Multistage analysis in astrostatistics

Abstract

A sequence of statistical analyses often needs to be conducted by different groups of researchers where the output of each analysis feeds into subsequent analyses. The statistical and systematic uncertainties of estimated quantities, especially high dimensional quantities, are hard to quantify and difficult to carry forward into subsequent analyses. In practice, uncertainty is often ignored and the estimated quantities are often treated as fixed and known, leading to erroneous interpretation of the data and underestimation of uncertainties. An astrophysical example occurs when we use the spectra of the solar and stellar coronae to estimate its density, temperature, and other composition. This requires the use of the results of atomic physical experiments and calculations. In the example, the interaction of statistical and atomic uncertainties in the context of a spectral model inform the analysis of the images of solar and stellar physics. The interpretation of the spectral observations in relation to atomic data and their statistical uncertainties is necessary for deriving meaningful uncertainties on the solar and stellar coronal plasma parameters like electron density and temperature. Understanding how uncertainties in the underlying atomic physics propagates to the uncertainties in the inferred plasma parameters is an essential component of this analysis. We propose a principled multistage analysis to carry forward the model-generated atomic data uncertainties and statistical uncertainties obtained from preliminary analyses to a primary analysis based on the observed spectral lines under a Bayesian framework. Besides the Bayesian methodology that considers the atomic data uncertainties as fully specified and uncorrectable (the so-called pragmatic Bayesian method), we allow for the ob- served data to update the atomic data uncertainties (the fully Bayesian method). The former generally increases the uncertainties on the inferred parameters compared with models that incorporate only statistical uncertainties. In contrast, the latter reduces the uncertainties on the inferred parameters. To incorporate uncertainties into a primary analysis, we summarize a Monte Carlo sample of the atomic data that represents its statistical uncertainty by treating these samples as equally likely. We also consider a degenerated multivariate Gaussian model derived via a principal component analysis as a low dimensional summary of the uncertainty in the atomic data. Markov Chain Monte Carlo based model fitting is implemented including Multi-step Monte Carlo Gibbs Sampler and Hamiltonian Monte Carlo. The multistage analysis is able to cope with case studies of different levels of complexity. Two-stage analysis is used to infer the plasma parameters in spectral analysis with case studies on the density-sensitive only Fe XIII lines and the temperature-sensitive only Fe XVII lines. Three-stage analysis is used on the density- and temperature-sensitive O VII lines in the X-ray regime considering one more parameter and its corresponding source of statistical uncertainties. This principled multistage analysis has the potential to be applied on complicated models with a variety of sources of uncertainties.