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From Wkipedia's article on hyperprior:

In Bayesian statistics, a hyperprior is a prior distribution on a hyperparameter, that is, on a parameter of a prior distribution.

There will be some parameters for the hyperprior and there is nothing stopping us from defining prior distribution on these parameters too. Practical considerations stop us from doing this inductive step too many times.

Q1. Is there any theoretical result showing that the effect of these hyperpriors diminishes as we keep on defining them?

Q2. Is there a more principled way (than computational and practical considerations) to decide when to stop this chain of priors?

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    $\begingroup$ Not really answering your question, but ultimately the number of levels in your model is really a modelling question, and at some point it would become difficult to interpret the parameters. $\endgroup$
    – fool
    Commented Oct 7, 2020 at 21:55
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    $\begingroup$ Yes, I agree. This partially answers Q2 since the hierarchical model incorporates your modeling assumptions. But my modeling assumptions could be influenced by the answer to Q1. $\endgroup$
    – Aditya
    Commented Oct 7, 2020 at 22:54

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