  In this thesis we start by providing some detail regarding how we arrived at
our present understanding of probabilities and how we manipulate them - the
product and addition rules by Cox. We also discuss the modern view of entropy
and how it relates to known entropies such as the thermodynamic entropy and the
information entropy. Next, we show that Skilling's method of induction leads us
to a unique general theory of inductive inference, the ME method and precisely
how it is that other entropies such as those of Renyi or Tsallis are ruled out
for problems of inference. We then explore the compatibility of Bayes and ME
updating. We show that ME is capable of producing every aspect of orthodox
Bayesian inference and proves the complete compatibility of Bayesian and
entropy methods. The realization that the ME method incorporates Bayes' rule as
a special case allows us to go beyond Bayes' rule and to process both data and
expected value constraints simultaneously. We discuss the general problem of
non-commuting constraints, when they should be processed sequentially and when
simultaneously. The generic "canonical" form of the posterior distribution for
the problem of simultaneous updating with data and moments is obtained. This is
a major achievement since it shows that ME is not only capable of processing
information in the form of constraints, like MaxEnt and information in the form
of data, as in Bayes' Theorem, but also can process both forms simultaneously,
which Bayes and MaxEnt cannot do alone. Finally, we illustrate some potential
applications for this new method by applying ME to potential problems of
interest.
