double yolk priors [a reply from the authors]

[Here is an email I received from Subhadeep Mukhopadhyay, one of the authors of the paper I discussed yesterday.}
Thank for discussing our work. Let me clarify the technical point that you raised:
– The difference between Legj(u)_j and Tj=Legj(G(θ)). One is orthonormal polyn of L2[0,1] and the other one is L2[G]. The second one is poly of rank-transform G(θ).
– As you correctly pointed out there is a danger in directly approximating the ratio. We work on it after taking the quantile transform: evaluate the ratio at g⁻¹(θ), which is the d(u;G,F) over unit interval. Now, this new transformed function is a proper density.
-Thus the ratio now becomes d(G(θ)) which can be expended into (NOT in Leg-basis) in T_j, in eq (2.2), as it lives in the Hilbert space L2(G)
– For your last point on Step 2 of our algo, we can also use the simple integrate command.
-Unlike traditional prior-data conflict here we attempted to answer three questions in one-shot: (i) How compatible is the pre-selected g with the given data? (ii) In the event of a conflict, can we also inform the user on the nature of misfit–finer structure that was a priori unanticipated? (iii) Finally, we would like to provide a simple, yet formal guideline for upgrading (repairing) the starting g.
Hopefully, this will clear the air. But thanks for reading the paper so carefully. Appreciate it.

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