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首页天文数据分析指南:Bayesian模型与Python、R实现
"《Bayesian Models for Astrophysical Data》是一本深入探讨天文学数据的贝叶斯建模方法的专业书籍,作者Joseph M. Hilbe是NASA喷气推进实验室的太阳系大使,同时也是亚利桑那州立大学的统计学兼职教授。这本书结合Python、R、JAGS和Stan等编程语言,提供了全面的实战指导和代码示例,旨在帮助读者直接应用或改编到自己的天文数据分析中。" 本书首先从频率主义和贝叶斯视角对比分析了正态模型,然后逐步引入一系列的贝叶斯广义线性模型、混合或层次模型,以及如ABC(Approximate Bayesian Computation)和INLA(Integrated Nested Laplace Approximation)等其他类型的模型。书中提供的代码大部分在其他地方难以找到,这对于实践者来说是一个宝贵的资源。 为了使读者能够轻松将模型应用到自己的数据中,作者首先以合成数据的形式介绍模型,然后再将这些模型应用于实际的天文数据。全书的核心在于动手建模、数据分析和科学问题的解释,这使得它不仅对天文学家来说是必备读物,同时也对自然科学领域的研究人员具有广泛的吸引力。 通过阅读本书,读者可以学习如何利用贝叶斯方法处理天文学中的复杂数据,理解不同模型的优缺点,以及如何评估和解释模型结果。无论是对于初学者还是经验丰富的研究者,这都是一本能够提升数据驱动科学理解力的优秀教材。无论你是R语言的爱好者,还是Python的热衷者,或者想要通过JAGS和Stan进行贝叶斯推断,这本书都能提供你需要的知识和工具,帮助你在天文学领域进行更深入的研究。
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1 Astrostatistics
1.1 The Nature and Scope of Astrostatistics
Astrostatistics is at the same time one of the oldest disciplines, and one of the youngest. The Ionian
Greek philosopher Thales of Miletus is credited with correctly predicting a total solar eclipse in
central Lydia, which he had claimed would occur in May of 585 BCE. He based this prediction on an
examination of records maintained by priests throughout the Mediterranean and Near East. The fact
that his prediction was apparently well known, and the fact that the Lydians were engaged in a war
with the Medes in Central Lydia during this period, brought his prediction notice and fame. Thales
was forever after regarded as a sage and even today he is named the father of philosophy and the
father of science in books dealing with these subjects.
Thales’ success spurred on others to look for natural relationships governing the motions of
astronomical bodies. Of particular note was Hipparchus (190–120 BCE) who, following on the earlier
work of Aristarchus of Samos (310–230 BCE and Eratosthenes (276–147 BCE), is widely regarded as
the first to clearly apply statistical principles to the analysis of astronomical events. Hipparchus also
is acknowledged to have first developed trigonometry, spherical trigonometry, and trigonometric
tables, applying these to the motions of both the moon and sun. Using the size of the moon’s parallax
and other data from the median percent of the Sun covered by the shadow of the Earth at various
sites in the area, he calculated the distance from the Earth to the Moon as well as from the Earth to
the Sun in terms of the Earth’s radius. His result was that the median value is 60.5 Earth radii. The
true value is 60.3. He also calculated the length of the topical year to within six minutes per year of
its true value.
Others in the ancient world, as well as scientists until the early nineteenth century, also used
descriptive statistical techniques to describe and calculate the movements and relationships between
the Earth and astronomical bodies. Even the first application of a normal or ordinary least squares
regression was to astronomy. In 1801 Hungarian Franz von Zach applied the new least squares
regression algorithm developed by Carl Gauss for predicting the position of Ceres as it came into
view from its orbit behind the Sun.
The development of the first inferential statistical algorithm by Gauss, and its successful
application by von Zach, did not immediately lead to major advances in inferential statistics.
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Astronomers by and large seemed satisfied to work with the Gaussian, or normal, model for
predicting astronomical events, and statisticians turned much of their attention to deriving various
probability functions. In the early twentieth century William Gosset, Karl Pearson, and Ronald Fisher
made the most advances in statistical modeling and hypothesis testing. Pearson developed the
mathematics of goodness-of-fit and of hypothesis testing. The Pearson test is still used as an
assessment of frequentist based model fit.
1
In addition, Pearson developed a number of tests related
to correlation analysis, which is important in both frequentist and Bayesian modeling. Fisher (1890–
1962) is widely regarded as the father of modern statistics. He is basically responsible for the
frequentist interpretation of hypothesis testing and of statistical modeling. He developed the theories
of maximum likelihood estimation and analysis of variance and the standard way that statisticians
understood p-values and confidence intervals until the past 20 years. Frequentists still employ his
definition of the p-value in their research. His influence on twentieth century statistics cannot be
overestimated.
It is not commonly known that Pierre-Simon Laplace (1749–1827) is the person foremost
responsible for bringing attention to the notion of Bayesian analysis, which at the time meant
employing inverse probability to the analysis of various problems. We shall discuss Bayesian
methodology in a bit more detail in Chapter 3. Here we can mention that Thomas Bayes (1702–
1761) developed the notion of inverse probability in unpublished notes he made during his lifetime.
These notes were discovered by Richard Price, Bayes’s literary executor and a mathematician, who
restructured and presented Bayes’ paper to the Royal Society. It had little impact on British
mathematicians at the time, but it did catch the attention of Laplace. Laplace fashioned Bayes’ work
into a full approach to probability, which underlies current Bayesian statistical modeling. It would
probably be more accurate to call Bayesian methodology Bayes–Laplace methodology, but
simplification has given Bayes the nominal credit for this approach to both probability and statistical
modeling.
After enthusiastically promoting inverse probability, Laplace abandoned this work and returned
to researching probability theory from the traditional perspective. He also made major advances in
differential equations, including his discovery of Laplace transforms, which are still very useful in
mathematics. Only a few adherents to the Bayesian approach to probability carried on the tradition
throughout the next century and a half. Mathematicians such as Bruno de Finetti (Italy) and Harold
Jeffreys (UK) promoted inverse probability and Bayesian analysis during the early part of the
twentieth century, while Dennis Lindley (UK), Leonard J Savage (US), and Edwin Jaynes, a US
physicist, were mainstays of the tradition in the early years of the second half of the twentieth
century. But their work went largely unnoticed.
The major problem with Bayesian analysis until recent times has been related to the use of
priors. Priors are distributions representing information from outside the model data that is
incorporated into the model. We shall be discussing priors in some detail later in the text. Briefly
though, except for relatively simple models the mathematics of calculating so-called posterior
distributions was far too difficult to do by hand, or even by most computers, until computing speed
and memory became powerful enough. This was particularly the case for Bayesian models, which
are extremely demanding on computer power. The foremost reason why most statisticians and
analysts were not interested in implementing Bayesian methods into their research was that
computing technology was not advanced enough to execute more than fairly simple problems. This
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was certainly the case with respect to the use of Bayesian methods in astronomy.
We shall provide a brief overview of Bayesian methodology in Chapter 3. Until such methods
became feasible, however, Fisherian or frequentist methodology was the standard way of doing
statistics. For those who are interested in understanding the history of this era of mathematics and
statistics, we refer you to the book Willful Ignorance: The Mismeasure of Uncertainty
(Weisberg, 2014).
Astronomy and descriptive statistics, i.e., the mathematics of determining mean, median, mode,
range, tabulations, frequency distributions, and so forth, were closely tied together until the early
nineteenth century. Astronomers have continued to employ descriptive statistics, as well as basic
linear regression, to astronomical data. But they did not concern themselves with the work being
done in statistics during most of the twentieth century. Astronomers found advances in telescopes
and the new spectroscope, as well as in calculus and differential equations in particular, to be much
more suited to understanding astrophysical data than hypothesis testing and other frequentist
methods. There was a near schism between astronomers and statisticians until the end of the last
century.
As mentioned in the Preface, astrostatistics can be regarded as the statistical analysis of
astronomical data. Unlike how astronomers utilized statistical methods in the past, primarily focusing
on descriptive measures and to a moderate extent on linear regression from within the frequentist
tradition, astrostatistics now entails the use, by a growing number of astronomers, of the most
advanced methods of statistical analysis that have been developed by members of the statistical
profession. However, we still see astronomers using linear regression on data that to a statistician
should clearly be modeled by other, non-linear, means. Recent texts on the subject have been aimed
at informing astronomers of these new advanced statistical methods.
It should be mentioned that we have incorporated astroinformatics under the general rubric of
astrostatistics. Astroinformatics is the study of the data gathering and computing technology needed
to gather astronomical data. It is essential to statistical analysis. Some have argued that
astroinformatics incorporates astrostatistics, which is the reverse of the manner in which we
envision it. But the truth is that gathering information without an intent to analyze it is a fairly useless
enterprise. Both must be understood together. The International Astronomical Union (IAU) has
established a new commission on astroinformatics and astrostatistics, seeming to give primacy to the
former, but how the order is given depends somewhat on the interests of those who are establishing
such names. Since we are focusing on statistics, although we are also cognizant of the important
role that informatics and information sciences bring to bear on statistical analysis, we will refer to
the dual studies of astrostatistics and astroinformatics as simply astrostatistics.
In the last few decades the size and complexity of the available astronomical information has
closely followed Moore’s law, which states roughly that computing processing power doubles every
two years. However, our ability to acquire data has already surpassed our capacity to analyze it.
The Sloan Digital Sky Survey (SDSS), in operation since 2000, was one of the first surveys to face
the big data challenge: in its first data release it observed 53 million individual objects. The Large
Synoptic Survey Telescope (LSST) survey is aiming to process and store 30 terabytes of data each
night for a period of ten years. Not only must astronomers (astroinformaticists) deal with such
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massive amounts of data but also it must be stored in a form in which meaningful statistical analysis
can proceed. In addition, the data being gathered is permeated with environmental noise (due to
clouds, moon, interstellar and intergalactic dust, cosmic rays), instrumental noise (due to distortions
from telescope design, detector quantum efficiency, a border effect in non-central pixels, missing
data), and observational bias (since, for example, brighter objects have a higher probability of being
detected). All these problem areas must be dealt with prior to any attempt to subject the data to
statistical analysis. The concerns are immense, but nevertheless this is a task which is being handled
by astrostatisticians and information scientists.
1.2 The Recent Development of Astrostatistics
Statistical analysis has developed together with the advance in computing speed and storage.
Maximum likelihood and all such regression procedures require that a matrix be inverted. The size
of the matrix is based on the number of predictors and parameters in the model, including the
intercept. Parameter estimates can be determined using regression for only very small models if
done by hand. Beginning in the 1960s, larger regression and multivariate statistical procedures could
be executed on mainframe computers using SAS, SPSS, and other statistical and data management
software designed specifically for mainframe systems. These software packages were ported to the
PC environment when PCs with hard drives became available in 1983. Statistical routines requiring
a large number of iterations could take a long time to converge. But as computer speeds became
ever faster, new regression procedures could be developed and programmed that allowed for the
rapid inversion of large matrices and the solution to complex modeling projects.
Complex Bayesian modeling, if based on Markov chain Monte Carlo (MCMC) sampling, was
simply not feasible until near the turn of the century. By 2010 efficient Bayesian sampling algorithms
were implemented into statistical software, and the computing power by then available allowed
astronomers to analyze complex data situations using advanced statistical techniques. By the closing
years of the first decade of this century astronomers could take advantage of the advances in
statistical software and of the training and expertise of statisticians.
As mentioned before, from the early to middle years of the nineteenth century until the final
decade of the twentieth century there was little communication between astronomers and
statisticians. We should note, though, that there were a few astronomers who were interested in
applying sophisticated statistical models to their study data. In 1990, for example, Thomas Loredo
wrote his PhD dissertation in astrophysics at the University of Chicago on “From Laplace
Supernova SN 1987A: Bayesian inference in astrophysics” (see the book Feigelson and
Babu, 2012a), which was the first thorough application of Bayesian modeling to astrophysical data.
It was, and still is, the seminal work in the area and can be regarded as one of the founding articles
in the new discipline of astrostatistics.
In 1991 Eric Feigelson and Jogesh Babu, an astronomer and statistician respectively at
Pennsylvania State University, collaboratively initiated a conference entitled Statistical Challenges
in Modern Astronomy. The conference was held at their home institution and brought together
astronomers and a few statisticians for the purpose of collaboration. The goal was also to find a
forum to teach astronomers how to use appropriate statistical analysis for their study projects.
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These conferences were held every five years at Penn State until 2016. The conference site has
now shifted to Carnegie Mellon University under the direction of Chad Schafer. During the 1990s
and 2000s a few other conferences, workshops, and collaborations were held that aimed to provide
statistical education to astronomers. But they were relatively rare, and there did not appear to be
much growth in the area.
Until 2008 there were no astrostatistics working groups or committees authorized under the
scope of any statistical or astronomical association or society. Astrostatistics was neither recognized
by the IAU nor recognized as a discipline in its own right by the principal astronomical and statistical
organizations. However, in 2008 the first interest group in astrostatistics was initiated under the
International Statistical Institute (ISI), the statistical equivalent of the IAU for astronomy. This
interest group, founded by the first author of the book, and some 50 other interested astronomers
and statisticians, met together at the 2009 ISI World Statistics Congress in Durban, South Africa.
The attendees voted to apply for the creation of a standing committee on astrostatistics under the
auspices of the ISI. The proposal was approved at the December 2009 meeting of the ISI executive
board. It was the first such astrostatistics committee authorized by an astronomical or statistical
society.
In the following month the ISI committee expanded to become the ISI Astrostatistics Network.
Network members organized and presented two invited sessions and two special theme sessions in
astrostatistics at the 2011 World Statistics Congress in Dublin, Ireland. Then, in 2012, the
International Astrostatistics Association (IAA) was formed from the Network as an independent
professional scientific association for the discipline. Also in 2012 the Astrostatistics and
Astroinformatics Portal (ASAIP) was instituted at Pennsylvania State under the editorship of Eric
Feigelson and the first author of this volume. As of January 2016 the IAA had over 550 members
from 56 nations, and the Portal had some 900 members. Working Groups in astrostatistics and
astroinformatics began under the scope of IAU, the American Astronomical Society, and the
American Statistical Association. In 2014 the IAA sponsored the creation of its first section, the
Cosmostatistics Initiative (COIN), led by the second author of this volume. In 2015 the IAU
working group became the IAU Commission on Astroinformatics and Astrostatistics, with Feigelson
as its initial president. Astrostatistics, and astroinformatics, is now a fully recognized discipline.
Springer has a series on astrostatistics, the IAA has begun agreements with corporate partnerships,
Cambridge University Press is sponsoring IAA awards for contributions to the discipline, and
multiple conferences in both astrostatistics and astroinformatices are being held – all to advance the
discipline. The IAA headquarters is now at Brera Observatory in Milan. The IAA website is
located at: http://iaa.mi.oa-brera.inaf.it and the ASAIP Portal URL is https://asaip.psu.edu. We
recommend the readers of this volume to visit these websites for additional resources on
astrostatistics.
1.3 What is a Statistical Model?
Statistics and statistical modeling have been defined in a variety of ways. We shall define it in a
general manner as follows:
Statistics may be generically understood as the science of collecting and analyzing data for the purpose of classification,
prediction, and of attempting to quantify and understand the uncertainty inherent in phenomena underlying data
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