In the latest edition of Nous, (subscription required) Cory Juhl replies to Roger White’s discussion of the fine-tuning argument. While I am sympathetic to Juhl’s side of the debate, I think there are some missteps in his application of probability theory here. Many more details below the fold.
Juhl starts off with three cases.
Case 1: Suppose that we are told that a fair coin will be flipped. If the result
of the flip is heads, a roulette wheel will be spun exactly once on that night. If the result is tails, the wheel will be spun hundreds of times on that night. We then learn that the result of some spin was 3 (i.e., that there exists a spin whose result was 3). We are asked to bet on whether many spins were made (whether the coin landed tails), or whether the wheel was spun only once.
For concreteness, let’s say the wheel will be spun exactly 136 times if the coin lands tails. Juhl argues, correctly, that in this case we should conclude that the coin probably landed tails. In fact, our posterior credence that the coin landed tails should be almost exactly 37/38. (For the sake of numerical simplicity, I’ll pretend that the probability of at least one spin landing 3 in 136 tries is 37/38. In fact it’s closer to 73/75. We can ignore these complications in what follows.) So far so good. Now for the second example.
Let us suppose that after the gambler is shown a spin resulting in 3, he is shuffled out of the room and told that a fair coin was tossed, and if it came up heads, there was only one spin that night, whereas if it came up tails, many spins would take place. He is then asked to bet on whether one or many spins would take place (whether they might have taken place before, took place during, or would occur afterwards will be left unspecified). The gambler reasons as before: a 3 result would be unlikely if only one spin occurs that night, but highly probable otherwise, so he infers that many spins take place. But the reasoning here seems fallacious. What the gambler witnessed was not merely that some spin resulted in a 3, but that a particular spin, call it s, resulted in 3. We can stipulate (and have the gambler be given the information) that whether s results in 3 is probabilistically independent of whether other spins take place. What this comes to is that
(1) P(s3 | M) = P(s3 | O)
where M is the ‘many-spins’ hypothesis, O is the ‘one-spin’ hypothesis, and
s3 is the event that spin s results in a 3.When we stipulate this, it is clear that
the posteriors of M and O (conditionalizing on s3) remain the same as their
priors, namely 1/2.
This is already too quick, for reasons we’ll get back to below. But let’s wait and see what follows.
Now here is a really counterintuitive (initially) result. The gambler is first told, as in case 1, that a 3 resulted on some spin. He correctly infers that many spins took place (that is, the inference is not a fallacy; it could of course turn out nevertheless that the conclusion of the inference is false). Then someone says, ‘You want to see one? We have it on video.’ He is shown a video in which s results in 3. This extra information s3 defeats his first inference, so to speak, given his commitment to (1) (we are ignoring complications as to whether (1) simply has to be true or whether the gambler must know that (1), be justified in believing that (1), and so on). The gambler can no longer justifiably assign greater posterior weight to hypothesis M than he had prior to any further data, if he takes (1) to hold. (For simplicity, we are taking the (relevant) information acquired to be s3, and not, for example, that the gambler was shown a video showing that s3.)
Surprisingly, intuition is correct here. (This is surprising because in tricksy probabilistic cases, intuition is normally reliably counter-indicative.) Being shown the video can’t make a difference to the case. What has gone wrong? One hint is in the last line.
Juhl wants to assume that the gambler being shown the video is not relevant information. But this assumption simply can’t be made, at least without changing earlier parts of the case. It usually is crucial to bear in mind the difference between your evidence being that p, and your evidence being that someone is telling you (in a particular situation) that p, and this is no exception. We can’t just assume this away, any more than we could assume away, the fact that 38 is not prime. What evidence is and isn’t relevant is not a free variable in a story that we can just adjust to suit our purposes.
This distinction (between finding out that p, and finding out that someone is telling you that p) is fairly obvious to folk epistemology, but it is worth noting how striking its probabilistic consequences are. Here’s an old example (not one of mine) illustrating this.
A certain card game consists of Jill being dealt two cards from a deck consisting of just three cards, the ace of hearts, the ace of diamonds and the two of clubs. Jack is trying to figure out what cards Jill has. They have the following conversation.
Jill: I have at least one ace.
Jack: I knew that already – every possible hand includes at least one ace.
Jill: I can show you if you like.
Jack: It won’t tell you much.
Jill: That’s what you think
and Jill shows Jack the ace of hearts she is holding.
We need a dictionary to keep track of all the relevant propositions.
A1 = Jill has at least one ace
A2 = Jill has two aces
AH = Jill has the ace of hearts
AD = Jill has the ace of diamonds
SAH = Jill shows the ace of hearts
SAD = Jill shows the ace of diamonds
The relevant probabilities go as follows
Pr(A1) = 1
Pr(A2) = 1/3
Pr(A2 | A1) = 1/3
Pr(A2 | AH) = 1/2
Pr(A2 | AD) = 1/2
Hopefully that should all be self-explanatory. So when Jack sees the ace of hearts, should his credence in A2 go up to 1/2? No, because the relevant probability is not Pr(A2 | AH), but Pr(A2 | SAH). And that can be calculated as follows (assuming Jill is going to show an ace one way or the other, and will pick an ace at random to show if she has both aces.)
Pr(SAH | AH & ~A2) = 1
Pr(SAH | AD & ~A2) = 0
Pr(SAH | A2) = 1/2
Pr(A2) = Pr(AH & ~A2) = Pr(AD & ~A2) = 1/3
Hence, Pr(A2 | SAH) = 1/3
So although AH moves Jack’s credences, SAH does nothing at all to Jack’s credences. It would be hard to know what one would make of the following story if I told it, and added in an afterthought “Assume the only relevant evidence is AH, and not SAH.” I think if I did that I’d be contradicting myself, and there’s not much to say (in this context) about contradictory stories.
Is this what is going on in Juhl’s case? It’s part of the story, though I suspect there’s also something deeper going on. Let’s get back to the second case. Why exactly is what the gambler sees not meant to be evidence for tails? Before we answer that, we need to be told a little more about the story. Compare the following story.
The gambler is told the initial setup – heads means one spin, tails means many spins. He has an 0.5 credence that there has been at least one spin already, and this is probabilistically independent of heads or tails. He is then shown a spin of the wheel, which happens to land on 3. What should his credence be that the coin landed tails?
Here’s the argument for this. Let H =heads, T = tails, E = there was an earlier spin. Here are the priors.
Pr(H&E) = Pr(T&E) = Pr(H&~E) = Pr(T&~E) = 0.25.
When the gambler sees the spin, no matter what it lands, he sees that H&E is not true, because then there wouldn’t be another spin to see. So he should conditionalise on ~(H&E), and Pr(T | H&~E) = 2/3.
Juhl says he wants to
stipulate (and have the gambler be given the information) that whether s results in 3 is probabilistically independent of whether other spins take place.
I think this is ambiguous. In one sense, it might mean that the objective chance of the spin resulting in 3 is probabilistically independent of whether other spins take place. But that’s not particularly relevant to the way the gambler shoudl update his credences. What Juhl needs is that whether s results in 3, and hence whether s exists, is independent of whether other spins take place. And by a generalisation of the previous case, we can see that that’s only true if the gambler’s prior credence in E is 0.
So the stipulation amounts to the claim that the gambler is told that this is the first spin of the wheel. And given that information, it’s true that the gambler shouldn’t change his credences in H and T.
The same goes for the spin the gambler is shown in case 3. Unless the gambler’s prior credence that the spin is the first spin is 1, the fact that the spin exists is evidence (potentially strong evidence) in favour of T. Juhl later says that we should “Suppose, for simplicity, that s occurs no matter what”, and s’s occurence has probability 1, but how on earth this is supposed to be possible unless we are told in advance that s is the first spin is a mystery.
So as far as I can tell, there is no way to support Juhl’s contentions about these cases unless (a) we ignore the distinction between the evidence’s existing and our being shown it, and (b) we are told explicitly that s is the first spin of the wheel. Without those stipulations, the intuitive reaction is correct.
This generalises to his discussion of the fine-tuning argument. We only get out of the argument this way if (a) we ignore the anthropic argument point that there is an important epistemic difference between the universe being fine-tuned and our perceiving that it is fine-tuned, and (b) we are told this is the first universe that was created. Given those constraints, I’m sure most people will agree the fine-tuning argument doesn’t work.
Having said all that, I don’t think much of the fine-tuning argument. It seems to me to turn on some symmetry principles of dubious coherence, and dubious plausibility given coherence. But that’s a story for different day.