The Solar System with its variety of planets and minor bodies, is it an exceptional entity in the universe, or is it perfectly normal to find planets orbiting around stars? Until the announcement of 51~Pegasi~b in 1995---the first solid discovery of a planet orbiting a main-sequence star other than Sun---the first answer was considered the most likely. The discovery gave birth to a new field of astrophysics,
the research of extrasolar planets and planetary systems.

Our understanding of the formation and evolution of planets and planetary systems has expanded rapidly since 1995, and this expansion is only accelerating.
Today we know that planets are common around main-sequence stars, and that small, Earth-sized, planets are especially plentiful. We have learned that the variety of planets is much wider than what was expected based on the planets in the Solar System. Gas-giants on extremely short-period orbits, planetary systems with almost all possible stable orbits occupied, planets on highly eccentric orbits, bloated planets with low densities, and small evaporating comet-like objects with rapidly varying dust halos and tails, were all unanticipated before they were first discovered.

The new discoveries have emerged with the help of the latest improvements in the analysis methods and instrumentation. Dedicated ground- and space-based instruments have yielded a vast number of planet discoveries, and we have currently reached the point where we can already start making inferences based on the known exoplanet population as a whole. The space-based CoRoT and Kepler missions have provided us with a clear picture of occurrence rates of planets with orbital periods from several hours to a year. Further, long-lasting radial velocity search programs with observations over tens of years are finding long-period planets not detectable by other methods, and the direct imaging searches have the potential of finding planets on even wider orbits.

In order to understand the variety and evolution of planetary systems based on the population of known extrasolar planets, we first need to understand the properties of the individual planets. Further, we need to be secure that the planets included into that population are indeed planets, and not something else. Several astrophysical phenomena, such as eclipsing binaries diluted by light from an unresolved third star, can mimic signals used to identify planets. Candidate confirmation is traditionally done using radial velocity measurements, which often require remarkable amounts of observation time and the use of large telescopes. Alternative confirmation methods could increase the number of planets available for population-based studies, and probabilistic methods combining different forms of evidence have lately gained wide acceptance.

This thesis has two principal scientific themes and covers three major goals. The two themes are the planet characterisation using Bayesian methods and the candidate confirmation and assessment using methods complementary to radial velocity observations.
The goals, as stated, are threefold. My first goal is to contribute to the exoplanet science in itself, focusing on the physical properties of extrasolar planets and their host stars. My second goal is to further advance the use of Bayesian statistics in astrophysical parameter estimation and model selection problems. The theory and basic methods of Bayesian statistics are well-established, but their application in astrophysics has only started gathering momentum during the last decade. Finally, my third goal is to improve the numerical methods needed in the analyses presented here. Bayesian methods are often computationally heavy, and fast, well-designed numerical methods can make a significant difference when deciding on the complexity of the analysis, given the computation and time constraints.

In this work, I first present an introduction to extrasolar planets and Bayesian statistics. I continue with separate chapters dedicated on different science cases, each based on one or more peer-reviewed publications. First, I consider the use of Bayesian model selection in the search for secondary eclipses in photometric time series. These are an example of weak astrophysical signals buried within non-normal correlated noise from many different sources. In the next chapters I present the characterisation of several new exoplanets, based on a joint analysis combining photometric time series, radial velocity observations, and information from spectroscopic stellar characterisation as well as theoretical modelling. Along with the main scientific theme of planet characterisation, I study how different assumptions made about the noise properties affect the analyses, consider improvements to an eclipse search method based on Bayesian model selection, search for star spot crossings and transit timing variations, and study the use of multicolour photometry in planet candidate confirmation and blending analysis.