You seem to be exaggerating the generality of the causal Markov condition (CMC) when you say it is deeper and more general than the second law of thermodynamics. In a big world, failures of the CMC abound. Let’s say the correlation between the psychic cousin’s predictions and the top card of the deck is explained by the person performing the test being a stooge, who is giving some non-verbal indication to the purported psychic about the top card. So here we have a causal explanation of the correlation, as the CMC would lead us to expect. But since we are in a big world, there are a massive number of Boltzmann brains out there, outside our light cone, whose brain states correlate with the top card in the same way that the cousin’s does. But there is no causal explanation for this correlation, it’s just the kind of thing one would expect to happen, even non-causally, in a sufficiently large world. So the CMC isn’t a universal truth.
Now, the CMC is a remarkably accurate rule if we restrict it to our local environment. But it’s pretty plausible that this is just because our local environment is monotonically entropy-increasing towards the future and entropy-decreasing towards the past. Because of this feature of our environment, local interventions produce correlations that propagate out spatially towards the future, but not towards the past. When you drop a rock into a pond, waves originate at the point the rock hit the water and travel outwards towards the future, eventually producing spatially distant correlations (like fish at either end of the pond being disturbed from their slumber).
Imagine that there is a patch somewhere in the trackless immensity of spacetime that looks exactly like our local environment, but time-reversed. Here we would have a pond with a rock initially lying at its bottom. Spontaneously, the edges of the pond fluctuate so as to produce a coherent inward-directed wave, which closes in on the rock, transferring to it sufficient energy to make it shoot out of the pond. If you don’t allow backward causation, then it seems that the initial correlated fluctuation that produced the coherent wave has no causal explanation, a violation of the CMC.
The second law is often read as a claim about the condition of the early universe (or some patch of the universe), specifically that there were no correlations between different degrees of freedom (such as the positions and velocities of particles) except for those imposed by the macroscopic state. There were no sneaky microscopic correlations that could later produce macroscopic consequences (see this paper). Entropy increase follows from that, the story goes, and, plausibly, the success of the CMC follows from that as well. There is a strong case to be made that the second law is prior to the CMC in the order of explanation.
I have doubts about how meaningful it is to talk of correlating things that are outside each other’s light cones.
Besides that, suppose there really are an astronomical number of Boltzmann Brains that you could say are non-causally correlated with the top card of a particular deck of cards. Calling this a failure of the Causal Markov Condition is begging the question because the only thing identifying this set is selection based on the correlation itself. The set you should consider, of all Boltzmann Brains that you could test for correspondence with the top card, will not be correlated with it at all.
Entropy increase follows from that, the story goes...
I have doubts about how meaningful it is to talk of correlating things that are outside each other’s light cones.
I don’t see why you would have these doubts. Whether or not two variables are correlated is a purely mathematical condition. Why do you think it matters where in space-time the physical properties those variables describe are instantiated?
Besides that, suppose there really are an astronomical number of Boltzmann Brains that you could say are non-causally correlated with the top card of a particular deck of cards. Calling this a failure of the Causal Markov Condition is begging the question because the only thing identifying this set is selection based on the correlation itself. The set you should consider, of all Boltzmann Brains that you could test for correspondence with the top card, will not be correlated with it at all.
Wait, why is the relevant reference class the class of all and only Boltzmann brains? It seems more natural to pick a reference class that includes all brains (or brain-states). But in that case, the probabilities of the Boltzmann brain being in the states that it is in will be exactly the same as the probabilities of the psychic cousin being in the states that he is in (since the states are the same by hypothesis), so if the psychic’s brain states are correlated with the top card the BB’s will be as well.
Follows from it causally, like? :)
Sure, if you want. I’m not denying here that causality is prior to the second law. I’m denying that the causal Markov condition is prior to the second law.
OK. wrt the light cones, I was posting without my brain switched on. Obviously two events can be outside each others light cones and yet a correlation between them still be observed where their light cones overlap in the future. I was thinking fairly unclearly about whether you could be in an epistemic state to consider correlation between things outside your own light cone, but this is kind of irrelevant, so please disregard.
the probabilities of the Boltzmann brain being in the states that it is in will be exactly the same as the probabilities of the psychic cousin being in the states that he is in (since the states are the same by hypothesis)
Just because the states are the same doesn’t mean the probability of being in that state are the same. It’s only meaningful to discuss the probability of an outcome in terms of a probability distribution over possible outcomes. If you pick a set of conditions such as “Boltzmann brains in the same state as that of the psychic cousin” you are creating the hypothetical correlation yourself by the way you define it. To my mind, that’s not a thought experiment that can tell you anything.
Just because the states are the same doesn’t mean the probability of being in that state are the same. It’s only meaningful to discuss the probability of an outcome in terms of a probability distribution over possible outcomes.
In my example, I specified that the BB is in a reference class with all other brains, including the psychic cousin’s. Given that they are both in the reference class, the fact that the BB and the cousin share the same cognitive history implies that the probabilities of their cognitive histories relative to this reference class are the same. The reference class is what fixes the probability distribution over possible outcomes if you’re determining probabilities by relative frequencies, and if they are in the same reference class, they will have the same probability distribution.
I suspect Eliezer was thinking of a different probability distribution over brain states when he said the psychic’s brain state is correlated with the deck of cards. The probabilities he is referring to are something like the relative frequencies of brain states (or brain state types) in a single observer’s cognitive history (ETA: Or perhaps more accurately for a Bayesian, the probabilities you get when you conditionalize some reasonable prior on the sequence of instantiated brain states). Even using this distribution, the BB’s brain state will be correlated with the top card.
Even if the BB and the psychic are in causally disconnected parts of your model, them having the same probability of being correlated with the card doesn’t imply that the Causal Markov Condition is broken. In order to show that, you would need to specify all of the parent nodes to the BB in your model, calculate the probability of it being correlated with the card, and then see whether having knowledge of the psychic would change your probability for the BB. Since all physics currently is local in nature, I can’t think of anything that would imply this is the case if the psychic is outside of the past light cone of the BB. Larger boundary conditions on the universe as a whole that may or may not make them correlate have no effect on whether the CMC holds.
I’m having trouble parsing this comment. You seem to be granting that the BB’s state is correlated with the top card (I’m assuming this is what you mean by “having the same probability”), that there is no direct causal link between the BB and the psychic, and that there are no common causes, but saying that this still doesn’t necessarily violate the CMC. Am I interpreting you right? If I’m not, could you tell me which one of those premises does not hold in my example?
If I am interpreting you correctly, then you are wrong. The CMC entails that if X and Y are correlated, X is not a cause of Y and Y is not a cause of X, then there are common causes of X and Y such that the variables are independent conditional on those common causes.
The CMC is not strictly violated in physics as far as we know. If you specify the state of the universe for the entire past light cone of some event, then you uniquely specify the event. The example that you gave of the rock shooting out of the pond indeed does not violate the laws of physics- you simply shoved the causality under the rug by claiming that the edge of the pond fluctuated “spontaneously”. This is not true. The edge of the pond fluctuating was completely specified by the past light cone of that event. This is the sense in which the CMC runs deeper than the 2nd law of thermodynamics- because the 2nd “law” is probabilistic, you can find counterexamples to it in an infinite universe. If you actually found a counterexample to the CMC, it would make physics essentially impossible.
I meant “spontaneous” in the ordinary thermodynamic sense of spontaneity (like when we say systems spontaneously equilibriate, or that spontaneous fluctuations occur in thermodynamic systems), so no violation of microphysical law was intended. Spontaneous here just means there is no discernable macroscopic cause of the event. Now it is true that everything that happened in the scenario I described was microscopically determined by physical law, but this is not enough to satisfy the CMC. What we need is some common cause account of the macroscopic correlation that leads to a coherent inward-directed wave, and simply specifying that the process is law-governed does not provide such an account. I guess you could just say that the common cause is the initial conditions of the universe, or something like that. If that kind of move is allowed, then the CMC is trivially satisfied for every correlation. But when people usually appeal to the CMC they intend something stronger than this. They’re usually talking about a spatially localized cause, not an entire spatial hypersurface.
If you allow entire hypersurfaces as nodes in your graph, you run into trouble. In a deterministic world, any correlation between two properties isn’t just screened off by the contents of past hypersurfaces, it’s also screened off by the contents of future hypersurfaces. But a future hypersurface can’t be a common cause of the correlated properties, so we have a correlation screened off by a node that doesn’t d-separate the correlated variables. This doesn’t violate the CMC per se, but it does violate the Faithfulness Condition, which says that the only conditional independencies in nature are the ones described by the CMC. If the Faithfulness Condition fails, then the CMC becomes pretty useless as a tool for discerning causation from correlation. The lessons of Eliezer’s posts would no longer apply. So to rule out radical failure of the Faithfulness Condition in a deterministic setting, we have to disallow the contents of an entire hypersurface from being treated as a single node in a causal graph. Nodes should correspond to sufficiently locally instantiated properties. But then that re-opens the possibility that the correlation described in my example violates the CMC. There is no locally instantiated common cause.
If there is some past screener-off of the correlation in the time-reversed patch, its counterpart would also be a future screener-off of the correlation in our patch. If we want to say that the Faithfulness Condition holds in our patch (or at least in this example), we have to rule out future screeners-off, but that also implies that the CMC fails in the time-reversed patch.
Indexically, though, you wouldn’t expect to be talking to a mind that just happened to issue something it called predictions, which just happened to be correlated with some unobserved cards, would you? I think the CMC doesn’t say that a mind can never be right without being causally entangled with the system it’s trying to be right about; just that if it is right, it’s down to pure chance.
I think the CMC doesn’t say that a mind can never be right without being causally entangled with the system it’s trying to be right about; just that if it is right, it’s down to pure chance.
No, the CMC says that if you conditionalize on all of the direct causes of some variable A in some set of variables, then A will be probabilistically independent of all other variables in that set except its effects. This rules out chance correlation. If there were some other variable in the set that just happened to be correlated with A without any causal explanation, then conditionalizing on A’s direct causes would not in general eliminate this correlation.
You seem to be exaggerating the generality of the causal Markov condition (CMC) when you say it is deeper and more general than the second law of thermodynamics. In a big world, failures of the CMC abound. Let’s say the correlation between the psychic cousin’s predictions and the top card of the deck is explained by the person performing the test being a stooge, who is giving some non-verbal indication to the purported psychic about the top card. So here we have a causal explanation of the correlation, as the CMC would lead us to expect. But since we are in a big world, there are a massive number of Boltzmann brains out there, outside our light cone, whose brain states correlate with the top card in the same way that the cousin’s does. But there is no causal explanation for this correlation, it’s just the kind of thing one would expect to happen, even non-causally, in a sufficiently large world. So the CMC isn’t a universal truth.
Now, the CMC is a remarkably accurate rule if we restrict it to our local environment. But it’s pretty plausible that this is just because our local environment is monotonically entropy-increasing towards the future and entropy-decreasing towards the past. Because of this feature of our environment, local interventions produce correlations that propagate out spatially towards the future, but not towards the past. When you drop a rock into a pond, waves originate at the point the rock hit the water and travel outwards towards the future, eventually producing spatially distant correlations (like fish at either end of the pond being disturbed from their slumber).
Imagine that there is a patch somewhere in the trackless immensity of spacetime that looks exactly like our local environment, but time-reversed. Here we would have a pond with a rock initially lying at its bottom. Spontaneously, the edges of the pond fluctuate so as to produce a coherent inward-directed wave, which closes in on the rock, transferring to it sufficient energy to make it shoot out of the pond. If you don’t allow backward causation, then it seems that the initial correlated fluctuation that produced the coherent wave has no causal explanation, a violation of the CMC.
The second law is often read as a claim about the condition of the early universe (or some patch of the universe), specifically that there were no correlations between different degrees of freedom (such as the positions and velocities of particles) except for those imposed by the macroscopic state. There were no sneaky microscopic correlations that could later produce macroscopic consequences (see this paper). Entropy increase follows from that, the story goes, and, plausibly, the success of the CMC follows from that as well. There is a strong case to be made that the second law is prior to the CMC in the order of explanation.
I have doubts about how meaningful it is to talk of correlating things that are outside each other’s light cones.
Besides that, suppose there really are an astronomical number of Boltzmann Brains that you could say are non-causally correlated with the top card of a particular deck of cards. Calling this a failure of the Causal Markov Condition is begging the question because the only thing identifying this set is selection based on the correlation itself. The set you should consider, of all Boltzmann Brains that you could test for correspondence with the top card, will not be correlated with it at all.
Follows from it causally, like? :)
I don’t see why you would have these doubts. Whether or not two variables are correlated is a purely mathematical condition. Why do you think it matters where in space-time the physical properties those variables describe are instantiated?
Wait, why is the relevant reference class the class of all and only Boltzmann brains? It seems more natural to pick a reference class that includes all brains (or brain-states). But in that case, the probabilities of the Boltzmann brain being in the states that it is in will be exactly the same as the probabilities of the psychic cousin being in the states that he is in (since the states are the same by hypothesis), so if the psychic’s brain states are correlated with the top card the BB’s will be as well.
Sure, if you want. I’m not denying here that causality is prior to the second law. I’m denying that the causal Markov condition is prior to the second law.
OK. wrt the light cones, I was posting without my brain switched on. Obviously two events can be outside each others light cones and yet a correlation between them still be observed where their light cones overlap in the future. I was thinking fairly unclearly about whether you could be in an epistemic state to consider correlation between things outside your own light cone, but this is kind of irrelevant, so please disregard.
Just because the states are the same doesn’t mean the probability of being in that state are the same. It’s only meaningful to discuss the probability of an outcome in terms of a probability distribution over possible outcomes. If you pick a set of conditions such as “Boltzmann brains in the same state as that of the psychic cousin” you are creating the hypothetical correlation yourself by the way you define it. To my mind, that’s not a thought experiment that can tell you anything.
In my example, I specified that the BB is in a reference class with all other brains, including the psychic cousin’s. Given that they are both in the reference class, the fact that the BB and the cousin share the same cognitive history implies that the probabilities of their cognitive histories relative to this reference class are the same. The reference class is what fixes the probability distribution over possible outcomes if you’re determining probabilities by relative frequencies, and if they are in the same reference class, they will have the same probability distribution.
I suspect Eliezer was thinking of a different probability distribution over brain states when he said the psychic’s brain state is correlated with the deck of cards. The probabilities he is referring to are something like the relative frequencies of brain states (or brain state types) in a single observer’s cognitive history (ETA: Or perhaps more accurately for a Bayesian, the probabilities you get when you conditionalize some reasonable prior on the sequence of instantiated brain states). Even using this distribution, the BB’s brain state will be correlated with the top card.
Even if the BB and the psychic are in causally disconnected parts of your model, them having the same probability of being correlated with the card doesn’t imply that the Causal Markov Condition is broken. In order to show that, you would need to specify all of the parent nodes to the BB in your model, calculate the probability of it being correlated with the card, and then see whether having knowledge of the psychic would change your probability for the BB. Since all physics currently is local in nature, I can’t think of anything that would imply this is the case if the psychic is outside of the past light cone of the BB. Larger boundary conditions on the universe as a whole that may or may not make them correlate have no effect on whether the CMC holds.
I’m having trouble parsing this comment. You seem to be granting that the BB’s state is correlated with the top card (I’m assuming this is what you mean by “having the same probability”), that there is no direct causal link between the BB and the psychic, and that there are no common causes, but saying that this still doesn’t necessarily violate the CMC. Am I interpreting you right? If I’m not, could you tell me which one of those premises does not hold in my example?
If I am interpreting you correctly, then you are wrong. The CMC entails that if X and Y are correlated, X is not a cause of Y and Y is not a cause of X, then there are common causes of X and Y such that the variables are independent conditional on those common causes.
The CMC is not strictly violated in physics as far as we know. If you specify the state of the universe for the entire past light cone of some event, then you uniquely specify the event. The example that you gave of the rock shooting out of the pond indeed does not violate the laws of physics- you simply shoved the causality under the rug by claiming that the edge of the pond fluctuated “spontaneously”. This is not true. The edge of the pond fluctuating was completely specified by the past light cone of that event. This is the sense in which the CMC runs deeper than the 2nd law of thermodynamics- because the 2nd “law” is probabilistic, you can find counterexamples to it in an infinite universe. If you actually found a counterexample to the CMC, it would make physics essentially impossible.
I meant “spontaneous” in the ordinary thermodynamic sense of spontaneity (like when we say systems spontaneously equilibriate, or that spontaneous fluctuations occur in thermodynamic systems), so no violation of microphysical law was intended. Spontaneous here just means there is no discernable macroscopic cause of the event. Now it is true that everything that happened in the scenario I described was microscopically determined by physical law, but this is not enough to satisfy the CMC. What we need is some common cause account of the macroscopic correlation that leads to a coherent inward-directed wave, and simply specifying that the process is law-governed does not provide such an account. I guess you could just say that the common cause is the initial conditions of the universe, or something like that. If that kind of move is allowed, then the CMC is trivially satisfied for every correlation. But when people usually appeal to the CMC they intend something stronger than this. They’re usually talking about a spatially localized cause, not an entire spatial hypersurface.
If you allow entire hypersurfaces as nodes in your graph, you run into trouble. In a deterministic world, any correlation between two properties isn’t just screened off by the contents of past hypersurfaces, it’s also screened off by the contents of future hypersurfaces. But a future hypersurface can’t be a common cause of the correlated properties, so we have a correlation screened off by a node that doesn’t d-separate the correlated variables. This doesn’t violate the CMC per se, but it does violate the Faithfulness Condition, which says that the only conditional independencies in nature are the ones described by the CMC. If the Faithfulness Condition fails, then the CMC becomes pretty useless as a tool for discerning causation from correlation. The lessons of Eliezer’s posts would no longer apply. So to rule out radical failure of the Faithfulness Condition in a deterministic setting, we have to disallow the contents of an entire hypersurface from being treated as a single node in a causal graph. Nodes should correspond to sufficiently locally instantiated properties. But then that re-opens the possibility that the correlation described in my example violates the CMC. There is no locally instantiated common cause.
If there is some past screener-off of the correlation in the time-reversed patch, its counterpart would also be a future screener-off of the correlation in our patch. If we want to say that the Faithfulness Condition holds in our patch (or at least in this example), we have to rule out future screeners-off, but that also implies that the CMC fails in the time-reversed patch.
Indexically, though, you wouldn’t expect to be talking to a mind that just happened to issue something it called predictions, which just happened to be correlated with some unobserved cards, would you? I think the CMC doesn’t say that a mind can never be right without being causally entangled with the system it’s trying to be right about; just that if it is right, it’s down to pure chance.
No, the CMC says that if you conditionalize on all of the direct causes of some variable A in some set of variables, then A will be probabilistically independent of all other variables in that set except its effects. This rules out chance correlation. If there were some other variable in the set that just happened to be correlated with A without any causal explanation, then conditionalizing on A’s direct causes would not in general eliminate this correlation.
If coincidences were a violation of the CMC, it wouldn’t be a truth at all, would it?
Well, one could still say it was true in certain environments, or true like the Ideal Gas Law is true.