Plausibly, a Christian commitment prohibits hedging. Thus in some
sense even if one’s own credence in Christianity is less than 100%, one
should act “as if it is 100%”, without hedging one’s bets. One shouldn’t
have a backup plan if Christianity is false.

Understanding what this exactly means is difficult. Suppose Alice has
Christian commitment, but her credence in Christianity is 97%. If
someone asks Alice her credence in Christianity, she should not lie and
say “100%”, even though that is literally acting “as if it is 100%”.

Here is a more controversial issue. Suppose Alice has a 97% credence
in Christianity, but has the opportunity to examine a piece of evidence
which will settle the question one way or the other—it will make her
100% certain Christianity is true or 100% certain it’s not. (Maybe she
has an opportunity for a conversation with God.) If she were literally
acting as if her credence were 100%, there would be no point to looking
at any more evidence. But that seems the wrong answer. It seems to be a
way of being scared that the evidence will refute Christianity, but that
kind of a fear is opposed to the no-hedge attitude.

Here is a suggestion about how no-hedge decision-making should work.
When I think about my credences, say in the context of decision-making,
I can:

think about the credences as psychological facts about me,
or

regulate my epistemic and practical behavior by the credences
(use them to compute expected values, etc.).

The distinction between these two approaches to my credences is
really clear from a third-person perspective. Bob, who is Alice’s
therapist, thinks about Alice’s credences as psychological facts about
her, but does not regulate his own behavior by these credences: Alice’s
credences have a psychologically descriptive role for Bob but not a
regulative role for Bob in his actions. In fact, they probably don’t
even have a regulative role for Bob when he thinks about what actions
are good for Alice. If Alice has a high credence in the danger of
housecats, and Bob does not, Bob will not encourage Alice to avoid
housecats—on the contrary, he may well try to change Alice’s credence,
in order to get Alice to act more normally around them.

So, here is my suggestion about no-hedging commitments. When you have
a no-hedging commitment to a set of claims, you regulate your behavior
by them as if the claims had credence 100%, but when you take the
credences into account as psychological facts about you, you give them
the credence they actually have.

(I am neglecting here a subtle issue. Should we regulate our behavior
by our credences or by our opinion about our credences? I suspect that
it is by our credences—else a regress results. If that’s right, then
there might be a very nice way to clarify the distinction between
taking credences into account as psychological facts and taking them
into account as regulative facts. When we take them into account as
psychological facts, our behavior is regulated by our credences
*about the credences*. When we take them into account
regulatively, our behavior is directly regulated by the credences. If I
am right about this, the whole story becomes neater.)

Thus, when Alice is asked what her credence in Christianity is, her
decision of how to answer depends on the credence *qua*
psychological fact. Hence, she answers “97%”. But when Alice decides
whether or not to engage in Christian worship in a time of persecution,
her decision on how to answer would normally depend on the credence
*qua* regulative, and so she does not take into account the 3%
probability of being wrong about Christianity—she just acts as if
Christianity were certain.

Similarly, when Alice considers whether to look at a piece of
evidence that might raise or lower her credence in Christianity, she
does need to consider what her credence is as a *psychological*
fact, because her interest is in what might happen to her actual
psychological credence.

Let’s think about this in terms of epistemic utilities (or accuracy
scoring rules). If Alice were proceeding “normally”, without any
no-hedge commitment, when she evaluates the expected epistemic value of
examining some piece of evidence—after all, it may be practically costly
to examine it (it may involve digging in an archaeological site, or
studying a new language)—she needs to take her credences into account in
two different ways: psychologically when calculating the potential for
epistemic gain from her credence getting closer to the truth and
potential for epistemic loss from her credence getting further from the
truth, and regulatively when calculating the expectations as well as
when thinking about what is or is not true.

Now on to some fun technical stuff. Let *Ï•*(*r*,*t*) be the
epistemic utility of having credence *r* in some fixed hypothesis of
interest *H* when the truth
value is *t* (which can be 0 or 1).
Let’s suppose there is no as-if stuff going on, and I am evaluating the
expected epistemic value of examining whether some piece of evidence
*E* obtains. Then if *P* indicates my credences, the
expected epistemic utility of examining the evidence is:

*V*_{E} = *P*(*H*)(*P*(*E*|*H*)*Ï•*(*P*(*H*|*E*),1)+*P*(∼*E*|*H*)*Ï•*(*P*(*H*|*E*),1)) + *P*(∼*H*)(*P*(*E*|∼*H*)*Ï•*(*P*(*H*|*E*),0)+*P*(∼*E*|∼*H*)*Ï•*(*P*(*H*|∼*E*),0)).

Basically, I am partitioning logical space based on whether *H* and *E* obtain.

Now, in the as-if case, basically the agent has two sets of
credences: psychological credences and regulative credences, and they
come apart. Let *Î¨* and *R* be the two. Then the formula above
becomes:

*V*_{E} = *R*(*H*)(*R*(*E*|*H*)*Ï•*(*Î¨*(*H*|*E*),1)+*R*(∼*E*|*H*)*Ï•*(*Î¨*(*H*|∼*E*),1)) + *R*(∼*H*)(*R*(*E*|∼*H*)*Ï•*(*Î¨*(*H*|*E*),0)+*R*(∼*E*|∼*H*)*Ï•*(*Î¨*(*H*|∼*E*),0)).

The no-hedging case that interests us makes *R*(*H*) = 1: we regulatively
ignore the possibility that the hypothesis is false. Our expected value
of examining whether *E* obtains
is then:

*V*_{E} = *R*(*E*|*H*)*Ï•*(*Î¨*(*H*|*E*),1) + *R*(∼*E*|*H*)*Ï•*(*Î¨*(*H*|∼*E*),1).

Let’s make a simplifying assumption that the doctrines that we are
as-if committed to do not affect the likelihoods *P*(*E*|*H*) and *P*(*E*|∣*H*) (granted
the latter may be a bit fishy if *P*(*H*) = 1, but let’s suppose
we have Popper functions or something like that to take care of that),
so that *R*(*E*|*H*) = *Î¨*(*E*|*H*)
and *R*(*E*|∣*H*) = *Î¨*(*E*|∣*H*).

We then have:

*Î¨*(*H*|*E*) = *Î¨*(*H*)*R*(*E*|*H*)/(*R*(*E*|*H*)*Î¨*(*H*)+*R*(*E*|∼*H*)*Î¨*(∼*H*)).

*Î¨*(*H*|∼*E*) = *Î¨*(*H*)*R*(∼*E*|*H*)/(*R*(∼*E*|*H*)*Î¨*(*H*)+*R*(∼*E*|∼*H*)*Î¨*(∼*H*)).

Assuming Alice has a preferred scoring rule, we now have a formula
that can guide Alice what evidence to look at: she can just check
whether *V*_{E}
is bigger than *Ï•*(*Î¨*(*H*),1), which
is her current score regulatively evaluated, i.e., evaluated in the
as-if *H* is true way. If *V*_{E} is bigger,
it’s worth checking whether *E*
is true.

One might hope for something really nice, like that if the scoring
rule *Ï•* is strictly proper,
then it’s always worth looking at the evidence. Not so, alas.

It’s easy to see that *V*_{E} beats the
current epistemic utility when *E* is perfectly correlated with *H*, assuming *Ï•*(*x*,1) is strictly
monotonic increasing in *x*.

Surprisingly and sadly, numerical calculations with the Brier score
*Ï•*(*x*,*t*) = − (*x*−*t*)^{2}
show that if Alice’s credence is 0.97, then unless the Bayes’ factor is
very far from 1, current epistemic
utility beats *V*_{E}, and so
no-hedging Alice should not look at the evidence, except in rare cases
where the evidence is extreme. Interestingly, though, if Alice’s current
credence were 0.5, then Alice should *always* look at the
evidence. I suppose the reason is that if Alice is at 0.97, there is not
much room for her Brier score to go up assuming the hypothesis is
correct, but there is a lot of room for her score to go down. If we took
seriously the possibility that the hypothesis could be false, it would
be worth examining the evidence just in case the hypothesis is false.
But that would be a form of hedging.