I am going through some lecture notes that unfortunatly do not contain any examples.
"However, the DIC relies on the posterior mean as a point estimate, is not invariant with respect to transformation of the parameters, and can have negative effective dimension $p_D$."
with $p_D$ being the posterior expected deviance. I've looked at some examples that illustrate how the posterior mean is generally not invariant under transformation (e.g. a beta-binomial model whose posterior follows a beta prime distribution). Yet I'm struggling understanding how DIC relies on the posterior mean. From the definition I see that it depends on the posterior expectation and the posterior expected deviance. How does this depend on the mean?
The part 'can have negative effective dimension $p_D$' also remains unclear to me. Could you help me understand what exactly this part means and if and how this corresponds to the invariance property?
The lecture notes further state:
"The drawback of the DIC can be avoided by the widely applicable information criterion (WAIC)"
Does this mean that the WAIC does not rely on the posterior mean? How does this solve the problem of the "negative effective dimension $p_D$"?
I am unsure if I chose a fitting title, please let me now if I should edit it. Thank you for reading, I'd greatly appreciate any hint!