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The discovery of thousands of transiting exoplanet candidates by NASA's Kepler mission revolutionized the study of exoplanets, shifting the field beyond the characterization of individual systems towards mapping the true distribution of all planetary systems and probing population-level trends. Inferring the underlying architectures of planetary systems from the Kepler data requires a detailed understanding of the detection pipeline and statistical methods. In this dissertation talk, I combine a forward modeling framework (SysSim) with approximate Bayesian computation to develop and test population models for the intrinsic distributions of planetary systems.
 
The properties of both single and multi-transiting systems can be combined to make powerful inferences on the underlying inter- and intra-system correlations. I define a series of distance functions for comparing models to the Kepler data, including the distributions of observed multiplicities, period ratios, transit depth ratios, and transit duration ratios, as well as complexity metrics designed to capture system-level patterns. I also train a Gaussian process emulator for rapidly constraining model posterior distributions. In separating detection biases from the physical patterns, I show evidence for the "peas-in-a-pod" patterns: that planets in multi-planet systems tend to be highly clustered in their radii, show a preferential size ordering, and exhibit remarkably uniform spacings. I also show that inner planetary systems are more common around later type stars across FGK dwarfs, although their architectures may be similar.
 
By parameterizing the mutual inclination distribution using Rayleigh distributions, I show that a mixture of a low and a high mutual inclination population is necessary to fit the Kepler-observed multiplicity distribution. I then describe a method of drawing clustered planetary systems at the angular momentum deficit (AMD) stability limit. This maximum AMD model produces additional correlations between the orbital excitations (eccentricities and mutual inclinations) and the multiplicities of each system. These results lead to observational predictions that are seen in the transit duration ratios and the number of transit duration variation detections, clarifying the long-standing "Kepler dichotomy" problem (an apparent excess of transiting singles) and providing a link to its dynamical nature.
 
Finally, I demonstrate how our knowledge of population-wide architectures can be leveraged to make predictions about the presence of additional planets conditioned on a given (e.g., transiting) planet. While unseen planets may be discovered with additional methods and provide a more complete picture of a system's architecture, they can also hinder the interpretation of known planets in radial velocity (RV) data. I perform simulations of RV surveys to quantify how many additional observations are needed to accurately measure the semi-amplitude of transiting planets, which is the standard strategy for follow-up characterization of planets from the TESS mission.

Statistical Inference on the Distribution of Exoplanetary Systems: Correlations in Planetary System Architectures (PhD defense talk)