Information: Measuring the Missing, Using the Observed, and Approximating the Complete

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Information: Measuring the Missing, Using the Observed, and Approximating the Complete

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Title: Information: Measuring the Missing, Using the Observed, and Approximating the Complete
Author: Jones, David Edward ORCID  0000-0002-8152-6818
Citation: Jones, David Edward. 2016. Information: Measuring the Missing, Using the Observed, and Approximating the Complete. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences.
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Abstract: In this thesis, we present three topics broadly connected to the concept and use of statistical information, and specifically regarding the problems of hypothesis testing and model selection, astronomical image analysis, and Monte Carlo integration.

The first chapter is inspired by the work of DeGroot (1962) and Nicolae et al. (2008) and is the most directly focused on the theme of statistical information. DeGroot (1962) developed a general framework for constructing Bayesian measures of the expected information that an experiment will provide for estimation. We propose an analogous framework for measures of information for hypothesis testing, and illustrate how these measures can be applied in experimental design. In contrast to estimation information measures that are typically used in experimental design for surface estimation, test information measures are more useful in experimental design for hypothesis testing and model selection. Indeed, one test information measure suggested by our framework is probability based, and in design contexts where decision problem are of interest, it has more appealing properties than variance based measures. The underlying intuition of our design proposals is straightforward: to distinguish between two or more models we should collect data from regions of the covariate space for which the models differ most. Nicolae et al. (2008) give an asymptotic equivalence between their test information measures and Fisher information. We extend this result to all test information measures under our framework, and hence further our understanding of the links between test and estimation information measures.

In the second chapter, we present a powerful new algorithm that combines both spatial and spectral (energy) information to separate photons from overlapping sources (e.g., stars) in an astronomical image. We use Bayesian statistical methods to simultaneously infer the number of overlapping sources, to probabilistically separate the photons among the sources, and to fit the parameters describing the individual sources. Using the Bayesian joint posterior distribution, we are able to coherently quantify the uncertainties associated with all these parameters. The advantages of combining spatial and spectral information are demonstrated through a simulation study. The utility of the approach is then illustrated by analysis of observations of the sources FK Aqr and FL Aqr with the XMM-Newton Observatory and the central region of the Orion Nebula Cluster with the Chandra X-ray Observatory. In this chapter we make additional effort to explain relevant standard statistical ideas and methods in order to make the exposition more accessible to astronomers unfamiliar with statistics.

The last chapter extends the maximum likelihood theory developed by Kong et al. (2003) for deriving Monte Carlo estimators of normalizing constants. Kong et al. (2003) had the fundamental idea of treating the baseline measure as an unknown quantity to be estimated, and found that this suggested a maximum likelihood method for estimating integrals of interest. Their work shows that sub-models of the baseline measure can be used to incorporate some of our knowledge of the true measure, thus allowing greater statistical precision to be gained at the expense of more function evaluations, but without the need for more Monte Carlo samples. Our contribution is to introduce a simple extension of this framework which greatly increases its flexibility for trading off statistical and computational efficiency. As a result, we gain an appealing maximum likelihood interpretation of the very effective warp transformations proposed by Meng and Schilling (2002). We additionally investigate the open problem of optimally choosing parameters for sub-models of the baseline measure.
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