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.
The LBT $Y_{\rm p}$ Project I: An Improved Determination of the Primordial Helium Abundance -- Project Description, Sample Selection, Observations, and Methodology
University of Minnesota（明尼苏达大学）；The Ohio State University（俄亥俄州立大学）；Gonzaga University（贡萨加大学）；Northwestern University（西北大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
Fifty years of primordial helium abundances: A statistical reanalysis
Indian Institute of Science Education and Research (IISER) Mohali（印度科学教育研究所）；Macquarie University（麦考瑞大学）；International Space Science Institute–Beijing（国际空间科学研究所北京分部）；Università di Roma Tor Vergata（罗马托尔维加塔大学）；INAF, Rome Astronomical Observatory（意大利国家天体物理研究所罗马天文台）
The LBT Y$_\mathrm{p}$ Project IV: A New Value of the Primordial Helium Abundance
Gonzaga University（冈萨加大学）；University of Minnesota（明尼苏达大学）；The Ohio State University（俄亥俄州立大学）；Northwestern University（西北大学）；The University of Texas at Austin（得克萨斯大学奥斯汀分校）；Indiana University（印第安纳大学）
EMPRESS. XV. A New Determination of the Primordial Helium Abundance Suggesting a Moderately Low $Y_\mathrm{P}$ Value
Institute for Cosmic Ray Research, The University of Tokyo（东京大学宇宙射线研究所）；Department of Physics, Graduate School of Science, The University of Tokyo（东京大学理学研究科物理学系）；National Astronomical Observatory of Japan, National Institutes of Natural Sciences（日本国立天文台，国立自然科学研究院）；Department of Astronomical Science, SOKENDAI (The Graduate University for Advanced Studies)（SOKENDAI（ advanced studies研究生院）天体科学系）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo（东京大学 Kavli宇宙物理数学研究所 (WPI)）；Department of Physics, Tohoku University（东北大学物理学系）
Degeneracies in Sky Localisation Determination from a Spinning Coalescing Binary through Gravitational Wave Observations: a Markov-Chain Monte-Carlo Analysis for two Detectors
