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This one took me way too long to get intuition for and I don't think OP's explanation would have helped me grok it. I wonder if it's like monads, in that the in

This one took me way too long to get intuition for and I don't think OP's explanation would have helped me grok it. I wonder if it's like monads, in that the intuition is really simple, but it's a very unique concept; so unusual that the intuition is hard to convey to the uninitiated. But the intuition really is simple, so the initiated feel compelled to try to convey how simple it is. My intuition for it is like a particular kind of spread out mixing. Imagine a giant bowl with a bunch of crazy high powered pinball bumpers [0] at the bottom, randomly jostling the pinballs around, sometimes kicking them out where they started, but they always come back eventually. If you roll a ball into it, it doesn't matter where you start. It'll get lost in the mix eventually. (it "mixes" sufficiently). And no matter where you start, those high powered bumpers will eventually happen to kick a pinball all the way out there again, given long enough. (The "mixing" spreads things out sufficiently and occasionally sends things all the way to any point in the bowl). By contrast, a bowl that's just high friction, where everything ends up stopped at the bottom wouldn't work, even though it makes it not matter where you started, because it doesn't "spread." It just sinks things to the same spot. An inverted bowl/a dome wouldn't work because starting on opposite sides means you'll just roll away from each other and never come together (no "mixing" at all). A bowl without bumpers would have you coming back where you started, but not "mixing" around to all the other spots. You need both elements. It has to not matter where you started specifically by getting back to where anyone started. Rereading my comment now, it really does come off as "a monad is like a burrito," doesn't it. But screw it, I'll hit post and maybe it helps somebody. [0] This kind of bumper: https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQa3gvhOr96M-82lDY8vaLj5XDyFj7cxLYTqMFA4iFhMQ6VWb0bk9bnqQdq221XUwvGfcA&usqp=CAU https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQa3gvh...

nvader 6y ago
I found this helpful. Not because it has helped me to grok ergodicity yet, but because it's another foothold on my way to climb the mountain. At some point it will click, and your pinball-bowl mixer will help, I'm sure. I'm also happy for the link to "a monad is like a burrito"!
ssivark 6y ago
Here’s the essence of non-ergodicity: the “typical” result (technically the mode, but practically the median is acceptable) is very different from the “mean” result. The language of “time” -vs- “ensemble” averages is simply about whether to include only the “typical” possibilities or to also include corner cases which have extremely low probability but extremely high payoffs. The technical point is that unless you get to play the game (“sample the process”) absurdly many times (comparable to the total number of possibilities), you will never access the corner cases, and so the averaging over the “typical” results corresponds better to reality. The “mixing” idea means that if you run the process “long enough” then there won’t be any corner cases — the extreme possibilities will thoroughly mix with the typical possibilities, so both kinds of averaging will lead to the same (correct) answer. If you grok this, everything else is technical detail and window dressing.

nvader 6y ago

I found this helpful. Not because it has helped me to grok ergodicity yet, but because it's another foothold on my way to climb the mountain. At some point it will click, and your pinball-bowl mixer will help, I'm sure. I'm also happy for the link to "a monad is like a burrito"!

ssivark 6y ago

Here’s the essence of non-ergodicity: the “typical” result (technically the mode, but practically the median is acceptable) is very different from the “mean” result. The language of “time” -vs- “ensemble” averages is simply about whether to include only the “typical” possibilities or to also include corner cases which have extremely low probability but extremely high payoffs. The technical point is that unless you get to play the game (“sample the process”) absurdly many times (comparable to the total number of possibilities), you will never access the corner cases, and so the averaging over the “typical” results corresponds better to reality. The “mixing” idea means that if you run the process “long enough” then there won’t be any corner cases — the extreme possibilities will thoroughly mix with the typical possibilities, so both kinds of averaging will lead to the same (correct) answer. If you grok this, everything else is technical detail and window dressing.