and/or their satellites. In the case of asteroid-asteroid perturbations, this
leads to an inverse problem in at least 13 dimensions where the aim is to
derive the mass of the perturbing asteroid(s) and six orbital elements for both
the perturbing asteroid(s) and the test asteroid(s) based on astrometric
observations. We have developed and implemented three different mass estimation
algorithms utilizing asteroid-asteroid perturbations: the very rough 'marching'
approximation, in which the asteroids' orbital elements are not fitted, thereby
reducing the problem to a one-dimensional estimation of the mass, an
implementation of the Nelder-Mead simplex method, and most significantly, a
Markov-chain Monte Carlo (MCMC) approach. We describe each of these algorithms
with particular focus on the MCMC algorithm, and present example results using
both synthetic and real data. Our results agree with the published mass
estimates, but suggest that the published uncertainties may be misleading as a
consequence of using linearized mass-estimation methods. Finally, we discuss
remaining challenges with the algorithms as well as future plans.
