Bayesian Methods and Machine Learning in Astrophysics
Creators: name: Higson, Edward John
Publisher: name: Apollo - University of Cambridge Repository
Year: 2019
Subjects: subject: Machine Learning
subject: Bayesian Inference
subject: Nested sampling
subject: Cosmology
subject: Black Holes
subject: Gravitational Waves
subject: Neural Networks
subject: Regression
subject: Astrophysics
subject: Sparsity
subject: Parameter Estimation
subject: Bayesian Evidence
subject: Bayesian
subject: Statistics
subject: FOS: Mathematics
subject: FOS: Mathematics
subject: Bayesian Sparse Reconstruction
subject: Computational Methods
subject: Error Analysis
subject: Dynamic Nested Sampling
subject: nestcheck
subject: perfectns
subject: dyPolyChord
subject: dynesty
subject: Image Processing
subject: Sparse Reconstruction
subject: Planck
subject: diagnostic tests
subject: error analysis
subject: PolyChord
subject: MultiNest
subject: Hubble Space Telescope
subject: Fitting
subject: Nonparametric statistics

description: This thesis is concerned with methods for Bayesian inference and their applications in astrophysics. We principally discuss two related themes: advances in nested sampling (Chapters 3 to 5), and Bayesian sparse reconstruction of signals from noisy data (Chapters 6 and 7). Nested sampling is a popular method for Bayesian computation which is widely used in astrophysics. Following the introduction and background material in Chapters 1 and 2, Chapter 3 analyses the sampling errors in nested sampling parameter estimation and presents a method for estimating them numerically for a single nested sampling calculation. Chapter 4 introduces diagnostic tests for detecting when software has not performed the nested sampling algorithm accurately, for example due to missing a mode in a multimodal posterior. The uncertainty estimates and diagnostics in Chapters 3 and 4 are implemented in the $\texttt{nestcheck}$ software package, and both chapters describe an astronomical application of the techniques introduced. Chapter 5 describes dynamic nested sampling: a generalisation of the nested sampling algorithm which can produce large improvements in computational efficiency compared to standard nested sampling. We have implemented dynamic nested sampling in the $\texttt{dyPolyChord}$ and $\texttt{perfectns}$ software packages. Chapter 6 presents a principled Bayesian framework for signal reconstruction, in which the signal is modelled by basis functions whose number (and form, if required) is determined by the data themselves. This approach is based on a Bayesian interpretation of conventional sparse reconstruction and regularisation techniques, in which sparsity is imposed through priors via Bayesian model selection. We demonstrate our method for noisy 1- and 2-dimensional signals, including examples of processing astronomical images. The numerical implementation uses dynamic nested sampling, and uncertainties are calculated using the methods introduced in Chapters 3 and 4. Chapter 7 applies our Bayesian sparse reconstruction framework to artificial neural networks, where it allows the optimum network architecture to be determined by treating the number of nodes and hidden layers as parameters. We conclude by suggesting possible areas of future research in Chapter 8.