This study discusses in detail the inversion of the Apollo lunar seismic data and the question of how to analyze the results. The well-known problem of estimating structural parameters (seismic velocities) and other parameters crucial to an understanding of a planetary body from a set of arrival times is strongly nonlinear. Here we consider this problem from the point of view of Bayesian statistics using a Markov chain Monte Carlo method. Generally, the results seem to indicate a somewhat thinner crust with a thickness around 45 km as well as a more detailed lunar velocity structure, especially in the middle mantle, than obtained in earlier studies. Concerning the moonquake locations, the shallow moonquakes are found in the depth range 50--220 km, and the majority of deep moonquakes are concentrated in the depth range 850--1000 km, with what seems to be an apparently rather sharp lower boundary. In wanting to further analyze the outcome of the inversion for specific features in a statistical fashion, we have used credible intervals, two-dimensional marginals, and Bayesian hypothesis testing. Using this form of hypothesis testing, we are able to decide between the relative importance of any two hypotheses given data, prior information, and the physical laws that govern the relationship between model and data, such as having to decide between a thin crust of 45 km and a thick crust as implied by the generally assumed value of 60 km. We obtain a Bayes factor of 4.2, implying that a thinner crust is strongly favored.