I came up with this problem: Let's consider two random variables $$ X \sim N(\mu, \sigma_x^2), Y \sim N(\mu, \sigma_y^2) $$ and I know that $$ Y = f(g(x)) $$ deterministic non linear functions composition. Is it possible to say something about a certain $ Z = g(x) $ if I know that: $$ p_x(x) = p_z(g(x))\bigg|\frac{\partial g(x)}{\partial x}\bigg|$$ $$ p_z(z) = p_y(f(z))\bigg|\frac{\partial f(z)}{\partial z}\bigg| $$ So, $$ \frac{p_x(x)}{p_y(f(g(x))} = \bigg|\frac{\partial f(g(x))}{\partial g(x)}\bigg|\bigg|\frac{\partial g(x)}{\partial x}\bigg|$$ I'd like to know if it is possible to come up with some interesting information about $Z = g(x)$ if it is a Gaussian as well and so on. Thank you in advance.
2026-04-08 00:21:31.1775607691
Latent space information from a Function composition of Gaussian Random Variables
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