We need a theory of anthropic measure binding

The pattern theory is wrong because nature is not lossily compressed in that way.

Speaking as one of the people you describe to whom the pattern theory comes naturally: I don't quite get how you know this. The following paragraphs do not seem to me to establish it. From the fact that a simulation could do something, why does it follow that "base-level reality" can not? Is there an implied argument here involving locality of effects, or similar?

I've always maintained that in order to solve this issue we must first solve the question of, what does it even mean to say that a physical system is implementing a particular algorithm? Does it make sense to say that an algorithm is only approximately implemented? What if the algorithm is something very chaotic such as prime-checking, where approximation is not possible? 

An algorithm should be a box that you can feed any input into, but in the real, causal world, there is no such choice, any impression that you "could" input anything into your pocket calculator is due to the counterfactuals your brain can consider purely because it has some uncertainty about the world (an omniscient being could not make any choice at all! -- assuming complete omniscience is possible, which I don't think it is, but let us imagine the universe as an omniscient being or something).

This leads me to believe that "anthropic binding" cannot be some kind of metaphysical primitive, since for it to be well-defined it needs to be considered by an embedded agent! Indeed, I claimed that recognizing algorithms "in the wild" requires the use of counterfactuals, and omniscient beings (such as "the universe") cannot use counterfactuals. Therefore I do not see how there could be a "correct" answer to the problem of anthropic binding.

What is your probability that you're the heavier brain?

Undefined.  It matters a lot what rent the belief is paying.  The specifics of how you'll resolve your probability (or at least what differential evidence would let you update) will help you pick the reference class(es) which matter, and inform your choice of prior (in this case, the amalgam of experience and models, and unidentified previous evidence you've accumulated).

Thank you for bringing attention to this issue— I think it's an under-appreciated problem. I agree with you that the "force" measure is untenable, and the "pattern" view, while better, probably can't work either.

Count-based measures seem to fail because they rely on drawing hard boundaries between minds. Also, there are going to be cases where it's not even clear whether a system counts as a mind or not, and if we take the "count" view we will probably be forced to make definitive decisions in those cases.

Mass/energy-based measures seem better because they allow you to treat anthropic measure as the continuous variable that it is, but I also don't think they can be the answer. In particular, they seem to imply that more efficient implementations of a mind (in terms of component size or power consumption or whatever) would have lower measure than less efficient ones, even if they have all the same experiences.

This is debatable, but it strikes me that anthropic measure and "degree of consciousness" are closely related concepts. Fundamentally, for a system to have any anthropic measure at all, it needs to be able to count as an "observer" or an "experiencer" which seems pretty close to saying that it's conscious on some level.

If we equate consciousness with a kind of information processing, then anthropic measure could be a function of "information throughput" or something like that. If a System A can "process" more bits of information per unit time than System B, then it can have more experiences than System B, and arguably should be given more anthropic measure. In other words, if you identify "yourself" with the set of experiences you're having in a given moment, then it's more likely that those experiences are being realized in a system with more computing power, more ability to have more experiences, than a system with less compute. Note that, on this view, the information that's being processed doesn't have to be compressed/deduplicated in any way; systems running the same computation on many threads in parallel would still have more measure than single-threaded systems ceteris paribus.

There's a lot that needs to be fleshed out with this "computational theory of anthropic measure" but it seems like the truth has to be something in this general direction.

Leinbiz's Law says that you cannot have separate objects that are indistinguishable from each other. It sounds like that is what you are doing with the 3 brains. That might be a good place to flesh out more to make progress on the question. What do you mean exactly by saying that the three brains are wired up to the same body and are redundant?

I've noticed another reason a person might need a binding theory, if they were the sort of agent who takes Rawlsian Veils to their greatest extent, the one who imagines that we are all the same eternal actor playing out every part in a single play, who follows a sort of timeless, placeless cooperative game theory that obligates them to follow a preference utilitarianism extending — not just over the other stakeholders in some coalition, as a transparent pragmatist might — but over every observer that exists.

They'd need a good way of counting and weighting observers, to reflect the extent to which their experiences are being had. That would be their weighting.

I also could suggest energy-used-for-computation-measure. First, assume that there is a minimal possible energy of computation, plank computation, which uses most theoretically effective computing. Each brain could be counterfactually sliced as sum of minimal computations. Now we could calculate which brain has more slices and conclude that I am more likely to be in such brain.

This measure is more probable than mass- or volume-based measure, as mass could be mass of computationally inert parts of the brain like bones.

Is this really an anthropic problem? It looks more like a problem of decision making under uncertainty.

The fundamental problem is that you have two hypotheses, and don't know which is true. There are no objective probabilities for the hypotheses, only subjective credences. There may even be observers in your epistemic situation for which neither hypothesis is true.

As an even more fun outcome, consider the situation where the mind states of the two types of entity are so exactly alike that they are guaranteed to think exactly the same thoughts and make the same decision, and only diverge afterward. Are there one, two, or four people making that decision? Does the distinction even make a difference, if the outcomes are exactly the same in each case?

A child is supposed to be transported to another town in a car by his father. His family has two cars. The first is a blue electric car with 3 passenger seats. The second car is red with only 1 passager seat and works on gas. Which car should the child expect to find themself in with which probability?

There can be possible situations where the information about the number of seats, the color of the car or its tipe of engine is relevant. Maybe his father like the color red more? Or maybe there is a law against transporting a child on a front seat? Maybe electric car isn't charged? Maybe childs father determined which seat to put him through random sample among all four seats? Maybe the father cares about environment and doesn't use the gas car anymore?

But if the child doesn't know anything about in which situation they are, anything about the actual causal process which would put them in the car, if all the information they have is just the fact that there are two cars with different characteristics, then the naive answer: 1/2 for every car due to the equiprobable prior is correct and any attempt to persuade themselves that "universe cares more" about red cars or cars with more seats, without having any evidence, is plain wrong.

The more I think about anthropics the more I realize there is no rational theory for anthropic binding. For the question "what is the probability that I am the heavy brain?" there really isn't a rational answer.  [LW · GW]

Compassion measure binds to anthropic binding weight because anthropic binding weight is compassion measure: there is no objective answer to "What is your probability" - probabilities are just numbers you multiply with utilities if you don't want to be endlessly pumped for money.

I like the thought experiment and it seems like an interesting way to split a bunch of philosophical models.

One thing I'm curious about is which (the small three or the big one) you would prefer to be, and whether that preference should factor into your beliefs here.

Probability is in the mind. Your question is meaningless. What is meaningful is what your expectations should be for specific experiences, and that's going to depend on the kind of evidence you have and Solomonoff induction - it's pretty trivial once you accept the relevant premises.

It's just not meaningful to say "X and Y exist", you need to reframe it in terms of how various pieces of evidence affect induction over your experiences.