Abstract
The ever increasing power and sophistication of today’s high energy astronomical instruments is opening a new realm of high quality data that is quickly pushing beyond the capabilities of the “classical” data-analysis methods in common use. In this chapter we discuss the use of highly structured models that not only incorporate the scientific model (e.g., for a source spectrum) but also account for stochastic components of data collection and the instrument (e.g., background contamination and pile up). Such hierarchical models when used in conjunction with Bayesian or likelihood statistical methods offer a systematic solution to many challenging data analytic problems (e.g., low count rates and pile up). Hierarchical models are becoming increasingly popular in physical and other scientific disciplines largely because of the recent development of sophisticated methods for statistical computation. Thus, we discuss such methods as the EM algorithm, data augmentation, and Markov chain Monte Carlo in the context of high energy high resolution low count data.
This paper is followed by a commentary by astronomer Michael Strauss.
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References
Carlin, B. P. and Louis, T. A. (1996). Bayes and Empirical Bayes Methods for Data Analysis. Chapman & Hall, London.
Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc., Ser. B, 39, 1–37.
Gelfand, A. E. and Smith, A. F. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85, 398–409.
Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian Data Analysis. Chapman & Hall, London.
Gilks, W. R., Richardson, S., and Spiegelhalter, D. J. (1996). Markov chain Monte Carlo in Practice. Chapman & Hall, London.
Hastings, W. K. (1970). Monte Carlo sampling methods usings Markov chains and their applications. Biometrika 57, 97–109.
Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., and Teller, E. (1953). Equation of state calculations by fast computing machines. J. Chem. Phy. 21, 1087–1092.
Protassov, R., van Dyk, D., Connors, A., Kashyap, V., and Siemiginowska, A. (2002). Statistics: Handle with care — detecting multiple model components with the likelihood ratio test. Astrophysical J. to appear.
Siemiginowska, A., Elivis, M., Alanna, C., Freeman, P., Kashyap, V., and Feigelson, E. (1997). in Statistical Challenges in Modern Astronomy II (eds. E. Feigelson and G. Babu), 241–253. Springer-Verlag, New York.
Tanner, M. A. and Wong, W. H. (1987). The calculation of posterior distributions by data augmentation (with discussion). J. Amer. Statist. Assoc. 82, 528–550.
Tierney, L. (1996). in Markov Chain Monte Carlo in Practice (eds. W. R. Gilks, S. Richardson, and D. J. Spiegelhalter). Chapman & Hall, London.
van Dyk, D. A., Connors, A., Kashyap, V., and Siemiginowska, A. (2001). Analysis of energy spectra with low photon counts via Bayesian posterior simulation. Astrophysical J. 548, 224–243.
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van Dyk, D.A. (2003). Hierarchical Models, Data Augmentation, and Markov Chain Monte Carlo. In: Statistical Challenges in Astronomy. Springer, New York, NY. https://doi.org/10.1007/0-387-21529-8_3
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DOI: https://doi.org/10.1007/0-387-21529-8_3
Publisher Name: Springer, New York, NY
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