In this paper we review the concepts of Bayesian evidence and Bayes factors, also known as log odds ratios, and their application to model selection. The theory is presented along with a discussion of analytic, approximate and numerical techniques. Specific attention is paid to the Laplace approximation, variational Bayes, importance sampling, thermodynamic integration, and nested sampling and its recent variants. Analogies to statistical physics, from which many of these techniques originate, are discussed in order to provide readers with deeper insights that may lead to new techniques. The utility of Bayesian model testing in the domain sciences is demonstrated by presenting four specific practical examples considered within the context of signal processing in the areas of signal detection, sensor characterization, scientific model selection and molecular force characterization.

EXONEST is an algorithm dedicated to detecting and characterizing the photometric signatures of exoplanets, which include reflection and thermal emission, Doppler boosting, and ellipsoidal variations. Using Bayesian inference, we can test between competing models that describe the data as well as estimate model parameters. We demonstrate this approach by testing circular versus eccentric planetary orbital models, as well as testing for the presence or absence of four photometric effects. In addition to using Bayesian model selection, a unique aspect of EXONEST is the potential capability to distinguish between reflective and thermal contributions to the light curve. A case-study is presented using Kepler data recorded from the transiting planet KOI-13b. By considering only the non-transiting portions of the light curve, we demonstrate that it is possible to estimate the photometrically-relevant model parameters of KOI-13b. Furthermore, Bayesian model testing confirms that the orbit of KOI-13b has a detectable eccentricity.

We present new ways to identify single and multiple moons around extrasolar planets using planetary transit timing variations (TTVs) and transit duration variations (TDVs). For planets with one moon, measurements from successive transits exhibit a hitherto undescribed pattern in the TTV-TDV diagram, originating from the stroboscopic sampling of the planet's orbit around the planet-moon barycenter. This pattern is fully determined and analytically predictable after three consecutive transits. The more measurements become available, the more the TTV-TDV diagram approaches an ellipse. For planets with multiple moons in orbital mean motion resonance (MMR), like the Galilean moon system, the pattern is much more complex and addressed numerically in this report. Exomoons in MMR can also form closed, predictable TTV-TDV figures, as long as the drift of the moons' pericenters is sufficiently slow. We find that MMR exomoons produce loops in the TTV-TDV diagram and that the number of these loops is equal to the order of the MMR, or the largest integer in the MMR ratio. We use a Bayesian model and Monte Carlo simulations to test the discoverability of exomoons using TTV-TDV diagrams with current and near-future technology. In a blind test, two of us (BP, DA) successfully retrieved a large moon from simulated TTV-TDV by co-authors MH and RH, which resembled data from a known Kepler planet candidate. Single exomoons with a 10% moon-to-planet mass ratio, like to Pluto-Charon binary, can be detectable in the archival data of the Kepler primary mission. Multi-exomoon systems, however, require either larger telescopes or brighter target stars. Complementary detection methods invoking a moon's own photometric transit or its orbital sampling effect can be used for validation or falsification. A combination of TESS, CHEOPS, and PLATO data would offer a compelling opportunity for an exomoon discovery around a bright star.

Planets emit thermal radiation and reflect incident light that they recieve from their host stars. As a planet orbits it's host star the photometric variations associated with these two effects produce very similar phase curves. If observed through only a single bandpass this leads to a degeneracy between certain planetary parameters that hinder the precise characterization of such planets. However, observing the same planet through two different bandpasses gives one much more information about the planet. Here, we develop a Bayesian methodology for combining photometry from both Kepler and the Transiting Exoplanet Survey Satellite (TESS). In addition, we demonstrate via simulations that one can disentangle the reflected and thermally emitted light from the atmosphere of a hot-Jupiter as well as more precisely constrain both the geometric albedo and dayside temperature of the planet. This methodology can further be employed using various combinations of photometry from the James Webb Space Telescope (JWST), the Characterizing ExOplanet Satellite (CHEOPS), or the PLATO mission.

Presented here is an independent re-analysis of the Kepler light curve of Kepler-91 (KIC 8219268). Using the EXONEST software package, which provides both Bayesian parameter estimation and Bayesian model testing, we were able to re-confirm the planetary nature of Kepler -91b. In addition to the primary and secondary eclipses of Kepler -91b, a third dimming event appears to occur approximately 60 o away (in phase) from the secondary eclipse, leading to the hypothesis that a Trojan planet may be located at the L4 or L5 Lagrange points. Here, we present a comprehensive investigation of four possibilities to explain the observed dimming event using all available photometric data from the Kepler Space Telescope, recently obtained radial velocity measurements, and N-body simulations. We find that the photometric model describing Kepler -91b and a Trojan planet is highly favored over the model involving Kepler -91b alone. However, it predicts an unphysically high temperature for the Trojan companion, leading to the conclusion that the extra dimming event is likely a false-postive.