This chapter presents a series of astronomical applications using some of the models presented earlier in the book. Each section concerns a specific type of astronomical data situation and the associated statistical model. The examples cover a large variety of topics, from solar (sunspots) to extragalactic (type Ia supernova) data, and the accompanying codes were designed to be easily modified in order to include extra complexity or alternative data sets. Our goal is not only to demonstrate how the models presented earlier can impact the classical approach to astronomical problems but also to provide resources which will enable both young and experienced researchers to apply such models in their daily analysis.

Following the same philosophy as in previous chapters, we provide codes in R/JAGS and Python/Stan for almost all the case studies. The exceptions are examples using type Ia supernova data for cosmological parameter estimation and approximate Bayesian computation (ABC). In the former we take advantage of the Stan ordinary differential equation solver, which, at the time of writing, is not fully functional within PyStan. Thus, we take the opportunity to show one example of how Stan can also be easily called from within R. The ABC approach requires completely different ingredients and, consequently, different software. Here we have used the cosmoabc Python package in order to demonstrate how the main algorithm works in a simple toy model. We also point the reader to the main steps towards using ABC for cosmological parameter inference from galaxy cluster number counts. This is considered an advanced topic and is presented as a glimpse of the potential of Bayesian analysis beyond the exercises presented in previous chapters. Many models discussed in this book represent a step forward from the types of models generally used by the astrophysical community. In the future we expect that Bayesian methods will be the predominant statistical approach to the analysis of astrophysical data.

Accessing Data

For all the examples presented in this chapter we will use publicly available astronomical data sets. These have been formatted to allow easy integration with our R and Python codes. All the code snippets in this chapter contain a path_to_data variable, which has the format path_to_data = ”˜ <some path>”, where the symbol ˜ should be substituted for the complete path to our GitHub repository.

Bayesian Models for Astrophysical Data provides those who are engaged in the Bayesian modeling of astronomical data with guidelines on how to develop code for modeling such data, as well as on how to evaluate a model as to its fit. One focus in this volume is on developing statistical models of astronomical phenomena from a Bayesian perspective. A second focus of this work is to provide the reader with statistical code that can be used for a variety of Bayesian models.

We provide fully working code, not simply code snippets, in R, JAGS, Python, and Stan for a wide range of Bayesian statistical models. We also employ several of these models in real astrophysical data situations, walking through the analysis and model evaluation. This volume should foremost be thought of as a guidebook for astronomers who wish to understand how to select the model for their data, how to code it, and finally how best to evaluate and interpret it. The codes shown in this volume are freely available online at www.cambridge.org/bayesianmodels. We intend to keep it continuously updated and report any eventual bug fixes and improvements required by the community. We advise the reader to check the online material for practical coding exercises.

This is a volume devoted to applying Bayesian modeling techniques to astrophysical data. Why Bayesian modeling? First, science appears to work in accordance with Bayesian principles. At each stage in the development of a scientific study new information is used to adjust old information. As will be observed when reviewing the examples later in this volume, this is how Bayesian modeling works. A posterior distribution created from the mixing of the model likelihood (derived from the model data) and a prior distribution (outside information we use to adjust the observed data) may itself be used as a prior for yet another enhanced model. New information is continually being used in models over time to advance yet newer models. This is the nature of scientific discovery. Yet, even if we think of a model in isolation from later models, scientists always bring their own perspectives into the creation of a model on the basis of previous studies or from their own experience in dealing with the study data.

Astrostatistics has only recently become a fully fledged scientific discipline. With the creation of the International Astrostatistics Association, the Astroinformatics & Astrostatistics Portal (ASAIP), and the IAU Commission on Astroinformatics and Astrostatistics, the discipline has mushroomed in interest and visibility in less than a decade.

With respect to the future, though, we believe that the above three organizations will collaborate on how best to provide astronomers with tutorials and other support for learning about the most up-to-date statistical methods appropriate for analyzing astrophysical data. But it is also vital to incorporate trained statisticians into astronomical studies. Even though some astrophysicists will become experts in statistical modeling, we cannot expect most astronomers to gain this expertise. Access to statisticians who are competent to engage in serious astrophysical research will be needed.

The future of astrostatistics will be greatly enhanced by the promotion of degree programs in astrostatistics at major universities throughout the world. At this writing there are no MS or PhD programs in astrostatistics at any university. Degree programs in astrostatistics can be developed with the dual efforts of departments of statistics and astronomy–astrophysics. There are several universities that are close to developing such a degree, and we fully expect that PhD programs in astrostatistics will be common in 20 years from now. They would provide all the training in astrophysics now given in graduate programs but would also add courses and training at the MS level or above in statistical analysis and in modeling in particular.

We expect that Bayesian methods will be the predominant statistical approach to the analysis of astrophysical data in the future. As computing speed and memory become greater, it is likely that new statistical methods will be developed to take advantage of the new technology. We believe that these enhancements will remain in the Bayesian tradition, but modeling will become much more efficient and reliable. We expect to see more non-parametric modeling taking place, as well as advances in spatial statistics – both twoand three-dimensional methods.

Finally, astronomy is a data-driven science, currently being flooded by an unprecedented amount of data, a trend expected to increase considerably in the next decade. Hence, it is imperative to develop new paradigms of data exploration and statistical analysis.

Introduction

Some of the history of the observations of the Cosmic Microwave Background radiation (CMB) has been recounted in the first part of this book (see Chapter 3). The culmination of the early efforts, the ‘first 30 years’, was perhaps the COBE DMR/FIRAS mission. COBE DMR produced the first all-sky maps of the CMB (Smoot et al., 1991), while the FIRAS experiment on the same satellite had established the remarkable accuracy of the Planckian form of the radiation spectrum (Mather and the COBE collaboration, 1990). Importantly, COBE had delivered a low resolution, albeit noisy, picture of the microwave sky that not only made substantial scientific advances, but also attracted widespread public attention and provided a vital scientific stimulus.

The manifest success of the COBE mission inspired a series of ground-based, balloon-based and and space observatories to look at the fluctuations with more sensitivity and at higher angular resolution. Satellite experiments are expensive and take a long time to build and so the competition to get the key results from lower cost ground based experiments before the space experiments could deliver was fierce, and in large part successful.

Classical Inference

In generic terms, the goal of statistical analysis is to discover how some factor Y responds to changes in some input, or set of inputs X. The observations (x, y) are paired, and while the values of x are generally under control of the experimentalist, the response y is subject to measurement errors ∊. The mechanism for doing this may take the form of establishing a model for the dependence of Y on X, or might be to discover which of the factors X, or which combination of the elements of X, are the main contributory factors to the response Y. Here we will consider only the first of these, fitting parameterised models.

Linear Regression

The simplest data model, linear regression is commonplace and involves determining parameters α and β in the relationship of the form

Y = AX + ∊, (24.1)

where the experiment consists of observing the values of Y, the response, resulting from treatments X. The observations Y are subject to errors ∊.

A Geometric Perspective

The space time of Einstein's theory is specified by its geometry. A gravitational field is thought of as distorting the space-time away from the no-gravity space-time of Minkowski. So, just as the Minkowski space of special relativity is entirely specified by a metric tensor or line element telling us what the space time separation of neighbouring points is, the space-time of the general theory is specified by a more general metric.

In the Aftermath of the CMB

The discovery of the CMB not only served to establish our cosmological paradigm as the Hot Big Bang theory, it also stimulated a growth in cosmology as a branch of physics. Fifty years later cosmology has reached a depth and precision of understanding that would have been inconceivable prior to 1965. The 50 years from 1965–2015 saw remarkable advances on both theoretical and data-acquisition and analysis fronts, which are described briefly in this chapter. One important consequence is the blurring of the boundary between theory and observation. Theoretical advances have driven experiments to gather and analyse data through which the theory could be exploited, while ever more sophisticated experiments demand improved methods of data analysis and impose constraints on the values of the parameters of the theories.

It has previously been said that the essence of physics, and science in general, is that it should be possible to perform experiments and to have other groups repeat those experiments, thus providing verification of the results and the consequent conclusions.

Why Bother with Newton?

Two Views of Gravity

The expansion of the Universe is dominated by the gravitational force. The best theory we have for the gravitational force is Einstein's Theory of General Relativity (Einstein, 1916a), which relates geometry with the material content of the space-time containing that matter.