$\newcommand{\Beta}{\operatorname{Beta}}$Consider $X \sim N(\mu,\sigma)$; I can reparameterize it by $X = \varepsilon\mu + \sigma; \varepsilon \sim N(0,I) $
But given Beta distribution $X \sim \Beta(\alpha,\beta)$; is there easy way (closed form transformation) to reparameterize $X$ with some very simple random variable (Normal, uniform )
My major goal is to do VAE such that my prior is Beta and my posterior is also Beta; so I'm thinking how to reparameterization trick for Beta. What I want to do, is instead of directly sampling from $\Beta(\alpha,\beta)$ (because couldn't do backpropagation), I want first generate $\varepsilon \sim \mathcal{Q}$ some easily sampled distribution, then apply some deterministic function that involves $\alpha, \beta$, such that after the transformation it follows $\Beta(\alpha,\beta)$.