Accounting for Source Uncertainties in Analyses of Astronomical Survey Data

I discuss an issue arising in analyzing data from astronomical surveys: accounting for measurement uncertainties in the properties of individual sources detected in a survey when making inferences about the entire population of sources. Source uncertainties require the analyst to introduce unknown ``incidental'' parameters for each source. The number of parameters thus grows with the size of the sample, and standard theorems guaranteeing asymptotic convergence of maximum likelihood estimates fail in such settings. From the Bayesian point of view, the missing ingredient in such analyses is accounting for the volume in the incidental parameter space via marginalization. I use simple simulations, motivated by modeling the distribution of trans-Neptunian objects surveyed in the outer solar system, to study the effects of source uncertainties on inferences. The simulations show that current non-Bayesian methods for handling source uncertainties (ignoring them, or using an ad hoc incidental parameter integration) produce incorrect inferences, with errors that grow more severe with increasing sample size. In contrast, accounting for source uncertainty via marginalization leads to sound inferences for any sample size.Comment: 12 pages, 5 figures; to appear in Bayesian Inference And Maximum Entropy Methods In Science And Engineering: 24th International Workshop, Garching, Germany, 2004; ed. Volker Dose et al. (AIP Conference Proceedings Series

Sines, steps and droplets: Semiparametric Bayesian modeling of arrival time series

I describe ongoing work developing Bayesian methods for flexible modeling of arrival time series data without binning, aiming to improve detection and measurement of X-ray and gamma-ray pulsars, and of pulses in gamma-ray bursts. The methods use parametric and semiparametric Poisson point process models for the event rate, and by design have close connections to conventional frequentist methods currently used in time-domain astronomy.Comment: 4 pages, 1 figure; to appear in the proceedings of IAU Symposium 285, "New Horizons in Time Domain Astronomy" (proceedings eds. Elizabeth Griffin, Bob Hanisch, and Rob Seaman), Cambridge University Press; see http://www.physics.ox.ac.uk/IAUS285

Bayesian Adaptive Exploration

I describe a framework for adaptive scientific exploration based on iterating an Observation--Inference--Design cycle that allows adjustment of hypotheses and observing protocols in response to the results of observation on-the-fly, as data are gathered. The framework uses a unified Bayesian methodology for the inference and design stages: Bayesian inference to quantify what we have learned from the available data and predict future data, and Bayesian decision theory to identify which new observations would teach us the most. When the goal of the experiment is simply to make inferences, the framework identifies a computationally efficient iterative ``maximum entropy sampling'' strategy as the optimal strategy in settings where the noise statistics are independent of signal properties. Results of applying the method to two ``toy'' problems with simulated data--measuring the orbit of an extrasolar planet, and locating a hidden one-dimensional object--show the approach can significantly improve observational efficiency in settings that have well-defined nonlinear models. I conclude with a list of open issues that must be addressed to make Bayesian adaptive exploration a practical and reliable tool for optimizing scientific exploration.Comment: 17 pages, 5 figure

Introduction to papers on astrostatistics

We are pleased to present a Special Section on Statistics and Astronomy in this issue of the The Annals of Applied Statistics. Astronomy is an observational rather than experimental science; as a result, astronomical data sets both small and large present particularly challenging problems to analysts who must make the best of whatever the sky offers their instruments. The resulting statistical problems have enormous diversity. In one problem, one may have to carefully quantify uncertainty in a hard-won, sparse data set; in another, the sheer volume of data may forbid a formally optimal analysis, requiring judicious balancing of model sophistication, approximations, and clever algorithms. Often the data bear a complex relationship to the underlying phenomenon producing them, much in the manner of inverse problems.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS234 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Pencil-Beam Surveys for Faint Trans-Neptunian Objects

We have conducted pencil-beam searches for outer solar system objects to a limiting magnitude of R ~ 26. Five new trans-neptunian objects were detected in these searches. Our combined data set provides an estimate of ~90 trans-neptunian objects per square degree brighter than ~ 25.9. This estimate is a factor of 3 above the expected number of objects based on an extrapolation of previous surveys with brighter limits, and appears consistent with the hypothesis of a single power-law luminosity function for the entire trans-neptunian region. Maximum likelihood fits to all self-consistent published surveys with published efficiency functions predicts a cumulative sky density Sigma(<R) obeying log10(Sigma) = 0.76(R-23.4) objects per square degree brighter than a given magnitude R.Comment: Accepted by AJ, 18 pages, including 6 figure

Bayesian Methods for Analysis and Adaptive Scheduling of Exoplanet Observations

We describe work in progress by a collaboration of astronomers and statisticians developing a suite of Bayesian data analysis tools for extrasolar planet (exoplanet) detection, planetary orbit estimation, and adaptive scheduling of observations. Our work addresses analysis of stellar reflex motion data, where a planet is detected by observing the "wobble" of its host star as it responds to the gravitational tug of the orbiting planet. Newtonian mechanics specifies an analytical model for the resulting time series, but it is strongly nonlinear, yielding complex, multimodal likelihood functions; it is even more complex when multiple planets are present. The parameter spaces range in size from few-dimensional to dozens of dimensions, depending on the number of planets in the system, and the type of motion measured (line-of-sight velocity, or position on the sky). Since orbits are periodic, Bayesian generalizations of periodogram methods facilitate the analysis. This relies on the model being linearly separable, enabling partial analytical marginalization, reducing the dimension of the parameter space. Subsequent analysis uses adaptive Markov chain Monte Carlo methods and adaptive importance sampling to perform the integrals required for both inference (planet detection and orbit measurement), and information-maximizing sequential design (for adaptive scheduling of observations). We present an overview of our current techniques and highlight directions being explored by ongoing research.Comment: 29 pages, 11 figures. An abridged version is accepted for publication in Statistical Methodology for a special issue on astrostatistics, with selected (refereed) papers presented at the Astronomical Data Analysis Conference (ADA VI) held in Monastir, Tunisia, in May 2010. Update corrects equation (3