Mapping systematic errors in helium abundance determinations using
  Markov Chain Monte Carlo
  Monte Carlo techniques have been used to evaluate the statistical and
systematic uncertainties in the helium abundances derived from extragalactic
H~II regions. The helium abundance is sensitive to several physical parameters
associated with the H~II region. In this work, we introduce Markov Chain Monte
Carlo (MCMC) methods to efficiently explore the parameter space and determine
the helium abundance, the physical parameters, and the uncertainties derived
from observations of metal poor nebulae. Experiments with synthetic data show
that the MCMC method is superior to previous implementations (based on flux
perturbation) in that it is not affected by biases due to non-physical
parameter space. The MCMC analysis allows a detailed exploration of
degeneracies, and, in particular, a false minimum that occurs at large values
of optical depth in the He~I emission lines. We demonstrate that introducing
the electron temperature derived from the [O~III] emission lines as a prior, in
a very conservative manner, produces negligible bias and effectively eliminates
the false minima occurring at large optical depth. We perform a frequentist
analysis on data from several "high quality" systems. Likelihood plots
illustrate degeneracies, asymmetries, and limits of the determination. In
agreement with previous work, we find relatively large systematic errors,
limiting the precision of the primordial helium abundance for currently
available spectra.
