Bayesian update on evidence *E* is transitioning from a credence
function *P* to the credence
function *P*(⋅∣*E*).
Anti-Bayesian update on *E* is
moving from *P* to *P*(⋅∣*E*^{c})
(where *E*^{c}
is the complement of *E*).
Whether one thinks that Bayesian update is rationally required, it is
clear that Bayesian update is better than anti-Bayesian update.

But here is a fun fact (assuming the Axiom of Choice). For any
scoring rule on an infinite space, there is a finitely additive
probability function *P* and an
event *E* such that 0 < *P*(*E*) < 1 where
*P*(⋅∣*E*) and *P*(⋅∣*E*^{c})
get exactly the same score everywhere in the probability space. It
follows that when dealing with finitely additive probabilities on
infinite spaces, a scoring rule will not always be able to distinguish
Bayesian update from anti-Bayesian update. This is a severe limitation
of scoring rules as a tool for evaluating the accuracy of a credence
function in infinite cases.

Here’s a proof of the fun fact. Let *s* be a scoring rule. Say that two
credence functions are maximally opinionated provided that they assign
0 or 1
to every event. It
is known that then there are two different maximally opinionated
finitely additive probability functions *p* and *q* such that *s*(*p*) = *s*(*q*)
everywhere. Let *P* = (*p*+*q*)/2 be
their average. Let *E* be an
event such that *p*(*E*) = 1 and *q*(*E*) = 0 (such an event
exists because *p* and *q* are maximally opinionated and yet
different). Then *P*(⋅∣*E*) = *p* and
*P*(⋅∣*E*^{c}) = *q*
while *P*(*E*) = 1/2.
Hence conditionalization on *E*
and *E*^{c} has
exactly the same score.

One might take this as some evidence that *finite* additivity
is not good enough.