I remember you linked me to Radford Neal’s paper (pdf) on Full Non-indexical Conditioning. I think FNC is a much nicer way to think about problems like these than SSA and SIA, but I guess you disagree?
To save others from having to wade through the paper, which is rather long, I’ll try to explain what FNC means:
First, let’s consider a much simpler instance of the Doomsday Argument: At the beginning of time, God tosses a coin. If heads then there will only ever be one person (call them “M”), who is created, matures and dies on Monday, and then the world ends. If tails then there will be two people, one (“M”) who lives and dies on Monday and another (“T”) on Tuesday. As this is a Doomsday Argument, we don’t require that T is a copy of M.
M learns that it’s Monday but is given no (other) empirical clues about the coin. M says to herself “Well, if the coin is heads then I was certain to find myself here on Monday, but if it’s tails then there was a 1⁄2 chance that I’d find myself experiencing a Tuesday. Applying Bayes’ theorem, I deduce that there’s a 2⁄3 chance that the coin is heads, and that the world is going to end before tomorrow.”
Now FNC makes two observations:
The event “it is Monday today” is indexical. However, an “indexical event” isn’t strictly speaking an event. (Because an event picks out a set of possible worlds, whereas an indexical event picks out a set of possible “centered worlds”.) Since it isn’t an event, it makes no sense to treat it as ‘data’ in a Bayesian calculation.
(But apart from that) the best way to do an update is to update on everything we know.
M takes these points to heart. Rather than updating on “it is Monday” she instead updates on “there once was a person who experienced [complete catalogue of M’s mental state] and that person lived on Monday.”
If we ignore the (at best) remote possibility that T has exactly the same experiences as M (prior to learning which day it is) then the event above is independent of the coin toss. Therefore M should calculate a posterior probability of 1⁄2 that the coin is heads.
On discovering that it’s Monday, M gains no evidence that the end of the world is nigh. Notice that we’ve reached this conclusion independently of decision theory.
If M is ‘altruistic’ towards T, valuing him as much as she values herself, then she should be prepared to part with one cube of chocolate in exchange for a guarantee that he’ll get two if he exists. If M is ‘selfish’ then the exchange rate changes from 1:2 to 1:infinity. These exchange rates are not probabilities. It would be very wrong to say something like “the probability that M gives to T’s existence only makes sense when we specify M’s utility function, and it in particular it changes from 1⁄2 to 0 if M switches from ‘altruistic’ to ‘selfish’”.
I used to be a great believer in FNC, but I’ve found it’s flawed. The main problem is that it’s not time-consistent.
For instance, if you start with some identical copies, and they are each going to flip a coin twenty times. Now FNC says that before they flip a coin, they should not believe that they are in a large universe, because they are identical.
However, after they have flipped, they will be nearly certainly very different, and so will believe that they are in a large universe.
So they know that after they flip the coin, their probability of being in a large universe will have increased, no matter what they see.
The problem isn’t just restricted to when you start with identical copies; whenever you increase your memory size by one bit, say, then FNC will be slightly inconsistent (because (1+e)^-n is approximately 1-ne for small e, but not exactly).
Yes, that is definitely a problem! The variation of FNC which I described in the final section of my UDT post has each person being allowed to help themselves to uniform random number in [0,1] - i.e. infinitely many random “coin flips”, as long as they don’t try to actually use the outcomes.
This solves the problem you mention, but others arise:
It’s hard to see how to give an independent justification of this trick.
More importantly, Eliezer’s tale of the Ebborians demonstrates that we can go continuously from one copy to two copies.
Actually, using (2), and variations alpha to gamma, I think I can construct a continuum of variations on Sleeping Beauty which stretch from one where the answer is unambiguously 1⁄3 (or 1⁄11 as in the link) to one where it’s unambiguously 1⁄2.
OK, I recant and denounce myself—the idea that any sensible variation of the Sleeping Beauty problem must have a ‘canonical’ answer is wrong, and FNC is broken.
OK, I recant and denounce myself—the idea that any sensible variation of the Sleeping Beauty problem must have a ‘canonical’ answer is wrong, and FNC is broken.
Very admirable stance to take :-) I wish I could claim I found the problem and immediately renounced SIA and FNC, but it was a long process :-)
Also, I don’t think probabilities are useful objects on their own; only the decisions that result from them. Different theories can get different probabilities but always the same decision...
I suppose I’m being obtuse about this, but please help me find my way through this argument.
The event “it is Monday today” is indexical. However, an “indexical event” isn’t strictly speaking an event. (Because an event picks out a set of possible worlds, whereas an indexical event picks out a set of possible “centered worlds”.) Since it isn’t an event, it makes no sense to treat it as ‘data’ in a Bayesian calculation.
Isn’t this argument confounded by the observation that an indexical event “It is Tuesday today”, in the process of ruling out several centered possible worlds—the ones occurring on Monday—also happens to rule out an entire uncentered world? If it’s not an event, how does it makes sense to treat it as data in a Bayesian calculation that rules out Heads? If that wasn’t the event that entered into the Bayesian calculation, what was?
If that wasn’t the event that entered into the Bayesian calculation, what was?
The Bayesian calculation only needs to use the event “Tuesday exists” which is non-indexical (though you’re right—it is entailed by “today is Tuesday”).
The problem with indexical events is that our prior is a distribution over possible worlds, and there doesn’t seem to be any non-arbitrary way of deriving a distribution over centered worlds from a distribution over uncentered ones. (E.g. Are all people equally likely regardless of lifespan, brain power, state of wakefulness etc.? What if people are copied and the copies diverge from one another? Where does the first ‘observer’ appear in the tree of life? etc.)
The Bayesian calculation only needs to use the event “Tuesday exists”
I can’t follow this. If “Tuesday exists” isn’t indexical, then it’s exactly as true on Monday as it is on Tuesday, and furthermore as true everywhere and for everyone as it is for anyone.
there doesn’t seem to be any non-arbitrary way of deriving a distribution over centered worlds from a distribution over uncentered ones.
Indeed, unless you work within the confines of a finite toy model. But why go in that direction? What non-arbitrary reason is there not to start with centered worlds and try to derive a distribution over uncentered ones? In fact, isn’t that the direction scientific method works in?
I can’t follow this. If “Tuesday exists” isn’t indexical, then it’s exactly as true on Monday as it is on Tuesday, and furthermore as true everywhere and for everyone as it is for anyone.
Well, in my toy model of the Doomsday Argument, there’s only a 1⁄2 chance that Tuesday exists, and the only way that a person can know that Tuesday exists is to be alive on Tuesday. Do you still think there’s a problem?
Indeed, unless you work within the confines of a finite toy model.
Even in toy models like Sleeping Beauty we have to somehow choose between SSA and SIA (which are precisely two rival methods for deriving centered from uncentered distributions.)
What non-arbitrary reason is there not to start with centered worlds and try to derive a distribution over uncentered ones? In fact, isn’t that the direction scientific method works in?
That’s a very good, philosophically deep question! Like many lesswrongers, I’m what David Chalmers would call a “Type-A materialist” which means that I deny the existence of “subjective facts” which aren’t in some way reducible to objective facts.
Therefore, I think that centered worlds can be regarded one of two ways: (i) as nonsense or (ii) as just a peculiar kind of uncentered world: A “centered world” really just means an “uncentered world that happens to contain an ontologically basic, causally inert ‘pointer’ towards some being and an ontologically basic, causally inert catalogue of its “mental facts”. However, because a “center” is causally inert, we can never acquire any evidence that the world has a “center”.
(I’d like to say more but really this needs a lot more thought and I can see I’m already starting to ramble...)
I’m what David Chalmers would call a “Type-A materialist” which means that I deny the existence of “subjective facts” which aren’t in some way reducible to objective facts.
The concerns Chalmers wrote about focused on the nature of phenomenal experience, and the traditional dichotomy between subjective and objective in human experience. That distinction draws a dividing line way off to the side of what I’m interested in. My main concern isn’t with ineffable consciousness, it’s with cognitive processing of information, information defined as that which distinguishes possibilities, reduces uncertainty and can have behavioral consequences. Consequences for what/whom? Situated epistemic agents, which I take as ubiquituous constituents of the world around us, and not just sentient life-forms like ourselves. Situated agents that process information don’t need to be very high on the computational hierarchy in order to be able to interact with the world as it is, use representations of the world as they take it to be, and entertain possibilities about how well their representations conform to what they are intended to represent. The old 128MB 286 I had in the corner that was too underpowered to run even a current version of linux, was powerful enough to implement an instantiation of a situated Bayesian agent. I’m completely fine with stipulating that it had about as much phenomenal or subjective experience as a chunk of pavement. But I think there are useful distinctions totally missed by Chalmers’ division (which I’m sure he’s aware of, but not concerned with in the paper you cite), between what you might call objective facts and what you might call “subjective facts”, if by the latter you include essentially indexical and contextual information, such as de se and de dicto information, as well as de re propositions.
Therefore, I think that centered worlds can be regarded one of two ways: (i) as nonsense or (ii) as just a peculiar kind of uncentered world: A “centered world” really just means an “uncentered world that happens to contain an ontologically basic, causally inert ‘pointer’ towards some being and an ontologically basic, causally inert catalogue of its “mental facts”. However, because a “center” is causally inert, we can never acquire any evidence that the world has a “center”.
(On Lewis’s account, centered worlds are generalizations of uncentered ones, which are contained in them as special cases.) From the point of view of a situated agent, centered worlds are epistemologically prior, about as patently obvious as the existence of “True”, “False” and “Don’t Know”, and the uncentered worlds are secondary, synthesized, hypothesized and inferred. The process of converting limited indexical information into objective, universally valid knowledge is where all the interesting stuff happens. That’s what the very idea of “calibration” is about. To know whether they (centered worlds or the other kind) are ontologically prior it’s just too soon for me to tell, but I feel uncomfortable prejudging the issue on such strict criteria without a more detailed exploration of the territory on the outside of the walled garden of God’s Own Library of Eternal Verity. In other words, with respect to that wall, I don’t see warrant flowing from inside out, I see it flowing from outside in. I suppose that’s in danger of making me an idealist, but I’m trying to be a good empiricist.
I remember you linked me to Radford Neal’s paper (pdf) on Full Non-indexical Conditioning. I think FNC is a much nicer way to think about problems like these than SSA and SIA, but I guess you disagree?
To save others from having to wade through the paper, which is rather long, I’ll try to explain what FNC means:
First, let’s consider a much simpler instance of the Doomsday Argument: At the beginning of time, God tosses a coin. If heads then there will only ever be one person (call them “M”), who is created, matures and dies on Monday, and then the world ends. If tails then there will be two people, one (“M”) who lives and dies on Monday and another (“T”) on Tuesday. As this is a Doomsday Argument, we don’t require that T is a copy of M.
M learns that it’s Monday but is given no (other) empirical clues about the coin. M says to herself “Well, if the coin is heads then I was certain to find myself here on Monday, but if it’s tails then there was a 1⁄2 chance that I’d find myself experiencing a Tuesday. Applying Bayes’ theorem, I deduce that there’s a 2⁄3 chance that the coin is heads, and that the world is going to end before tomorrow.”
Now FNC makes two observations:
The event “it is Monday today” is indexical. However, an “indexical event” isn’t strictly speaking an event. (Because an event picks out a set of possible worlds, whereas an indexical event picks out a set of possible “centered worlds”.) Since it isn’t an event, it makes no sense to treat it as ‘data’ in a Bayesian calculation.
(But apart from that) the best way to do an update is to update on everything we know.
M takes these points to heart. Rather than updating on “it is Monday” she instead updates on “there once was a person who experienced [complete catalogue of M’s mental state] and that person lived on Monday.”
If we ignore the (at best) remote possibility that T has exactly the same experiences as M (prior to learning which day it is) then the event above is independent of the coin toss. Therefore M should calculate a posterior probability of 1⁄2 that the coin is heads.
On discovering that it’s Monday, M gains no evidence that the end of the world is nigh. Notice that we’ve reached this conclusion independently of decision theory.
If M is ‘altruistic’ towards T, valuing him as much as she values herself, then she should be prepared to part with one cube of chocolate in exchange for a guarantee that he’ll get two if he exists. If M is ‘selfish’ then the exchange rate changes from 1:2 to 1:infinity. These exchange rates are not probabilities. It would be very wrong to say something like “the probability that M gives to T’s existence only makes sense when we specify M’s utility function, and it in particular it changes from 1⁄2 to 0 if M switches from ‘altruistic’ to ‘selfish’”.
I used to be a great believer in FNC, but I’ve found it’s flawed. The main problem is that it’s not time-consistent.
For instance, if you start with some identical copies, and they are each going to flip a coin twenty times. Now FNC says that before they flip a coin, they should not believe that they are in a large universe, because they are identical.
However, after they have flipped, they will be nearly certainly very different, and so will believe that they are in a large universe.
So they know that after they flip the coin, their probability of being in a large universe will have increased, no matter what they see.
The problem isn’t just restricted to when you start with identical copies; whenever you increase your memory size by one bit, say, then FNC will be slightly inconsistent (because (1+e)^-n is approximately 1-ne for small e, but not exactly).
Yes, that is definitely a problem! The variation of FNC which I described in the final section of my UDT post has each person being allowed to help themselves to uniform random number in [0,1] - i.e. infinitely many random “coin flips”, as long as they don’t try to actually use the outcomes.
This solves the problem you mention, but others arise:
It’s hard to see how to give an independent justification of this trick.
More importantly, Eliezer’s tale of the Ebborians demonstrates that we can go continuously from one copy to two copies.
Actually, using (2), and variations alpha to gamma, I think I can construct a continuum of variations on Sleeping Beauty which stretch from one where the answer is unambiguously 1⁄3 (or 1⁄11 as in the link) to one where it’s unambiguously 1⁄2.
OK, I recant and denounce myself—the idea that any sensible variation of the Sleeping Beauty problem must have a ‘canonical’ answer is wrong, and FNC is broken.
Very admirable stance to take :-) I wish I could claim I found the problem and immediately renounced SIA and FNC, but it was a long process :-)
Btw, a variant similar to your alpha to gamma was presented in my post http://lesswrong.com/lw/18r/avoiding_doomsday_a_proof_of_the_selfindication ; I found the problem with that in http://lesswrong.com/lw/4fl/dead_men_tell_tales_falling_out_of_love_with_sia/
Also, I don’t think probabilities are useful objects on their own; only the decisions that result from them. Different theories can get different probabilities but always the same decision...
I suppose I’m being obtuse about this, but please help me find my way through this argument.
Isn’t this argument confounded by the observation that an indexical event “It is Tuesday today”, in the process of ruling out several centered possible worlds—the ones occurring on Monday—also happens to rule out an entire uncentered world? If it’s not an event, how does it makes sense to treat it as data in a Bayesian calculation that rules out Heads? If that wasn’t the event that entered into the Bayesian calculation, what was?
The Bayesian calculation only needs to use the event “Tuesday exists” which is non-indexical (though you’re right—it is entailed by “today is Tuesday”).
The problem with indexical events is that our prior is a distribution over possible worlds, and there doesn’t seem to be any non-arbitrary way of deriving a distribution over centered worlds from a distribution over uncentered ones. (E.g. Are all people equally likely regardless of lifespan, brain power, state of wakefulness etc.? What if people are copied and the copies diverge from one another? Where does the first ‘observer’ appear in the tree of life? etc.)
I can’t follow this. If “Tuesday exists” isn’t indexical, then it’s exactly as true on Monday as it is on Tuesday, and furthermore as true everywhere and for everyone as it is for anyone.
Indeed, unless you work within the confines of a finite toy model. But why go in that direction? What non-arbitrary reason is there not to start with centered worlds and try to derive a distribution over uncentered ones? In fact, isn’t that the direction scientific method works in?
Well, in my toy model of the Doomsday Argument, there’s only a 1⁄2 chance that Tuesday exists, and the only way that a person can know that Tuesday exists is to be alive on Tuesday. Do you still think there’s a problem?
Even in toy models like Sleeping Beauty we have to somehow choose between SSA and SIA (which are precisely two rival methods for deriving centered from uncentered distributions.)
That’s a very good, philosophically deep question! Like many lesswrongers, I’m what David Chalmers would call a “Type-A materialist” which means that I deny the existence of “subjective facts” which aren’t in some way reducible to objective facts.
Therefore, I think that centered worlds can be regarded one of two ways: (i) as nonsense or (ii) as just a peculiar kind of uncentered world: A “centered world” really just means an “uncentered world that happens to contain an ontologically basic, causally inert ‘pointer’ towards some being and an ontologically basic, causally inert catalogue of its “mental facts”. However, because a “center” is causally inert, we can never acquire any evidence that the world has a “center”.
(I’d like to say more but really this needs a lot more thought and I can see I’m already starting to ramble...)
The concerns Chalmers wrote about focused on the nature of phenomenal experience, and the traditional dichotomy between subjective and objective in human experience. That distinction draws a dividing line way off to the side of what I’m interested in. My main concern isn’t with ineffable consciousness, it’s with cognitive processing of information, information defined as that which distinguishes possibilities, reduces uncertainty and can have behavioral consequences. Consequences for what/whom? Situated epistemic agents, which I take as ubiquituous constituents of the world around us, and not just sentient life-forms like ourselves. Situated agents that process information don’t need to be very high on the computational hierarchy in order to be able to interact with the world as it is, use representations of the world as they take it to be, and entertain possibilities about how well their representations conform to what they are intended to represent. The old 128MB 286 I had in the corner that was too underpowered to run even a current version of linux, was powerful enough to implement an instantiation of a situated Bayesian agent. I’m completely fine with stipulating that it had about as much phenomenal or subjective experience as a chunk of pavement. But I think there are useful distinctions totally missed by Chalmers’ division (which I’m sure he’s aware of, but not concerned with in the paper you cite), between what you might call objective facts and what you might call “subjective facts”, if by the latter you include essentially indexical and contextual information, such as de se and de dicto information, as well as de re propositions.
(On Lewis’s account, centered worlds are generalizations of uncentered ones, which are contained in them as special cases.) From the point of view of a situated agent, centered worlds are epistemologically prior, about as patently obvious as the existence of “True”, “False” and “Don’t Know”, and the uncentered worlds are secondary, synthesized, hypothesized and inferred. The process of converting limited indexical information into objective, universally valid knowledge is where all the interesting stuff happens. That’s what the very idea of “calibration” is about. To know whether they (centered worlds or the other kind) are ontologically prior it’s just too soon for me to tell, but I feel uncomfortable prejudging the issue on such strict criteria without a more detailed exploration of the territory on the outside of the walled garden of God’s Own Library of Eternal Verity. In other words, with respect to that wall, I don’t see warrant flowing from inside out, I see it flowing from outside in. I suppose that’s in danger of making me an idealist, but I’m trying to be a good empiricist.