How to evaluate a Poisson/NB CAR/IAR model

Comparison with LOO (or waic if you insist) is fine if you are fine with the leave-one-out prediction task. For spatial data sometimes people want instead predict for certain region and then you would need to take into account the spatial correlation. This is easier to understand in the case of time series with correlated residuals, and if you want to predict future, then you need to switch from LOO to leave-future-out-cross-validation.

See A quick note what I infer from p_loo and Pareto k values
and then tell how many Pareto k values are larger than 0.7 and how many parameters do you have in M_1. My guess is that you spatial prior is very weak, but with the above information I can be more certain.

LOO diagnostic is more reliable than waic diagnostic, and since there are no warnings from loo diagnostic for M_2, LOO is working just fine for that model.

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