In this thesis, we develop two novel statistical methods and software to
tackle outstanding astronomical and cosmological data analysis problems. In
particular, we deploy stochastic variability models as fundamental building
blocks in the conception and refinement of Bayesian methods to process
astronomical data and infer parameters of interest.
The first method, eBascs, concerns the analysis of individual X-ray sources
that appear in crowded fields, which can easily be compromised by the
misallocation of recorded events to their originating sources. We develop a Bayesian method designed to sift high-energy
photon events from multiple sources with overlapping point spread functions,
leveraging the differences in their spatial, spectral, and temporal signatures.
The method probabilistically assigns each event to a given source. Such a
disentanglement allows more detailed spectral or temporal analysis to focus on
the individual component in isolation, free of contamination from other sources
or the background. We are also able to compute source parameters of interest
like their locations, relative brightness, and background contamination, while
accounting for the uncertainty in event assignments. The second method, Td-Carma, is a Bayesian method designed to estimate cosmological time delays by modeling observed and irregularly sampled
light curves as realizations of a CARMA process. Our model accounts for heteroskedastic measurement errors and microlensing. The semiseparable structure of
the CARMA covariance matrix allows for fast and scalable likelihood computation using Gaussian process modeling. We obtain a sample from the
joint posterior distribution of the model parameters using a nested sampling
approach. This allows for ”painless” Bayesian computation, dealing with
the expected multimodality of the posterior distribution in a straightforward
manner and not requiring the specification of starting values or an initial
guess for the time delay. The proposed
sampling procedure automatically evaluates the Bayesian evidence, allowing
us to perform principled Bayesian model selection.