  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?
