Elga’s Highly Restricted Principle of Indifference

In the Sleeping Beauty paper, Elga tells us that “Since being in [Tails and Monday] is subjectively just like being in [Tails and Tuesday], and since exactly the same propositions are true whether you are in [Tails and Monday] or [Tails and Tuesday], even a highly restricted principle of indifference yields that you ought then to have equal credence in each”.

Recall that the unrestricted Principle of Indifference says that when your evidence doesn’t give you any more reason to believe one proposition rather than another, you should assign credence to the possibilities equally.

The more restricted principle Elga seems to be endorsing here is this:

Highly Restricted POI: If some collection of situations are subjectively identical and exactly the same uncentered propositions are true at them, one ought to divide one’s credences among them.

The regular POI is famously plighted by a worry that it delivers inconsistent dictates. Consider the Cube Factory example from Van Fraassen: All you know about some object is that it is a cube and is less than 2 units wide.  So, {It’s less than one unit wide, It’s not less than one unit wide} partitions the possibilities.  So, by POI, you should have .5 credence in it being less than one unit wide.  {Its side area is less than one unit, Its side area is at least one and less then 2 units, Its side area is at least 2 and less than 3 units, Its side area is at least 3 and less than 4 units} also partitions the space. So, by POI, you should have credence .25 that its side area is less than 1 unit.  But, of course, its side area is less than one unit iff it’s less than one unit wide, so POI instructs you to have distinct credences in the one proposition.

Choosing one partition over another to divide’s credences among would have the agent importing more information than she has, according to the Imprecise Bayesian (Levi (1980), Jeffrey (1983), Kaplan (1996), and Joyce (2005, p. 171)).   For this reason, the Imprecise Bayesian rejects POI.   I think the Imprecise Bayesian ought to reject Highly Restricted POI for the same reason they reject POI: employing it has the agent believe as though she has more information than she does.  This case shows that:

Actual Cube Factories: An agent is certain that she is at some point on a line of continuum-many cube factories where the side length of the cubes produced in the factories ranges from 0 meters on one end to 2 meters on the other (including every real number possibility).  She doesn’t know where she is on the line as all of the spots on the line look the same.  What should be the agent’s credence that the factory in front of her makes cubes of side length between 0 and 1 meter?

Here, the agent is certain of the uncentered propositions in play, but she is uncertain of where she is in those worlds.  Here, Highly Restricted POI gives contradictory dictates.  The centered possibility where she is in front of a factory making cubes of side lengths 0 to 1 is subjectively identical to the one in which she is in front of a factory making cubes of side lengths 1 to 2.  So, by Highly Restricted POI, she should have credence 1/2 in the first possibility.  But, the centered possibilities where she is in front of factories making cubes of side areas 0 to 1, 1 to 2, 2 to 3, 3 to 4 are also subjectively identical.  Hence she should also have credence 1/4 in the side area being between 0 and 1.  But, the side area between 0 and 1 is just the possibility where it has side length between 0 and 1.  So, the Highly Restricted POI gives also gives inconsistent dictates, and the Imprecise Bayesian ought to reject it for the same reason they reject the regular POI.

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11 Responses to Elga’s Highly Restricted Principle of Indifference

  1. Shen-yi Liao says:

    To start, I love that you made a post!

    That said, I don’t agree that, in your Actual Cube Factory case, the agent is certain of the uncentered proposition in play. Let me say why.

    Here is how Hajek sets up the paradox in the SEP article on probability:

    The following example (adapted from van Fraassen 1989) nicely illustrates how Bertrand-style paradoxes work. A factory produces cubes with side-length between 0 and 1 foot; what is the probability that a randomly chosen cube has side-length between 0 and 1/2 a foot? The tempting answer is 1/2, as we imagine a process of production that is uniformly distributed over side-length. But the question could have been given an equivalent restatement: A factory produces cubes with face-area between 0 and 1 square-feet; what is the probability that a randomly chosen cube has face-area between 0 and 1/4 square-feet? Now the tempting answer is 1/4, as we imagine a process of production that is uniformly distributed over face-area. This is already disastrous, as we cannot allow the same event to have two different probabilities (especially if this interpretation is to be admissible!). But there is worse to come, for the problem could have been restated equivalently again: A factory produces cubes with volume between 0 and 1 cubic feet; what is the probability that a randomly chosen cube has volume between 0 and 1/8 cubic-feet? Now the tempting answer is 1/8, as we imagine a process of production that is uniformly distributed over volume. And so on for all of the infinitely many equivalent reformulations of the problem (in terms of the fourth, fifth, … power of the length, and indeed in terms of every non-zero real-valued exponent of the length). What, then, is the probability of the event in question?

    A crucial component of the paradox — what allows the conflicting applications of POI — is that there are two possible ways the production process is uniformly distributed, over length and over area.

    Let’s get back to your Actual Cube Factories case.

    First, suppose the world is such that the factories are uniformly distributed over side length. Then the centered possibilities where the agent is in front of a factory making cubes of side lengths 0 to 1 is subjectively identical to the one in which she is in front of a factory making cubes of side lengths 1 to 2. However, it is not the case that the centered possibilities where she is in front of factories making cubes of side areas 0 to 1, 1 to 2, 2 to 3, 3 to 4 are also subjectively identical. Possible individuals in this world are not uniformly distributed across propositions about side length. The agent should have credence .5 in I am in front of a factory making cubes of side lengths 0 to 1 and credence .5 in I am in front of a factory making cubes of side area 0 to 1. No paradox.

    Second, suppose the world is such that the factories are uniformly distributed over side area. Then, the centered possibilities where the agent is in front of factories making cubes of side areas 0 to 1, 1 to 2, 2 to 3, 3 to 4 are subjectively identical. However, it is not the case that the centered possibilities where the agent is in front of a factory making cubes of side lengths 0 to 1 is subjectively identical to the one in which she is in front of a factory making cubes of side lengths 1 to 2. Possible individuals in this world are not uniformly distributed across propositions about side length. The agent should have credence .25 in I am in front of a factory making cubes of side area 0 to 1 and credence .25 in I am in front of a factory making cubes of side lengths 0 to 1. No paradox.

    Now, what seems to me to be the case is that the agent doesn’t know which world she is in. Is she in the uniform-distribution-over-side-length world or the uniform-distribution-over-side-area world? That is, she is not certain which of the corresponding uncentered proposition is true in her world.

    That is not to say there aren’t some other pressures on those who reject POI to also reject Highly Restricted POI, but I am not convinced the current example is forceful.

  2. Shen-yi Liao says:

    Boy, that was terse.

  3. jpkonek says:

    Sam,
    I’m a little unclear about the suppositions in each of your two cases. As Dan set up the problem, there are continuum-many cube factories and continuum-many individuals. I thought that was sufficient as a description of the state of the world. What are you supposing when you suppose that the factories are uniformly distributed over side length, or uniformly distributed over area? It seems like you want this to affect how many possible individuals there on certain portions of the line. But there are continuum many individuals on each such portion. There are continuum many individuals in front of factories making cubes of side lengths 0 to 1, continuum many individuals in front of factories making cubes of side lengths 1 to 2, continuum many individuals in front of factories making cubes of side lengths 0 to 1, continuum many individuals in front of factories making cubes of area 0 to 1, continuum many individuals in front of factories making cubes of area 1 to 2, and so on. Your distributions, then, don’t affect how many possible individuals there on different portions of the line. So what are they? Are they chance distributions? I’m just a little unclear.

    And even if this case is problematic in a way I don’t see, aren’t the cases easily multipliable? Suppose I know it’s between noon and 1, but have no idea whether it’s any of the continuum-many times in between. If I partition by seconds and distribute equally, I get one credal distribution. If I partition by seconds squared, I get another, incompatible distribution.

    • Shen-yi Liao says:

      Maybe I don’t understand POI, but I do think the uniform-distribution bit is important because of something in the neighborhood of “having the same number of individuals”, even though that’s not the technically right way to put it. Take Dan’s scenario. There are as many factories making cubes of side lengths 0 to 0.0000000001 and factories making cubes of side lengths 0.0000000001 to 2. Surely POI would not tell us we should place the same subjective probability in the two corresponding propositions? If it would, then it seems that the weirdness comes from distributing probability over infinity than anything else.

      I think I’d say the same thing about the time case, though it certainly sounds weirder. It depends on how moments are uniformly distributed. Over seconds, or over seconds squared?

    • Shen-yi Liao says:

      To put it another way: the paradox would be much easier to generate if it were just about the same number of individuals. There are same number of individuals who are standing in front of factories making cubes of side lengths 0 to 1 and making side lengths 1 to 2. But there are also the same number of individuals who are standing in front of factories making cubes of side lengths 0 to 1, making side lengths 1 to 1.5, and making side lengths 1.5 to 2. Paradox???

      That the original formulation of the paradox goes through things like side length vs side area gives us reason to think the paradox had better not be generated just from weirdness about infinity. It depends on how the individuals are distributed — but that is uncentered information.

    • Yeah, I think you’re right, Jason. The notion of distribution doesn’t make much sense here for the reasons you say.

      Sam, the problem may just be a problem that only occurs in infinite case, but why does it matter? There’s always going to be some property P such that all problematic says have property P (namely, being a problematic case). So, why, in this case, can we say that the weirdness is due to having that property instead of just the falsity of POI?

  4. jdmitrig says:

    Dan,

    The “Highly Restricted POI” isn’t the restricted POI from the Dr. Evil paper, right?

    It looks like you’re trying to charitably attribute some restricted principle of indifference to Elga on the basis of his off-the-cuff remark in the Sleeping Beauty paper. But then it seems to me that the principle isn’t restricted enough. Elga doesn’t need the principle to apply to random variables which take on continuum-many values, so why have it apply to them? Why not interpret Elga as leaning on the following (even more highly) restricted principle of indifference:

    If some *discrete* collection of situations are subjectively identical and exactly the same uncentered propositions are true at them, one ought to divide one’s credence among them evenly.

    And then the principle is weak enough that the typical Bertrad paradoxes don’t arise, yet strong enough to get the thirder answer in sleeping beauty.

    So why attribute the stronger principle to Elga, when it’s far more than he needs?

    ——————

    Also, I’m really confused by your counterexample. Aren’t the propositions about the size of the cube *uncentered* propositions? And if so, doesn’t that mean that the set of worlds you’re suggesting the agent spread their credence over differ with respect to what uncentered propositions are true at them? And doesn’t that mean that this set of worlds fails to satisfy the antecedent of the highly restricted principle of indifference?

    • jdmitrig says:

      Disregard my second comment. I see what you were going for now.

    • Yeah, Dave Chalmers also suggested this, that the right move is to restrict it only to finite cases. I’m worried that the move is ad hoc though. I also wouldn’t be surprised if a similar problem arises in the finite case. Dan Greco mentioned that someone has already shown that finite POI generates violations of additivity.

  5. jdmitrig says:

    Do you really think that it’s *possible* for there to be continuum-many factories, all lined up in a row?

  6. Shen-yi Liao says:

    @Dan

    You say, “Sam, the problem may just be a problem that only occurs in infinite case, but why does it matter? There’s always going to be some property P such that all problematic says have property P (namely, being a problematic case). So, why, in this case, can we say that the weirdness is due to having that property instead of just the falsity of POI?”

    I don’t think the infinity feature is merely accidental or derivative. Instead, I claim that the supposed problem is generated from infinity AND an underspecification of the distribution of chance. With finite possible individuals, it is not possible to underspecify the distribution of chance in the same way. The number of individuals under a certain description and the number of individuals total are enough to give you that information.

    Again, I don’t want to defend POI. I think it’s problematic. Whatever good story we can tell about distribution of chance, I think the information about distribution of chance is uncentered information. So, insofar as the agent is ignorant because she’s ignorant of the distribution of chance, that is not an issue for Elga’s HRPOI, as you’ve stated it.

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