On rereading Savage

TL;DR

This paper revisits Savage's axioms to clarify that subjective and objective probabilities are unified under his framework, emphasizing the importance of Bayesian axioms for statistical inference and learning.

Contribution

It provides a clear, accessible exposition of Savage's Bayesian axioms, highlighting their implications for subjective and objective probability integration.

Findings

Savage's axioms unify subjective and objective probability.

Bayesian updating is grounded in Savage's axioms.

Non-Bayesian views often reject the axiomatic basis of probability.

Abstract

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.