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Context: Given a scalar normal distribution $X\sim \mathrm{N}(\mu, \sigma^2)$, it is possible to express $X$ as a mixture of uniform distributions over intervals (compound probability distributions), i.e., $X | V=v \sim \mathrm{Unif}(\mu-\sigma \sqrt{v},\mu+\sigma \sqrt{v})$ and latent variable $V\sim \Gamma(3/2,1/2)$, as shown in Qin et al. "Scale Mixture Models with Applications to Bayesian Inference".

Question: is it possible to generalize to the multivariate normal distribution as a mixture of uniform distributions over high dimensional balls parameterized by some latent variable?

Kolakoski54
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PiePiePie
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  • Does it have to be a finite mixture? The Central Limit Theorem describes one way to construct a normal distribution from the limit of an infinite sum of uniform RVs – whpowell96 Sep 30 '24 at 16:21
  • It should be a continuous mixture. – PiePiePie Sep 30 '24 at 16:23
  • I think you are asking about a similar topic here:

    https://math.stackexchange.com/questions/1602003/why-is-x-x-2-uniformly-distributed-on-a-unit-sphere-when-x-is-n-dimensiona

     – BGM Sep 30 '24 at 16:27

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