  If we accept Savage's set of axioms, then all uncertainties must be treated
like ordinary probability. Savage espoused subjective probability, allowing,
for example, the probability of Donald Trump's re-election. But Savage's
probability also covers the objective version, such as the probability of heads
in a fair toss of a coin. In other words, there is no distinction between
objective and subjective probability. Savage's system has great theoretical
implications; for example, prior probabilities can be elicited from subjective
preferences, and then get updated by objective evidence, a learning step that
forms the basis of Bayesian computations. Non-Bayesians have generally refused
to accept the subjective aspect of probability or to allow priors in formal
statistical modelling. As demanded, for example, by the late Dennis Lindley,
since Bayesian probability is axiomatic, it is the non-Bayesians' duty to point
out which axioms are not acceptable to them. This is not a simple request,
since the Bayesian axioms are not commonly covered in our professional
training, even in the Bayesian statistics courses. So our aim is to provide a
readable exposition the Bayesian axioms from a close rereading Savage's classic
book.
