I don’t think any of these are “approximations of DT” in any interesting sense. If you declare by fiat that DT is the Theory of Everything, then you could try to force-fit it… but that’s going to come out awkward and unconvincing.
Aren't probabilities such terribly fuzzy things? Where oh where do the priors come from? And yet, this doesn't really make very much difference to a glass of hot water, and by being more careful we can phrase thermodynamics in a less subjective way.
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So thermodynamics, at its core, has a certain twist of subjectivity; it is relative to things an observer knows. But the fuzzy parts of the subjectivity factor out cleanly in almost all practice, and the tiniest injection of perspective suffices to animate the mathematical core.
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DT requires a larger injection of perspective; preference as well as probability. Like thermodynamics, you can inject this subjectivity in the wrong place and imagine that nobody comprehends the motions in a spinning cylinder and that the rotation is all waste heat.
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But once you go so far as to map certain parts of the system onto preference and belief, the coherence theorems hold like the theorems they are. In that sense they're more universal, and less physically informative, than thermodynamics.
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Most mappings on the system will be useless. If you regard a particular molecule as having compact preferences and beliefs, it will probably do very poorly. Human beings do have relatively compact preferences and beliefs, on the other hand, and applying it to humans make sense.
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This is not because humans obey the prescriptions of decision theory. It is because humans are making things that can be viewed as decisions. This is not because human beliefs are coherent. It's because humans can be well-viewed as believing things even if incoherently.
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It seems to me that you don't understand the very abstract use that is being made of decision theory. It's not a recipe or an algorithm. It's a generalization relating coherent or incoherent behavior to performance, making far more minimal assumptions than you seem to think.
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Well… I *think* I understand this; but I may be wrong! We clearly have very different cognitive styles, which makes it difficult for us to understand each other.
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Returning to your Facebook post, I take it that it is an attempt to characterize the difference (Law vs Toolbox). I don’t think that is accurate; I think I understand and use both of those ways of thinking. But, I can’t be sure. And I don’t have a good alternative model!
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