The basis of all of our development up to this point has been the cluster ensemble, a discrete ensemble that generates every possible distribution of integers i with fixed zeroth and first order moments. Thermodynamics arises naturally in this ensemble when M and N become very large. In this chapter we will reformulate the theory on a mathematical basis that is more abstract and also more general. The key idea is as follows. If we obtain a sample from a given distribution h
, the distribution of the sample may be, in principle, any distribution h that is defined in the same domain. This sampling process defines a phase space of distributions h generated by sampling distribution h
. We will introduce a sampling bias via a selection functional W to define a probability measure on this space and obtain its most probable distribution. When the generating distribution h
is chosen to be exponential, the most probable distribution obeys thermodynamics. Along the way we will make contact with Information Theory, Bayesian Inference, and of course Statistical Mechanics.