Bayesian Statistics in Astronomy

Abstract

In this thesis I investigate the properties of Markov Chain Monte Carlo methods including the Affine Invariant Ensemble Sampler (AIES) and the application of the combination of Diffusive Nested Sampling, Reversible Jump, and Metropolis-Hastings, and apply the latter to astronomical inference problems. The AIES has become popular in astronomy. To investigate its convergence properties the AIES was used to sample two test distributions with a known mean and covariance structure. The tests revealed that the mean and the variance of the target distribution do not match the known values, and the chain takes a very long time to converge, even though the trace plots might not show any convergence issues. A common practice in applied statistics is to use visual inspection of trace plots to evaluate convergence. These tests demonstrate that this practice is insufficient when the AIES is used. The combination of Diffusive Nested Sampling, Reversible Jump, and Metropolis-Hastings is then applied to two astronomy problems which require trans-dimensional transitions. The first problem is a gravitational lensing problem. The model applied consisted of a main galaxy model augmented with an unknown number of satellite galaxies and sources. After this the model is applied to three datasets, and the posterior distributions agreed well with the literature. The second problem is a galaxy morphology problem where the number of galaxies is unknown. Applying a multi-galaxy model to a dataset, produced models which fit the data sufficiently, confirmed by the standardised residuals. In general, a two-dimensional surface plot of the posterior distribution was able to identify areas of higher object density. With the application of a simple clustering algorithm I was able to group objects potentially associated with the same galaxies using the posterior distribution.