WAIC on integrated likelihood

Hi, I would like to use WAIC to do model comparison for a circumstance where I have manually (due to conjugacy) integrated out a few parameters from my joint distribution as this speeds things up immensely.

My question is, can I use the (semi-)marginalised likelihood rather than the full likelihood when calculating the log pointwise predictive density? Or, alternatively, should I sample the marginalised parameters for each set of the other parameters, and use the full likelihood as per usual?

Are these equivalent?

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They aren’t equivalent and the difference is important for what you are imagining being left out. Here is a StanCon presentation about it

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