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I'm specifically looking to apply the trick to a Dirichlet distribution. Kingma and Welling (2013) briefly talk about how the trick can be applied to non-Gaussian distributions, and state that the Dirichlet distribution can be expressed as a "weighted sum of Gamma variates", however I've not found any more details on this.

I know that a Dirichlet distribution with parameters $(\alpha_1, \ldots, \alpha_k)$ can be expressed as a normalised sum of $X_i \sim \Gamma(\alpha_i, 1)$, however as I understand it, the whole point of the reparameterisation trick is to ensure that the parameters of the distribution are independent of the stochasticity, meaning this method cannot be used.

Could anyone shed some light on this?

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