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Bayesian Astrostatistics: A Backward Look to the Future

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Astrostatistical Challenges for the New Astronomy

Part of the book series: Springer Series in Astrostatistics ((SSIA,volume 1))

Abstract

This “perspective” chapter briefly surveys (1) past growth in the use of Bayesian methods in astrophysics; (2) current misconceptions about both frequentist and Bayesian statistical inference that hinder wider adoption of Bayesian methods by astronomers; and (3) multilevel (hierarchical) Bayesian modeling as a major future direction for research in Bayesian astrostatistics.

This volume contains presentations from the first invited session on astrostatistics to be held at an International Statistical Institute (ISI) World Statistics Congress. This session was a major milestone for astrostatistics as an emerging cross-disciplinary research area. It was the first such session organized by the ISI Astrostatistics Committee, whose formation in 2010 marked formal international recognition of the importance and potential of astrostatistics by one of its information science parent disciplines. It was also a significant milestone for Bayesian astrostatistics, as this research area was chosen as a (non-exclusive) focus for the session.

As an early (and elder!) proponent of Bayesian methods in astronomy, I have been asked to provide a “perspective piece” on the history and status of Bayesian astrostatistics. I begin by briefly documenting the rapid rise in use of the Bayesian approach by astrostatistics researchers over the past two decades. Next, I describe three misconceptions about both frequentist and Bayesian methods that hinder wider adoption of the Bayesian approach across the broader community of astronomer data analysts. Then I highlight the emerging role of multilevel (hierarchical) Bayesian modeling in astrostatistics as a major future direction for research in Bayesian astrostatistics. I end with a provocative recommendation for survey data reporting, motivated by the multilevel Bayesian perspective on modeling cosmic populations.

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Notes

  1. 1.

    Notably, Sturrock [3] earlier introduced astronomers to the use of Bayesian probabilities for “bookkeeping” of subjective beliefs about astrophysical hypotheses, but he did not discuss statistical modeling of measurements per se.

  2. 2.

    A third paper [11] had some Bayesian content but focused on frequentist evaluation criteria, even for the one Bayesian procedure considered; these three presentations, with discussion, comprised the Bayesian session.

  3. 3.

    Roberto Trotta and Martin Hendry have shown similar plots in various venues, helpfully noting that the recent rate of growth apparent in Fig. 2.1 is much greater than the rate of growth in the number of all publications; i.e., not just the amount but also the prevalence of Bayesian work is rapidly rising.

  4. 4.

    I am not providing references to publications exhibiting the problem for diplomatic reasons and for a more pragmatic and frustrating reason: In the field where I have repeatedly encountered the problem—analysis of infrared exoplanet transit data—authors routinely fail to describe their analysis methods with sufficient detail to know what was done, let alone to enable readers to verify or duplicate the analysis. While there are clear signs of statistical impropriety in many of the papers, I only know the details from personal communications with exoplanet transit scientists.

  5. 5.

    Efron [16] describes some such cases by saying the frequentist result can be accurate but not correct. Put another way, the performance claim is valid, but the long-run performance can be irrelevant to the case-at-hand, e.g., due to the existence of so-called recognizable subsets in the sample space (see [9] and [16] for elaboration of this notion). This is a further example of how nontrivial the relationship between variability and uncertainty can be.

  6. 6.

    There are theorems linking single-case Bayesian probabilities and long-run performance in some general settings, e.g., establishing that, for fixed-dimension parametric inference, Bayesian credible regions with probability P have frequentist coverage close to P (the rate of convergence is \(o(1/\sqrt N )\) for flat priors, and faster for so-called reference priors). But the theorems do not apply in some interesting classes of problems, e.g., nonparametric problems.

  7. 7.

    The convention is to reserve the term for models with three or more levels of nodes, i.e., two or more levels of edges, or two or more levels of nodes for uncertain variables (i.e., unshaded nodes). The model depicted in panel (d) would be called a two-level model.

  8. 8.

    http://www.mrc-bsu.cam.ac.uk/bugs/

  9. 9.

    It is worth pointing out that this is not a uniquely Bayesian insight. Eddington, Malmquist, and Lutz and Kelker used frequentist arguments to justify their corrections; Eddington even offered adaptive corrections. The large and influential statistics literature on shrinkage estimators leads to similar conclusions; see [22] for further discussion and references.

  10. 10.

    I am tempted to recommend that, even in this regime, the likelihood summary be chosen so as to deter misuse as an estimate, say by tabulating the + 1σ and −2σ points rather than means and standard deviations. I am only partly facetious about this!

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Acknowledgments

I gratefully acknowledge NSF and NASA for support of current research underlying this commentary, via grants AST-0908439, NNX09AK60G and NNX09AD03G. I thank Martin Weinberg for helpful discussions on information propagation within multilevel models. Students of Ed Jaynes’s writings on probability theory in physics may recognize the last part of my title, borrowed from a commentary by Jaynes on the history of Bayesian and maximum entropy ideas in the physical sciences [53]. This bit of plagiarism is intended as a homage to Jaynes’s influence on this area — and on my own research and thinking.

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  1. Center for Radiophysics and Space Research, Cornell University, Ithaca, NY, 14853-6801, USA

    Thomas J. Loredo

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Correspondence to Thomas J. Loredo .

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    Joseph M. Hilbe

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Loredo, T.J. (2013). Bayesian Astrostatistics: A Backward Look to the Future. In: Hilbe, J. (eds) Astrostatistical Challenges for the New Astronomy. Springer Series in Astrostatistics, vol 1. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3508-2_2

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