  Massive black hole binaries are the primary source of gravitational waves
(GW) for the future eLISA observatory. The detection and parameter estimation
of these sources to high redshift would provide invaluable information on the
formation mechanisms of seed black holes, and on the evolution of massive black
holes and their host galaxies through cosmic time. The Fisher information
matrix has been the standard tool for GW parameter estimation in the last two
decades. However, recent studies have questioned the validity of using the
Fisher matrix approach. For example, the Fisher matrix approach sometimes
predicts errors of $\geq100\%$ in the estimation of parameters such as the
luminosity distance and sky position. With advances in computing power,
Bayesian inference is beginning to replace the Fisher matrix approximation in
parameter estimation studies. In this work, we conduct a Bayesian inference
analysis for 120 sources situated at redshifts of between $0.1\leq z\leq 13.2$,
and compare the results with those from a Fisher matrix analysis. The Fisher
matrix results suggest that for this particular selection of sources, eLISA
would be unable to localize sources at redshifts of $z\lesssim6$. In contrast,
Bayesian inference provides finite error estimations for all sources in the
study, and shows that we can establish minimum closest distances for all
sources. The study further predicts that we should be capable with eLISA, out
to a redshift of at least $z\leq13$, of predicting a maximum error in the chirp
mass of $\lesssim 1\%$, the reduced mass of $\lesssim20\%$, the time to
coalescence of 2 hours, and to a redshift of $z\sim5$, the inclination of the
source with a maximum error of $\sim60$ degrees.
