In the second paper of this series we extend our Bayesian reanalysis of the evidence for a cosmic variation of the fine structure constant to the semi-parametric modelling regime. By adopting a mixture of Dirichlet processes prior for the unexplained
errors in each instrumental subgroup of the benchmark quasar dataset we go some way towards freeing our model selection procedure from the apparent subjectivity of a fixed distributional form. Despite the infinite-dimensional domain of the error hierarchy so constructed we are able to demonstrate a recursive scheme for marginal likelihood estimation with prior-sensitivity analysis directly analogous to that presented in Paper I, thereby allowing the robustness of our posterior Bayes factors to hyper-parameter choice and model specification to be readily verified. In the course of this work we elucidate various similarities between unexplained error problems in the seemingly disparate fields of astronomy and clinical meta-analysis, and we highlight a number of sophisticated techniques for handling such problems made available by past research in the latter. It is our hope that the novel approach to semi-parametric model selection demonstrated herein may serve as a useful reference for others exploring this potentially difficult class of error model.
We review the evidence behind recent claims of spatial variation in the fine structure constant deriving from observations of ionic absorption lines in the light from distant quasars. To this end we expand upon previous non-Bayesian analyses limited
by the assumptions of an unbiased and strictly Normal distribution for the unexplained errors of the benchmark quasar dataset. Through the technique of reverse logistic regression we estimate and compare marginal likelihoods for three competing hypotheses---(i) the null hypothesis (no cosmic variation), (ii) the monopole hypothesis (a constant Earth-to-quasar offset), and (iii) the monopole+dipole hypothesis (a cosmic variation manifest to the Earth-bound observer as a North-South divergence)---under a variety of candidate parametric forms for the unexplained error term. Our analysis reveals weak support for a skeptical interpretation in which the apparent dipole effect is driven solely by systematic errors of opposing sign inherent in measurements from the two telescopes employed to obtain these observations. Throughout we seek to exemplify a best practice approach to Bayesian model selection with prior-sensitivity analysis; in a companion paper we extend this methodology to a semi-parametric framework using the infinite-dimensional Dirichlet process.
