Simple two-stage algorithms for fitting hierarchies of complex models with applications in astrophysics

Abstract

Hierarchical models are one of the most efficient and principled techniques to estimate the parameters of each object in a population. In astrophysics, we often aim to estimate a set of parameters for each of a number of objects in a population, which motivates the application of hierarchical models. However, fully modelling a population of objects within a hierarchical model re- quires substantial computational investment and often specialised computer code (black-box), especially for complicated problems. In this study, we develop novel methods to conveniently obtain the improved estimates available under a hierarchi- cal model. While taking advantage of the existing code for case-by-case analyses, our methods simultaneously estimate parameters of the individual objects and parameters that describe their population distribution. There are many possible applications of hierarchical models in astrophysics. In this research we deploy our novel computational algorithms to three projects: the determination of distance modulus to the Large Magellanic Cloud (LMC), the precise estimates of a group of Galactic halo white dwarfs (WDs), the reliable estimates of the initial-final mass relations (IFMRs). These three circumstances correspond to the applications of Bayesian hierarchical models to more precise determination of a common parameter, a set of parameters of one kind and a set of functions of one kind, respectively. The main contribution of this work lies in the last two–WDs and IFMRs–projects, which are built upon a few computer models. Our innovation is to fold these complex computer-based astrophysical models into a Bayesian hierarchical model that allows us to simultaneously fit individual models for multiple stars or even multiple clus- ters of stars to obtain estimators with superior statistical properties (e.g., reduced mean squared errors). Our easy-to-implement computational algorithms make these computationally challenging tasks tractable via taking advantage of case-by-case fits, which are potentially useful to many projects in other areas. Our contribution is mainly on the statistical side and all astronomical projects are meant to be testbeds for proofs of concept of my new algorithms. Our analyses are preliminary in the astronomy side, and all the astronomical results are not definitive.