Concern with using the DIC to compare Bayesian models

Question

The deviance information criterion (DIC) is a very popular tool for Bayesian model selection, due, in part, to its support by the BUGS platforms. However, there are some remaining limitations as discussed by Plummer (2008). One of these is that "the effective number of parameters, pD, must be small in relation to the sample size," otherwise the DIC under-penalizes the model (p. 535). Is there any knowledge regarding what the minimum ratio of parameters-to-observations needs to be for the DIC to be valid? Has this been formally explored?

Plummer M. Penalized loss functions for Bayesian model comparison. (2008). Biostatistics, 9, 523–539.

Answer 1

I cannot answer for DIC, but for WAIC (which Gelman et al. recommend over DIC - See BDA3 7.3), Vehtari and Gelman (2014, p. 11) indicate that if the number of effective parameters is greater than half the number of observations, the WAIC approximation is unreliable. Moreover, if any one parameter's contribution to p_effective is > 1, the approximation may be unreliable.