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The billiards player who picks up a cue takes careful aim and sends the billiard balls smacking into one another. So, if you want to become the best player in a game, then you will need to play and gain experience while training. So it works out that the summary statistics are basically the things which will propagate over a long distance. Daniel Filan: Okay. Yeah, it’s almost saying that like somehow the summary statistics are the things that are being transmitted somehow, and they’re minimal things, which will tell you anything else about the distribution. However, one thing you need to keep in mind is that when there are many different balls on the field, you will first have to put all the balls that fall into the hole and then get to the last white ball. Supposing that there exists a sufficient statistic, which means that we can aggregate all of these measurements into some fixed dimensional thing, no matter how many measurements we take, we can aggregate them all into this one aggregate of some limited dimension and that will summarize all of the information about these measurements, which is relevant to our estimates of the thing, right?

John Wentworth: Now, the reason that needed to be generalized is because the original Koopman-Pitman-Darmois theorem only applies when you have these repeated independent measurements of the same thing, right? John Wentworth: You can see how conceptually, this is a lot like the abstraction thing. The interesting question with exponential family distributions - why do we see them pop up so often? Daniel Filan: The original question is in a deterministic world, how does any of this probabilistic inference happen, given that it seems like my physical state sort of has to… Then the question is like, why this particular kind of constraint? John Wentworth: Then Koopman-Pitman-Darmois is saying, "Well, these things that are max entropic subject to this particular kind of constraint are the only things that have these nice summary statistics, the only things that abstract well." So not really answering the question, but it sure is putting a big ole spotlight on the question, isn’t it?

The weird thing is that it’s a very specific kind of constraint. Daniel Filan: Okay. A thing which you’ve also talked about as a different view on abstraction is the generalized Koopman-Pitman-Darmois theorem. Daniel Filan: Is the idea that natural abstraction hypothesis gets solved, and… Daniel Filan: Yeah. Or some function of my measurement as well. Daniel Filan: In particular, that’s because exponential family distributions are maximum entropy, which means that they’re as uncertain as you can be subject to some constraints, like the average energy. So this is this really nice class of distributions, which are very convenient to work with. That turned out to basically work. If you're thinking of hiring a professional pool table installation company to come out and help you set up your game, you're making a wise choice. 1⁄16 in with a tolerance of 0.05 mm) and they must weigh the same to a tolerance of 0.5 g within a set. Daniel Filan: Okay. So independent measurements and each of them takes the same random distribution. Daniel Filan: Yeah. What do you think we don’t know about alignment that you wish we did?

Problem solved. Do you think there’s anything wrong with that picture? I’ve got this chunk of the world - there’s this water bottle here, this is a chunk of the world. Daniel Filan: So if the real world isn’t random, right, then every physical system, there’s only one way it could have been, right? So if you’re thinking about information propagating far away, anything that’s optimizing stuff far away is mainly going to care about that information that’s propagating far away, right? The summary is something like abstraction, the way it works is that if I’m far away from some subsystem, how to play billiards there’s only a few bits which are reliably preserved about that subsystem that ever reach me and everything else is like shook out by noise or whatever. If you’re in a reasonably large world, most stuff is far away. You can’t know all the initial conditions because you’re embedded in the system.