model. This assumption does not always hold, and it is desirable to consider model assessment procedures in that scenario. Also, the observed data are used both to construct the posterior distribution and to evaluate the estimated models. Therefore, DIC tends to select over-fitted models. Extensions A resolution to the issues above was suggested by , with the proposal of the Bayesian predictive information criterion (BPIC). Ando (2010, Ch. 8) provided a discussion of various Bayesian model selection criteria. To avoid the over-fitting problems of DIC, developed Bayesian model selection criteria from a predictive view point. The criterion is calculated as The first term is a measure of how well the model fits the data, while the second term is a penalty on the model complexity. Note that the in this expression is the predictive distribution rather than the likelihood above. See also Akaike information criterion (AIC) Bayesian information criterion (BIC) Focused information criterion (FIC) Hannan-Quinn information criterion Kullback–Leibler divergence Jensen–Shannon divergence Watanabe–Akaike information criterion (WAIC) References Ando, T. (2010). Bayesian Model Selection and Statistical Modeling, CRC Press. Chapter 7. Claeskens, G, and Hjort, N.L. (2008). Model Selection and Model Averaging, Cambridge. Section 3.5. van der Linde, A. (2005). "DIC in variable selection", Statistica Neerlandica, 59: 45-56. doi:10.1111/j.1467-9574.2005.00278.x External links Bayesian statistics Regression variable selection Model selection