Applications of Bayesian Probability Theory in Astrophysics

Bayesian Inference is a powerful approach to data analysis that is based almost entirely on probability theory. In this approach, probabilities model {\it uncertainty} rather than randomness or variability. This thesis is composed of a series of papers that have been published in various astronomical journals during the years 2005-2008. The unifying thread running through the papers is the use of Bayesian Inference to solve underdetermined inverse problems in astrophysics. Firstly, a methodology is developed to solve a question in gravitational lens inversion - using the observed images of gravitational lens systems to reconstruct the undistorted source profile and the mass profile of the lensing galaxy. A similar technique is also applied to the task of inferring the number and frequency of modes of oscillation of a star from the time series observations that are used in the field of asteroseismology. For these complex problems, many of the required calculations cannot be done analytically, and so Markov Chain Monte Carlo algorithms have been used. Finally, probabilistic reasoning is applied to a controversial question in astrobiology: does the fact that life formed quite soon after the Earth constitute evidence that the formation of life is quite probable, given the right macroscopic conditions?

💡 Research Summary

The thesis brings together three distinct astrophysical investigations that share a common methodological backbone: Bayesian inference. The first study tackles the classic inverse problem of gravitational‑lens reconstruction. By treating the observed lensed image as a noisy projection of an unknown source brightness distribution and an unknown lens mass profile, the author builds a hierarchical Bayesian model. Priors encode physical expectations such as smoothness of the source, positivity of the mass density, and plausible ranges for lens ellipticity. The posterior distribution over source pixels and lens parameters is explored with a hybrid Metropolis‑Hastings/Gibbs Markov Chain Monte Carlo sampler. Convergence diagnostics and Bayesian evidence calculations allow the author to compare alternative mass models (e.g., singular isothermal sphere versus NFW) and to quantify uncertainties on both the reconstructed source and the lens mass map. The resulting reconstructions display higher fidelity and well‑characterized error bars compared with traditional regularized inversion techniques.

The second investigation applies Bayesian methods to asteroseismology, where the goal is to infer the number, frequencies, amplitudes, and phases of stellar oscillation modes from irregular, noisy time‑series photometry. The author formulates a parametric model that sums sinusoidal components, each described by a set of mode parameters. Priors reflect astrophysical knowledge about expected mode spacing and amplitude distributions. Model selection is performed by computing the Bayesian evidence for different numbers of modes, using either Reversible‑Jump MCMC or Nested Sampling. The posterior samples provide credible intervals for each mode’s frequency and amplitude, revealing low‑amplitude modes that would be missed by conventional Fourier analysis. The Bayesian evidence naturally penalizes over‑fitting, yielding an objective estimate of the most probable mode count.

The third part addresses a controversial question in astrobiology: does the rapid emergence of life on Earth imply that life is likely to arise given suitable planetary conditions? The author treats the probability of abiogenesis, (p), as an unknown parameter and adopts several plausible priors (uniform, log‑uniform, Beta distributions). The single observation that life appeared within a fraction (t_{\text{life}}/T) of the available habitable window is modeled as a Bernoulli trial. Applying Bayes’ theorem yields a posterior distribution for (p) that is highly sensitive to the chosen prior, illustrating the limited inferential power of a single data point. Sensitivity analyses and simulated exoplanet datasets demonstrate that, while the posterior median may suggest a moderate to high probability of life, the credible interval remains broad, underscoring the need for additional empirical constraints.

Across all three studies, the thesis showcases the power of hierarchical Bayesian modeling combined with advanced MCMC techniques to solve underdetermined inverse problems. By explicitly incorporating prior scientific knowledge, quantifying uncertainties, and using Bayesian evidence for model comparison, the author provides a coherent framework that outperforms traditional deterministic or frequentist approaches. The work not only delivers concrete scientific results—improved lens mass maps, more complete asteroseismic mode catalogs, and a nuanced probabilistic assessment of abiogenesis—but also sets a methodological precedent for future high‑precision astrophysical data analysis, especially as next‑generation observatories deliver ever larger and more complex datasets.