How would LOO with multiple imputations look like?

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Are you imputing just covariates or also target variable y?

Assuming you are imouting just teh covariates, and you are using R and loo package

  1. For each of the m imputed datasets compute loo as usual
  2. each loo object has pointwise log predictive densities loo_object$pointwise[,'elpd_loo']
  3. compute means of pointwise predictive densities (using exp(loo_object$pointwise[,'elpd_loo']) where means are over the m imputed datasets (the result has n pointwise predictive densities)
  4. elpd_loo is then sum of log of pointwise predictions (from step 3.)
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