In the latest Analysis, Darren
Bradley and Brandon Fitselsen argue that using confirmation theory can assist
understand what happens in the Monty Hall problem. The idea is that by assuming
that “the degree to which E confirms H is properly measured by the ration
Pr(H | E):Pr(H) of the posterior to the prior probability of H” we get
the nice conclusion that in all but a very special case Monty’s action
provides better evidence that you chose the wrong door than that you chose the
right door. (I assume some familiarity with the Monty Hall problem. If not,
read Bradley and Fitselsen’s paper — it’s a nice problem.)

The problem with all this is that we don’t need to
bring out anything so heavy duty as a contentious theory of confirmation to
get this. Assume that you choose door 3. There are four options remaining.

1. Monty will open door 1 and the prize is behind
   door 2.
2. Monty will open door 1 and the prize is behind
   door 3.
3. Monty will open door 2 and the prize is behind
   door 1.
4. Monty will open door 2 and the prize is behind
   door 3.

When Monty opens, say, door 2, options 1 and 2 are
removed. That is, one of the ways in which the prize could have been behind
door 3, one of the ways in which you could have made the right choice, is taken
away, but all of the ways in which the prize could have been behind (what is
now) the other door, door 1, are still live options. Unless that way in which
the prize could have been behind door 3, that is option 2, was not a live possibility,
which I suppose it isn’t ridiculous to equate with Pr(Option 2) = 0, it isn’t
at all surprising that Monty’s action better confirms the hypothesis that you’re
wrong than the hypothesis that you’re right.

(And, interestingly, Bradley and Fitselsen also note
that if Pr(Option 2) = 0 then by their lights Monty’s action doesn’t differentially
confrm the hypothesis that you’re wrong.)

So in this case I’m not sure what the heavy duty
confirmation theory is doing, except perhaps to confirm something entirely expected.

This approach is also held to validate a version
of the Doomsday argument, but not I think the most interesting version of the
Doomsday argument. What we care about in Doomsday is a de se proposition,
how long it is from now until Doom. What most versions of the argument, and
Bradley and Fitselsen are no different to most here, is show that evidence that
we are not that far from the beginning of time differentially confirms certain
pessimistic looking de dicto hypotheses about how many people that will
ever live. But, in their case at least, everything they say is consistent with
evidence that we are not that far from the beginning of time having no effect
whatsoever on our future-directed de se beliefs, and only affecting the
de dicto attitudes by altering our past-directed de se attitudes.

It isn’t too hard to see this is exactly what their
argument must show, when we notice that their reasoning in support of
Doomsday arguments would apply equally well to an agent who had been told by
God that there would be exactly 100 billion more people in the future of the
world as it would to agents like us with no such knowledge.