Inferences about the distribution, merger rate, and evolutionary processes of compact binaries from gravitational wave observations

We are living through the dawn of the era of gravitational wave astronomy. Our first glances through this new window upon the sky has revealed a new population of objects. Since it first began observing in late 2015, the advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) has detected gravitational waves three times, along with an additional strong candidate -- and there shall be orders of magnitude more in the years to come. In all four cases, the waveform\u27s signature is consistent with general relativity\u27s predictions for the merging of two black holes. Through parameter estimation studies, estimates on features such as the black holes\u27 masses and spins have been determined. At least two of the black hole pairs lie above the mass range spanned by comparable black holes observed through traditional means. This suggests they constitute a separate population, either too elusive or rare to be found with traditional telescopes. The most natural questions to ask about these black holes -- how did they form, how many of them are there, and how can they be categorized -- remain open ended. We know black holes can form when massive stars die, so it\u27s most natural to claim stars as their progenitors. Since we now know black holes merge into larger black holes, could it be the case that they formed from previous mergers? Were the two black holes part of a binary from their birth, or did they become coupled later on in life? The measurements provided by LIGO can help answer these questions and more. Throughout this thesis, I will describe and demonstrate results from a number of novel methods whose purpose is to better understand these black holes and their progenitors. At their heart, these methods give answers to a few, critical questions. a) What is the overall rate at which these objects merge? b) What is the range of values these objects\u27 properties can take, and how are they distributed? c) Given a number of physical models, how can we evaluate the performance of each relative to the others, and determine which gives the best description of reality