If anyone's confused by the post, here's my stab at a simpler version.
Assume we're measuring random variables A and B at time moments 1 and 2, so we have many samples containing four values (A1,A2,B1,B2) each. We want to figure out whether moment 1 came before or after moment 2, using Eliezer's assumption that the value of each variable at the later moment is independently probabilistically caused by the values of both variables at the earlier moment. Formally, if 1 came before 2, then A2 must be independent of B2 conditional on knowing A1 and B1, and if 2 came before 1, then A1 must be independent of B1 conditional on A2 and B2.
Let's run through a simple example and see if we can figure out the direction of time. Let random() be a function that returns 0 or 1 with probability 50% each, and all calls to random() are independent. Our variables will be defined as follows:
A1 = random()
B1 = random()
A2 = A1 + B1 + random()/10
B2 = A1 + B1 + random()/10
Clearly, if we know A1 and B1, then additionally knowing A2 doesn't give information about B2 or vice versa. But if we know, for example, that A2=1.1 and B2=1, then we know that A1+B1=1, so additionally knowing A1 would give information about B1 and vice versa. Therefore moment 1 must have come before moment 2.
Eliezer's assumption of independent probabilistic causation seems to be doing most of the work here. I don't know if that assumption applies very often in real life. For example, if A1 is allowed to probabilistically generate an intermediate value A1' that's used in the computation of both A2 and B2, we can't figure out the direction of time using the proposed method.
One scenario where the method would work is if A2 and B2 are generated by two spacelike-separated generators, each of which receives the true values of A1 and B1. Or at least it should work with classical probability. I don't know enough physics to tell if quantum entanglement tricks can screw up the method even when A2 and B2 are spacelike separated. Maybe someone could chime in?
To compute a consistent universe with a low-entropy terminal condition and high-entropy initial condition, you have to simulate lots and lots of universes, then throw away all but a tiny fraction of them that end up with low entropy at the end.
Look at your explanation of entropy again, Eliezer. It's an expression of the multiplicity of equivalent-as-far-as-you-can-tell states. So, if you know you're going to have a low-entropy final state, then the entropy of the initial state has to be that low already in the beginning.
I guess that means you don't know that it's going to end up low entropy; most universes don't end up low-entropy, so you expect this one won't as well.
equivalent-as-far-as-you-can-tell
Based on what information? If I compute a state that will evolve into a low entropy state and then forget that information about it and base my calculation of entropy on its temperature, pressure etc. then I'll have a high entropy state that evolves into a low entropy one. Indeed "entropy (based on some macro-scale parameters)" seems to be what people mean when they say "entropy".
Is there any clever maneuver we can use to distinguish between right and left causality, if it's assumed to be deterministic? Can we distinguish between right and left causality, under the following conditions:
We allow the functions 'from (L1,L2) to R1' and 'from (L1,L2) to R2' not to be identical (assuming rightward causality). In other words, the rule that the system uses to produce the next state of V1 by looking at the current states of V1 and V2, doesn't have to be the same rule as the one to produce the next state of V2 from the current states of V1 and V2.
Both functions are known to be surjective.
We don't know the functions.
Both V1 and V2 may have any natural number of states, and they need not have the same number of states.
(edit):
Those conditions are really just a suggestion, if you have better ones, use them. And share 'em too plz.
Today's post, Timeless Causality was originally published on 29 May 2008. A summary (taken from the LW wiki):
Using the modern, Bayesian formulation of causality, we can define causality without talking about time - define it purely in terms of relations. The river of time never flows, but it has a direction.
Discuss the post here (rather than in the comments to the original post).
This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was Timeless Beauty, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.
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