I’ve been reading through a lot of early papers on the
Sleeping Beauty problem recently. (Check out SleepingBeautyProblem.org!)One of the main issues is how the problem
relates to van Fraassen’s Reflection Principle. The principle states that your
present probability assignments should be the same as your known future
probability assignments (or at least lie within the range that spans what you
take your future assignments to possibly be).
So if you know that you’re going to watch a debate tomorrow on whether
P, and you’re going to come out thinking that probably P, you should already
think that probably P.
Moreover, your present probabilities conditional on having a
given future probability should be equal to those future probabilities:
Reflection: Pr(A|pr+) = Pr+(A) where ‘pr+’ = Pr+() is
your future probability assignment
Reflection has its faults, but there is something
intuitively right about it. It is odd
for there to be foreseeable rational changes in assignments – that is, it is
odd if there is some kind of evidence that I could get, such that I know that I
am going to get it and that it’s going to convince me of something, and not be
convinced of it already.
On the most popular answer to the Sleeping Beauty problem,
Thirdism, the Reflection Principle is violated. (It also violated by Lewis’s
version of Halfism). At some point Beauty should have some foreseen change of
probability – so Beauty knows that while she assigns ½ to HEADS on Sunday, she
knows she will later assign ½ (or ¾, for Halfers) to HEADS on Monday after
awakening (or after being told that it
is Monday).
Brad Monton suggested failure results from Beauty forgetting
her self-locating information. Van Fraassen’s statement of the Principle seems
to run into problems when we might forget things we had known previously. If
you know that you’re not going to remember what you ate for breakfast in a
year, that doesn’t mean you anything about what you should believe now. Beauty
does lose some kind of information. So Monton’s suggestion makes some sense.
However, I think that Monton missed an important part of the
story. There is something more substantive that can be said about why
Reflection fails in this case, and the failure highlights an important lesson
about deferential norms of probability.
A deferential norm of
probability tells you what probability to assign to a given proposition in light
of what you know about certain other special probability assignments. There are
three main deferential norms of probability, though the equal weight view in
the literature on disagreement might be seen as a fourth. The first is the Principal Principle. It tells you to
assign probabilities that align with known chances. If you know a coin toss
will land heads with a 50/50 chance, you should assign a 50% probability of it
doing so. The second is the Principle of
Reflection, which we’ve already seen. The third is the Rational Reflection principle. It tells you to assign probabilities
in line with known rational probabilities. If you believe that the rational
probability to assign to reality of global warming is .8, you should assign a
.8 probability.
Principal
Principle: Pr(A|ch) = Ch(A) where
‘ch’ = Ch() is the chance function
Reflection:
Pr(A|pr+) = Pr+(A) where
‘pr+’ = Pr+() is your future probability assignment
Rational
Reflection: Pr(A|rp) = RP(A) where ‘rp’ = RP() is the rational
probability assignment
The Principal Principle is known to face problems when
paired with certain reductive of objective chance. Suppose (for an over-simplified
example) that you reduce facts about chances to frequencies: a coin has a ½
chance of landing heads if it lands heads half the time. Then given that the
coin has a 50/50 chance of landing heads and will only be tossed twice, we can
conclude that it will land heads once and tails once. If it has already landed
heads, we know it will land tails. The Rational Reflection principle leads to interesting
difficulties in the clock puzzle.
There is a similar fix for both problems (see Elga, “The
Puzzle of the Unmarked Clock and the New Rational Reflection Principle” and Hall,
“Correcting the Guide to Objective Chance”), that properly understood will help
us understand why the Principle of Reflection fails in the Sleeping Beauty
case.
This fix is to make the
probability assignments conditional:
Prinicipal Principle: Pr(A|ch) =
Ch(A|ch) where ch is
the statement that Ch( ) is the chance function.
Rational
Reflection : Pr(A|rp) = RP(A|rp) where
rp is the statement that RP( ) is the rational probability assignment.
The general reason to be deferential to these probability
assignments is that they are more likely to be accurate. There is no reason to
withhold what we know from those whose judgment we should defer to. If we know
something about the chance function, we shouldn’t seclude it from the chance
function. If we know something about the rational probability function, we
shouldn’t keep it secret from the rational probability assumption.
This carries over to the Reflection Principle. It doesn’t
make sense to align one’s probabilities with ones known future probabilities if
one presently knows more than one will in the future. Instead, we should align
our present probabilities with known future probabilities conditional on what we now know. And moreover, what we assign in A conditional
on some probability assignment T being worthy of deference should equal T given that T is worthy of deference.
This may not be thought to help explain why Reflection fails
in Sleeping Beauty. After all, doesn’t Beauty know on Monday everything she
knew on Sunday? In one sense, this is correct: she doesn’t actually forget
anything between Sunday and Monday.
However, on Sunday Beauty knows more about her foreseen
situation-on-Monday than she does on Monday. On Monday, she doesn’t know that
it is Monday. On Sunday, she knows that the situation that she foresees that will
assign different probabilities than she presently assigns will be on Monday.
This isn’t something that she forgets. But she
later is unsure whether she is in the situation she foresaw she would be in.
The Reflection Principle must be amended to make sense of
the fact that we might now know more about the situation we foresee ourselves
being in than we know at the time. Let X summarize everything we know about our
status in that foreseen future situation and Y summarize everything else one
presently knows.
New Reflection:
Pr(A|pr+ in X) = Pr+(A |Y & X) where pr+ is the statement that Pr( ) is one’s
future probability assignment.
What happens to Sleeping Beauty? Well, Beauty only knows
that she will awaken on Monday, her present probability in HEADS conditional on
the fact that she will come to have a 1/3 probability shouldn’t be 1/3, because
in the only situation she foresees herself being in, she will have a ½ probability
in HEADS conditional on it being that situation (i.e. MONDAY).
Thus Pr(HEADS|pr+ on Monday) = Pr+(HEADS|MONDAY) = 1/2
One thing that is interesting about this explanation is that
it is available only to Thirders. For Halfers of the traditional Lewisian sort,
this explanation won’t be available. That is one more problem for Lewisian Halfers.
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