Bayesian Revisit of the Relationship between the Total Field Strength and the Volume Density of Interstellar Clouds

The Zeeman effect has been the only method to directly probe the magnetic field strength in molecular clouds. The Bayesian analysis of Zeeman measurements carried out by Crutcher et al. is the only reference for cloud magnetic field strength. Here we extended their model and Bayesian analysis of the relation between field strength ( B ) and volume density ( n ) in the following three directions based on the recent observational and theoretical development. First, we take R , the observational uncertainty of n , as a parameter to be estimated from data. Second, the restriction of α , the index of the B – n relationship, is relieved from [0, 0.75] to [0, 1]. Third, we allow f , the minimum-to-maximum B ratio, to vary with n . Our results show that taking R as a parameter provides a better fitting to the B – n relationship and much more reliable estimates on R , f, and the changing point of α . Arguably our most important finding is that α cannot be reliably estimated by any of the models studied here, either from us or Crutcher et al., if R  > 2, which is indeed the case from our estimate. This is the so-called errors-in-variables bias, a well known problem for statisticians.