Information criteria: suitable for different response distributions?

[avehtari]

@Homer

an information criterion like LOO-IC (or the estimated ELPD instead, but I guess that makes no difference)

Since what is badly name as LOO-IC is just -2*elpd, the main difference is whether I’m frowning or smiling when answering the question. See explanation why I don’t like multiplying by -2 in this thread https://twitter.com/avehtari/status/1227900322082906112

@Homer

In this example, (PSIS-)LOO-CV was used to compare a Poisson to a negative binomial model.

This is ok.

@Homer

In contrast, McElreath (2016, section 9.2.4, page 288) states:

Really, all you have to remember is to only compare models that all use the same type of likelihood.

That is unfortunate mistake by McElreath and he now knows better (Hopefully in the future all books will be in git repos and when finding a mistake like this we could make an issue or pull request to fix these).

My earlier answer is in the thread Can WAIC/LOOIC be used to compare models with different likelihoods? - #2 by avehtari

Please tell if that answer is not clear. I should include that answer to loo documentation and some books, so it helps if it can be clarified. I know some ways how to clarify it with more explanation, but is it useful in a short version?