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I really hate this kind of numerology. cosines don't have intimate relationship with euclidean distance - the dot product and the norm aren't equivalent https:/

I really hate this kind of numerology. cosines don't have intimate relationship with euclidean distance - the dot product and the norm aren't equivalent https://math.stackexchange.com/questions/528864/is-every-normed-vector-space-an-inner-product-space https://math.stackexchange.com/questions/528864/is-every-nor.... I also have no idea what you mean by normal distribucións being most continuous because CLT? and then what's linearity got to do with continuity? normal distribucións are convenient to work with simply because of the exponential (other exponential families are also nice to work with). again don't know what you mean by normals being nicely differentiable. lots of PDFs are differentiable (in fact most functions are easy to differentiate). finally you only have to farther than an undergrad book to see what most things are actually not high school trig (e.g. time dependent models like SDEs or Bayesian inference with intractable marginals)

nerdponx 8y ago
I'm not talking in generalities. I'm talking about the standard inner product space on R^n that most people use (either implicitly or explicitly) every day. You are right that the normal distribution is not continuous because of the CLT. That was a typo. I meant "most important continuous" distribution. Yes, other exponential distributions are convenient to work with, but the Gaussian maximum-likelihood estimate has a closed-form expression defined with just linear algebra. That's really damn nice in my opinion. The point is that distance (in most real-world applications) is equivalent to the length of the hypotenuse of a right triangle in Euclidean space. So in my opinion, yes, high-school trigonometry governs a lot of advanced mathematics, including in statistics and machine learning. Call it numerology if you want, I'm just trying to get people excited about math here.

nerdponx 8y ago

I'm not talking in generalities. I'm talking about the standard inner product space on R^n that most people use (either implicitly or explicitly) every day. You are right that the normal distribution is not continuous because of the CLT. That was a typo. I meant "most important continuous" distribution. Yes, other exponential distributions are convenient to work with, but the Gaussian maximum-likelihood estimate has a closed-form expression defined with just linear algebra. That's really damn nice in my opinion. The point is that distance (in most real-world applications) is equivalent to the length of the hypotenuse of a right triangle in Euclidean space. So in my opinion, yes, high-school trigonometry governs a lot of advanced mathematics, including in statistics and machine learning. Call it numerology if you want, I'm just trying to get people excited about math here.