latent variables for a hierarchical Poisson model

Answering a question on X validated about a rather standard hierarchical Poisson model, and its posterior Gibbs simulation, where observations are (d and w being a document and a word index, resp.)

N_{w,d}\sim\mathcal P(\textstyle\sum_{1\le k\le K} \pi_{k,d}\varphi_{k,w})\qquad(1)

I found myself dragged into an extended discussion on the validation of creating independent Poisson latent variables

N_{k,w,d}\sim\mathcal P(\pi_{k,d}\varphi_{k,w})\qquad(2)

since observing their sum in (1) was preventing the latent variables (2) from being independent. And then found out that the originator of the question had asked on X validated an unanswered and much more detailed question in 2016, even though the notations differ. The question does contain the solution I proposed above, including the Multinomial distribution on the Poisson latent variables given their sum (and the true parameters). As it should be since the derivation was done in a linked 2014 paper by Gopalan, Hofman, and Blei, later published in the Proceedings of the 31st Conference on Uncertainty in Artificial Intelligence (UAI). I am thus bemused at the question resurfacing five years later in a much simplified version, but still exhibiting the same difficulty with the conditioning principles…

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