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.
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.