Why is it that for heavy-tailed distributions, the higher moments do not exist or do not converge? What is the characteristic function of a distribution? Why is it useful when we already have the PDF and MGF to define a distribution?

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The reparameterization trick was first introduced when I first learned about the Variational Autoencoder(VAE). It shamefully took me multiple years from when I first learned about it to when I fully internalize why it is needed, and how I can turn any inference problem into one solved by variational inference. Here, I write about my general approaches to inference, with a focus on Variational Inference, and detailed explanation for why the reparameterization trick is needed.

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I came across situations where I had to find the distribution of certain variables based on those of others. The approach of derivation through the probability density function (PDF) did not work, but moment generating functions come to the rescue. Here, I present such a situation and show how to apply moment generating functions to draw conclusion on about the distribution of a random variable.

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We should not take any distribution for granted, as in treating them as entities with googleable formula. Here, I derive the Beta distribution from the Uniform distribution.

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