When, if ever, are there analytic solutions to hierarchical models? For example, under what conditions can we find an analytic form for the distribution of $X$ in the network below? $X$ has one parameter, which in turn has one hyperparameter.

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For my actual application, I'm using a lognormal distribution where the parameters are themselves uncertain. That is,

\begin{align} X &\sim \ln(\mu, \sigma^2) \\ &\quad\text{ and} \\ \mu &\sim D_1(x, y) \\ \sigma &\sim D_2(a, b) \end{align}

Where $D_1$ and $D_2$ are some unspecified distributions. The distributions of the mu and sigma are suggested in some sources to also be lognormal distributions. However, consider a free hand for the distribution of the hyperparameters if it makes an analytic form possible. Can we find parameters mu' and sigma' which are point values, such that $X \sim \ln(\mu', \sigma'^2)$ (or perhaps for some another distribution which isn't lognormal)?


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