Negative probability thread! (it's the tweet version of this blog post, as an experiment: https://drossbucket.wordpress.com/2019/08/01/negative-probability/ …)
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Instead I'm going to go through an example that shows *how* they work. I got this one from another excellent blog post, by
@sigfpe: http://blog.sigfpe.com/2008/04/negative-probabilities.html … I like it because the numbers are nice and simple.Show this thread
So we have the following setup (from that post): "a machine produces boxes with (ordererd) pairs of bits in them, each bit viewable through its own door"
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Now we ask 3 questions: 1. Is the first box in state 0? 2. Is the second box in state 0? 3. Are the boxes both in the same state? Obviously these overlap: e.g. to be consistent, if you answered Y to 1. and N to 2., you'd also answer N to 3.
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So, for YNN, we learn that box 1 is definitely in state 0 and box 2 is definitely in state 1. We can represent this as the following probability distribution over possible pairs of states:pic.twitter.com/7XV6lwH2Cc
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OK, nothing very clever here! Now for an inconsistent set, NNN. Both boxes are in state 0, but... they're also in different states? Turns out you can still assign a 'probability' distribution. Sort of. Scare quotes because one 'probability' is negative:pic.twitter.com/P5v6tjbway
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Given this, it all works. E.g. for question 1 P(first 0) = P(first 0, second 0) + P(first 0, second 1) = -½ + ½ = 0 So answer is N. Same for the others.
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Now, where that -½ comes from was a bit opaque to me. I mean I could follow the algebra, just didn't have much intuition for how it got there. So I played around for a while and came up with the following.
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Motivating idea: NNN is inconsistent, so all four possible assignments don't work. But {first box 0, second box 0} is *especially* bad. It's the wrong answer to all three questions. Other three boxes are only the wrong answer to one question.
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So... we penalise bad answers, and, because {first box 0, second box} is particularly bad, it gets clobbered three times instead of just once, driving it negative. That's vague. But I'll outline a precise version.
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Start with the consistent, YNN example, and ask the questions in turn. Before you start you are completely ignorant - probability ¼ of all boxes. After first question you narrow it to two boxes. View this as adding on a correction term:pic.twitter.com/HiqFHfcil8
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You can add one of these correction terms for the other two questions as well, ending up with the same as before:pic.twitter.com/58Fzyh4IvC
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And it also works for the inconsistent, NNN example! You can see how the bottom left box gets clobbered by the successive -¼s, as in my vague motivation at the start.pic.twitter.com/8mgW5CvKT2
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Is this trick any use? I really don't know! But I definitely find this breakdown more illuminating than just plugging through the algebra.
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Now the reason I'm interested in this is the link to quantum physics. This toy model is very similar to phase space for a qubit. In some sense it's slightly 'worse', as -½ is more negative than anything that comes up there. But the simple numbers make it easy to play with.
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For more information on how all this relates to quantum physics, see my follow up post: https://drossbucket.wordpress.com/2019/08/01/negative-probability-now-with-added-equations/ … May try a thread version of that one, too, but that'll be harder work. This is enough for today!
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End of conversation
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