A major focus of this paper is the demonstration of several
significant problems with the degree-of-belief interpretation (dob) of
probability as commonly employed. Some of them can be synopsized in
the following five points:
i) The dob interpretation is not necessary, mathematically. This is
because (for example) quantum mechanics provides us with perfectly
consistent, objective, non-dob probabilities.
ii) Because there exist those objective physical probabilities, and
since in our universe those probabilities are the ones that ultimately
determine the utility value associated with any action, it *is*
logically necessary that any mathematics involving the dob
interpretation distinguish between the objective probabilities and the
dob probabilities.
Failure to do so is perhaps the single most significant
oversight of conventional Bayesian statistics.
iii) The mathematics ensuing from (ii) proves that in any dob-based
exercise in statistical inference, expected utility is given by a
(usually non-Euclidean) inner product. This product is between the
posterior of the objective probabilities and the posterior of one's
dob probabilties.
iv) Even within the dob interpretation, there is no reason to
encapsulate one's "prior knowledge" in the prior. It is just as
legitimate (i.e., just as consistent with Bayes' theorem) to
encapsulate it directly in the posterior.
v) (iv) and (iii) provide the reconciliation between "Bayesian" and
"Non-Bayesian" statistics: Non-Bayesians implicitly encapsulate their
prior knowledge directly into the posterior rather than the prior (as
do those who call themselves "Bayesians"). This will result in
better/worse performance than a particular Bayesian alternative based
solely on the two associated inner products (cf. (iii). In other
words, whether a "Bayesian" or a "Non-Bayesian" wins in any particular
bake-off is determined by how well the associated encapsulations of
prior knowledge "line up" with the posterior over the objective
probabilities determining the actual outcome.
David Wolpert
Data Understanding Group
NASA Ames Research Center
MS 269-1, Moffett Field, CA 94035, USA
dhw@ptolemy.arc.nasa.gov
650-604-3362 (v)
650-604-3594 (f)