That gif is wildly intriguing to me. Probability theory is the bread and butter of mainstream science. Have people been doing it wrong this whole time?
I want to dig deeper into this but two things come to mind:
1. George Sugihara's and Floris Takens' works on quantifying causation in non-linear systems. Correlation doesn't imply causation, so why not look at causation? Here's a writeup on it: https://www.quantamagazine.org/chaos-theory-in-ecology-predi...
2. Andrei Khrennikov argues that quantum models can be expressed with Kolmogorovian (classical) probability theory. He points out that many probability theorists have glossed over an entire subtlety of Kolmogorov's work: that probability spaces need not be finite dimensional, that they can be subspaces of higher dimensional, even infinite dimensional probability spaces. Khrennikov's philosophical convictions are not widely shared by quantum theorists, but his math is solid (he seems to be in pursuit of quantum theories with "hidden variables", like Einstein was looking for). http://www.sciencedirect.com/science/article/pii/S0375960103...
Even in the gif shown in this article, you might suppose that each point set could be modeled in higher dimensional terms, for example adding some kind of discrete curl-like measure. But even without going there: something as simple as a changing to a polar basis should reveal different statistics.
And isn't this kind of what deep learning is all about? Going between higher and lower dimensional representations?
I want to dig deeper into this but two things come to mind:
1. George Sugihara's and Floris Takens' works on quantifying causation in non-linear systems. Correlation doesn't imply causation, so why not look at causation? Here's a writeup on it: https://www.quantamagazine.org/chaos-theory-in-ecology-predi...
2. Andrei Khrennikov argues that quantum models can be expressed with Kolmogorovian (classical) probability theory. He points out that many probability theorists have glossed over an entire subtlety of Kolmogorov's work: that probability spaces need not be finite dimensional, that they can be subspaces of higher dimensional, even infinite dimensional probability spaces. Khrennikov's philosophical convictions are not widely shared by quantum theorists, but his math is solid (he seems to be in pursuit of quantum theories with "hidden variables", like Einstein was looking for). http://www.sciencedirect.com/science/article/pii/S0375960103...
Even in the gif shown in this article, you might suppose that each point set could be modeled in higher dimensional terms, for example adding some kind of discrete curl-like measure. But even without going there: something as simple as a changing to a polar basis should reveal different statistics.
And isn't this kind of what deep learning is all about? Going between higher and lower dimensional representations?