The starting point in modern Bayesian probability theory is that probability is interpreted as a degree of belief (for bibliographic notes, see [7,57]). Richard Cox showed that certain very general requirements for the calculus of beliefs result in the rules of probability theory . Decision theory also leads to the same rules [129,114,102] with the same interpretation. There are other domains, most notably measure theory, where the same rules appear, but from the point of view of learning systems and decisions in the face of uncertainty, degree of belief is the appropriate interpretation.
Beliefs are always subjective, and therefore all the probabilities appearing in Bayesian probability theory are conditional. In particular, under the belief interpretation probability is not an objective property of some physical setting, but is conditional to the prior assumptions and experience of the learning system. It is completely reasonable to talk about ``the probability that there is a tenth planet in the solar system'' although this planet either exists or does not exist and there is no sense in interpreting the probability as a frequency of observing a tenth planet. Sometimes the probabilities can be roughly equated with empirical frequencies, but this can be considered as a special case of the belief interpretation as was shown by Cox .
Accessible introductions to practical applications of Bayesian probability theory can be found, for instance, in [75,103,28].